Calculus Examples

$f (t) = t \sqrt{25 - t^{2}}$

Step 1

Find the first derivative of the function.

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Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{25 - t^{2}}$ as ${(25 - t^{2})}^{\frac{1}{2}}$ .

$\frac{d}{d t} [t {(25 - t^{2})}^{\frac{1}{2}}]$

Step 1.2

Differentiate using the Product Rule which states that $\frac{d}{d t} [f (t) g (t)]$ is $f (t) \frac{d}{d t} [g (t)] + g (t) \frac{d}{d t} [f (t)]$ where $f (t) = t$ and $g (t) = {(25 - t^{2})}^{\frac{1}{2}}$ .

$t \frac{d}{d t} [{(25 - t^{2})}^{\frac{1}{2}}] + {(25 - t^{2})}^{\frac{1}{2}} \frac{d}{d t} [t]$

Step 1.3

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = t^{\frac{1}{2}}$ and $g (t) = 25 - t^{2}$ .

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Step 1.3.1

To apply the Chain Rule, set $u$ as $25 - t^{2}$ .

$t (\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d t} [25 - t^{2}]) + {(25 - t^{2})}^{\frac{1}{2}} \frac{d}{d t} [t]$

Step 1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{1}{2}$ .

$t (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d t} [25 - t^{2}]) + {(25 - t^{2})}^{\frac{1}{2}} \frac{d}{d t} [t]$

Step 1.3.3

Replace all occurrences of $u$ with $25 - t^{2}$ .

$t (\frac{1}{2} {(25 - t^{2})}^{\frac{1}{2} - 1} \frac{d}{d t} [25 - t^{2}]) + {(25 - t^{2})}^{\frac{1}{2}} \frac{d}{d t} [t]$

Step 1.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$t (\frac{1}{2} {(25 - t^{2})}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d t} [25 - t^{2}]) + {(25 - t^{2})}^{\frac{1}{2}} \frac{d}{d t} [t]$

Step 1.5

Combine $- 1$ and $\frac{2}{2}$ .

$t (\frac{1}{2} {(25 - t^{2})}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d t} [25 - t^{2}]) + {(25 - t^{2})}^{\frac{1}{2}} \frac{d}{d t} [t]$

Step 1.6

Combine the numerators over the common denominator.

$t (\frac{1}{2} {(25 - t^{2})}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d t} [25 - t^{2}]) + {(25 - t^{2})}^{\frac{1}{2}} \frac{d}{d t} [t]$

Step 1.7

Simplify the numerator.

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Step 1.7.1

Multiply $- 1$ by $2$ .

$t (\frac{1}{2} {(25 - t^{2})}^{\frac{1 - 2}{2}} \frac{d}{d t} [25 - t^{2}]) + {(25 - t^{2})}^{\frac{1}{2}} \frac{d}{d t} [t]$

Step 1.7.2

Subtract $2$ from $1$ .

$t (\frac{1}{2} {(25 - t^{2})}^{\frac{- 1}{2}} \frac{d}{d t} [25 - t^{2}]) + {(25 - t^{2})}^{\frac{1}{2}} \frac{d}{d t} [t]$

Step 1.8

Combine fractions.

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Step 1.8.1

Move the negative in front of the fraction.

$t (\frac{1}{2} {(25 - t^{2})}^{- \frac{1}{2}} \frac{d}{d t} [25 - t^{2}]) + {(25 - t^{2})}^{\frac{1}{2}} \frac{d}{d t} [t]$

Step 1.8.2

Combine $\frac{1}{2}$ and ${(25 - t^{2})}^{- \frac{1}{2}}$ .

$t (\frac{{(25 - t^{2})}^{- \frac{1}{2}}}{2} \frac{d}{d t} [25 - t^{2}]) + {(25 - t^{2})}^{\frac{1}{2}} \frac{d}{d t} [t]$

Step 1.8.3

Move ${(25 - t^{2})}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$t (\frac{1}{2 {(25 - t^{2})}^{\frac{1}{2}}} \frac{d}{d t} [25 - t^{2}]) + {(25 - t^{2})}^{\frac{1}{2}} \frac{d}{d t} [t]$

Step 1.8.4

Combine $\frac{1}{2 {(25 - t^{2})}^{\frac{1}{2}}}$ and $t$ .

$\frac{t}{2 {(25 - t^{2})}^{\frac{1}{2}}} \frac{d}{d t} [25 - t^{2}] + {(25 - t^{2})}^{\frac{1}{2}} \frac{d}{d t} [t]$

Step 1.9

By the Sum Rule, the derivative of $25 - t^{2}$ with respect to $t$ is $\frac{d}{d t} [25] + \frac{d}{d t} [- t^{2}]$ .

$\frac{t}{2 {(25 - t^{2})}^{\frac{1}{2}}} (\frac{d}{d t} [25] + \frac{d}{d t} [- t^{2}]) + {(25 - t^{2})}^{\frac{1}{2}} \frac{d}{d t} [t]$

Step 1.10

Since $25$ is constant with respect to $t$ , the derivative of $25$ with respect to $t$ is $0$ .

$\frac{t}{2 {(25 - t^{2})}^{\frac{1}{2}}} (0 + \frac{d}{d t} [- t^{2}]) + {(25 - t^{2})}^{\frac{1}{2}} \frac{d}{d t} [t]$

Step 1.11

Add $0$ and $\frac{d}{d t} [- t^{2}]$ .

$\frac{t}{2 {(25 - t^{2})}^{\frac{1}{2}}} \frac{d}{d t} [- t^{2}] + {(25 - t^{2})}^{\frac{1}{2}} \frac{d}{d t} [t]$

Step 1.12

Since $- 1$ is constant with respect to $t$ , the derivative of $- t^{2}$ with respect to $t$ is $- \frac{d}{d t} [t^{2}]$ .

$\frac{t}{2 {(25 - t^{2})}^{\frac{1}{2}}} (- \frac{d}{d t} [t^{2}]) + {(25 - t^{2})}^{\frac{1}{2}} \frac{d}{d t} [t]$

Step 1.13

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 2$ .

$\frac{t}{2 {(25 - t^{2})}^{\frac{1}{2}}} (- (2 t)) + {(25 - t^{2})}^{\frac{1}{2}} \frac{d}{d t} [t]$

Step 1.14

Combine fractions.

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Step 1.14.1

Multiply $2$ by $- 1$ .

$\frac{t}{2 {(25 - t^{2})}^{\frac{1}{2}}} (- 2 t) + {(25 - t^{2})}^{\frac{1}{2}} \frac{d}{d t} [t]$

Step 1.14.2

Combine $- 2$ and $\frac{t}{2 {(25 - t^{2})}^{\frac{1}{2}}}$ .

$\frac{- 2 t}{2 {(25 - t^{2})}^{\frac{1}{2}}} t + {(25 - t^{2})}^{\frac{1}{2}} \frac{d}{d t} [t]$

Step 1.14.3

Combine $\frac{- 2 t}{2 {(25 - t^{2})}^{\frac{1}{2}}}$ and $t$ .

$\frac{- 2 t \cdot t}{2 {(25 - t^{2})}^{\frac{1}{2}}} + {(25 - t^{2})}^{\frac{1}{2}} \frac{d}{d t} [t]$

Step 1.15

Raise $t$ to the power of $1$ .

$\frac{- 2 (t^{1} t)}{2 {(25 - t^{2})}^{\frac{1}{2}}} + {(25 - t^{2})}^{\frac{1}{2}} \frac{d}{d t} [t]$

Step 1.16

Raise $t$ to the power of $1$ .

$\frac{- 2 (t^{1} t^{1})}{2 {(25 - t^{2})}^{\frac{1}{2}}} + {(25 - t^{2})}^{\frac{1}{2}} \frac{d}{d t} [t]$

Step 1.17

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{- 2 t^{1 + 1}}{2 {(25 - t^{2})}^{\frac{1}{2}}} + {(25 - t^{2})}^{\frac{1}{2}} \frac{d}{d t} [t]$

Step 1.18

Add $1$ and $1$ .

$\frac{- 2 t^{2}}{2 {(25 - t^{2})}^{\frac{1}{2}}} + {(25 - t^{2})}^{\frac{1}{2}} \frac{d}{d t} [t]$

Step 1.19

Factor $2$ out of $- 2 t^{2}$ .

$\frac{2 (- t^{2})}{2 {(25 - t^{2})}^{\frac{1}{2}}} + {(25 - t^{2})}^{\frac{1}{2}} \frac{d}{d t} [t]$

Step 1.20

Cancel the common factors.

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Step 1.20.1

Factor $2$ out of $2 {(25 - t^{2})}^{\frac{1}{2}}$ .

$\frac{2 (- t^{2})}{2 ({(25 - t^{2})}^{\frac{1}{2}})} + {(25 - t^{2})}^{\frac{1}{2}} \frac{d}{d t} [t]$

Step 1.20.2

Cancel the common factor.

$\frac{2 (- t^{2})}{2 {(25 - t^{2})}^{\frac{1}{2}}} + {(25 - t^{2})}^{\frac{1}{2}} \frac{d}{d t} [t]$

Step 1.20.3

Rewrite the expression.

$\frac{- t^{2}}{{(25 - t^{2})}^{\frac{1}{2}}} + {(25 - t^{2})}^{\frac{1}{2}} \frac{d}{d t} [t]$

Step 1.21

Move the negative in front of the fraction.

$- \frac{t^{2}}{{(25 - t^{2})}^{\frac{1}{2}}} + {(25 - t^{2})}^{\frac{1}{2}} \frac{d}{d t} [t]$

Step 1.22

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 1$ .

$- \frac{t^{2}}{{(25 - t^{2})}^{\frac{1}{2}}} + {(25 - t^{2})}^{\frac{1}{2}} \cdot 1$

Step 1.23

Multiply ${(25 - t^{2})}^{\frac{1}{2}}$ by $1$ .

$- \frac{t^{2}}{{(25 - t^{2})}^{\frac{1}{2}}} + {(25 - t^{2})}^{\frac{1}{2}}$

Step 1.24

To write ${(25 - t^{2})}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{{(25 - t^{2})}^{\frac{1}{2}}}{{(25 - t^{2})}^{\frac{1}{2}}}$ .

$- \frac{t^{2}}{{(25 - t^{2})}^{\frac{1}{2}}} + \frac{{(25 - t^{2})}^{\frac{1}{2}} {(25 - t^{2})}^{\frac{1}{2}}}{{(25 - t^{2})}^{\frac{1}{2}}}$

Step 1.25

Combine the numerators over the common denominator.

$\frac{- t^{2} + {(25 - t^{2})}^{\frac{1}{2}} {(25 - t^{2})}^{\frac{1}{2}}}{{(25 - t^{2})}^{\frac{1}{2}}}$

Step 1.26

Multiply ${(25 - t^{2})}^{\frac{1}{2}}$ by ${(25 - t^{2})}^{\frac{1}{2}}$ by adding the exponents.

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Step 1.26.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{- t^{2} + {(25 - t^{2})}^{\frac{1}{2} + \frac{1}{2}}}{{(25 - t^{2})}^{\frac{1}{2}}}$

Step 1.26.2

Combine the numerators over the common denominator.

$\frac{- t^{2} + {(25 - t^{2})}^{\frac{1 + 1}{2}}}{{(25 - t^{2})}^{\frac{1}{2}}}$

Step 1.26.3

Add $1$ and $1$ .

$\frac{- t^{2} + {(25 - t^{2})}^{\frac{2}{2}}}{{(25 - t^{2})}^{\frac{1}{2}}}$

Step 1.26.4

Divide $2$ by $2$ .

$\frac{- t^{2} + {(25 - t^{2})}^{1}}{{(25 - t^{2})}^{\frac{1}{2}}}$

Step 1.27

Simplify ${(25 - t^{2})}^{1}$ .

$\frac{- t^{2} + 25 - t^{2}}{{(25 - t^{2})}^{\frac{1}{2}}}$

Step 1.28

Subtract $t^{2}$ from $- t^{2}$ .

$\frac{- 2 t^{2} + 25}{{(25 - t^{2})}^{\frac{1}{2}}}$

Step 1.29

Reorder terms.

$\frac{- 2 t^{2} + 25}{{(- t^{2} + 25)}^{\frac{1}{2}}}$

Step 2

Find the second derivative of the function.

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Step 2.1

Differentiate using the Quotient Rule which states that $\frac{d}{d t} [\frac{f (t)}{g (t)}]$ is $\frac{g (t) \frac{d}{d t} [f (t)] - f (t) \frac{d}{d t} [g (t)]}{{g (t)}^{2}}$ where $f (t) = - 2 t^{2} + 25$ and $g (t) = {(- t^{2} + 25)}^{\frac{1}{2}}$ .

$f'' (t) = \frac{{(- t^{2} + 25)}^{\frac{1}{2}} \frac{d}{d t} (- 2 t^{2} + 25) - (- 2 t^{2} + 25) \frac{d}{d t} {(- t^{2} + 25)}^{\frac{1}{2}}}{{({(- t^{2} + 25)}^{\frac{1}{2}})}^{2}}$

Step 2.2

Multiply the exponents in ${({(- t^{2} + 25)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f'' (t) = \frac{{(- t^{2} + 25)}^{\frac{1}{2}} \frac{d}{d t} (- 2 t^{2} + 25) - (- 2 t^{2} + 25) \frac{d}{d t} {(- t^{2} + 25)}^{\frac{1}{2}}}{{(- t^{2} + 25)}^{\frac{1}{2} \cdot 2}}$

Step 2.2.2

Cancel the common factor of $2$ .

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Step 2.2.2.1

Cancel the common factor.

$f'' (t) = \frac{{(- t^{2} + 25)}^{\frac{1}{2}} \frac{d}{d t} (- 2 t^{2} + 25) - (- 2 t^{2} + 25) \frac{d}{d t} {(- t^{2} + 25)}^{\frac{1}{2}}}{{(- t^{2} + 25)}^{\frac{1}{2} \cdot 2}}$

Step 2.2.2.2

Rewrite the expression.

$f'' (t) = \frac{{(- t^{2} + 25)}^{\frac{1}{2}} \frac{d}{d t} (- 2 t^{2} + 25) - (- 2 t^{2} + 25) \frac{d}{d t} {(- t^{2} + 25)}^{\frac{1}{2}}}{- t^{2} + 25}$

Step 2.3

Simplify.

$f'' (t) = \frac{{(- t^{2} + 25)}^{\frac{1}{2}} \frac{d}{d t} (- 2 t^{2} + 25) - (- 2 t^{2} + 25) \frac{d}{d t} {(- t^{2} + 25)}^{\frac{1}{2}}}{- t^{2} + 25}$

Step 2.4

Differentiate.

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Step 2.4.1

By the Sum Rule, the derivative of $- 2 t^{2} + 25$ with respect to $t$ is $\frac{d}{d t} [- 2 t^{2}] + \frac{d}{d t} [25]$ .

$f'' (t) = \frac{{(- t^{2} + 25)}^{\frac{1}{2}} (\frac{d}{d t} (- 2 t^{2}) + \frac{d}{d t} (25)) - (- 2 t^{2} + 25) \frac{d}{d t} {(- t^{2} + 25)}^{\frac{1}{2}}}{- t^{2} + 25}$

Step 2.4.2

Since $- 2$ is constant with respect to $t$ , the derivative of $- 2 t^{2}$ with respect to $t$ is $- 2 \frac{d}{d t} [t^{2}]$ .

$f'' (t) = \frac{{(- t^{2} + 25)}^{\frac{1}{2}} (- 2 \frac{d}{d t} t^{2} + \frac{d}{d t} (25)) - (- 2 t^{2} + 25) \frac{d}{d t} {(- t^{2} + 25)}^{\frac{1}{2}}}{- t^{2} + 25}$

Step 2.4.3

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 2$ .

$f'' (t) = \frac{{(- t^{2} + 25)}^{\frac{1}{2}} (- 2 (2 t) + \frac{d}{d t} (25)) - (- 2 t^{2} + 25) \frac{d}{d t} {(- t^{2} + 25)}^{\frac{1}{2}}}{- t^{2} + 25}$

Step 2.4.4

Multiply $2$ by $- 2$ .

$f'' (t) = \frac{{(- t^{2} + 25)}^{\frac{1}{2}} (- 4 t + \frac{d}{d t} (25)) - (- 2 t^{2} + 25) \frac{d}{d t} {(- t^{2} + 25)}^{\frac{1}{2}}}{- t^{2} + 25}$

Step 2.4.5

Since $25$ is constant with respect to $t$ , the derivative of $25$ with respect to $t$ is $0$ .

$f'' (t) = \frac{{(- t^{2} + 25)}^{\frac{1}{2}} (- 4 t + 0) - (- 2 t^{2} + 25) \frac{d}{d t} {(- t^{2} + 25)}^{\frac{1}{2}}}{- t^{2} + 25}$

Step 2.4.6

Add $- 4 t$ and $0$ .

$f'' (t) = \frac{{(- t^{2} + 25)}^{\frac{1}{2}} (- 4 t) - (- 2 t^{2} + 25) \frac{d}{d t} {(- t^{2} + 25)}^{\frac{1}{2}}}{- t^{2} + 25}$

Step 2.5

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = t^{\frac{1}{2}}$ and $g (t) = - t^{2} + 25$ .

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Step 2.5.1

To apply the Chain Rule, set $u$ as $- t^{2} + 25$ .

$f'' (t) = \frac{{(- t^{2} + 25)}^{\frac{1}{2}} (- 4 t) - (- 2 t^{2} + 25) (\frac{d}{d u} (u^{\frac{1}{2}}) \frac{d}{d t} (- t^{2} + 25))}{- t^{2} + 25}$

Step 2.5.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{1}{2}$ .

$f'' (t) = \frac{{(- t^{2} + 25)}^{\frac{1}{2}} (- 4 t) - (- 2 t^{2} + 25) (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d t} (- t^{2} + 25))}{- t^{2} + 25}$

Step 2.5.3

Replace all occurrences of $u$ with $- t^{2} + 25$ .

$f'' (t) = \frac{{(- t^{2} + 25)}^{\frac{1}{2}} (- 4 t) - (- 2 t^{2} + 25) (\frac{1}{2} \cdot {(- t^{2} + 25)}^{\frac{1}{2} - 1} \frac{d}{d t} (- t^{2} + 25))}{- t^{2} + 25}$

Step 2.6

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f'' (t) = \frac{{(- t^{2} + 25)}^{\frac{1}{2}} (- 4 t) - (- 2 t^{2} + 25) (\frac{1}{2} \cdot {(- t^{2} + 25)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d t} (- t^{2} + 25))}{- t^{2} + 25}$

Step 2.7

Combine $- 1$ and $\frac{2}{2}$ .

$f'' (t) = \frac{{(- t^{2} + 25)}^{\frac{1}{2}} (- 4 t) - (- 2 t^{2} + 25) (\frac{1}{2} \cdot {(- t^{2} + 25)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d t} (- t^{2} + 25))}{- t^{2} + 25}$

Step 2.8

Combine the numerators over the common denominator.

$f'' (t) = \frac{{(- t^{2} + 25)}^{\frac{1}{2}} (- 4 t) - (- 2 t^{2} + 25) (\frac{1}{2} \cdot {(- t^{2} + 25)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d t} (- t^{2} + 25))}{- t^{2} + 25}$

Step 2.9

Simplify the numerator.

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Step 2.9.1

Multiply $- 1$ by $2$ .

$f'' (t) = \frac{{(- t^{2} + 25)}^{\frac{1}{2}} (- 4 t) - (- 2 t^{2} + 25) (\frac{1}{2} \cdot {(- t^{2} + 25)}^{\frac{1 - 2}{2}} \frac{d}{d t} (- t^{2} + 25))}{- t^{2} + 25}$

Step 2.9.2

Subtract $2$ from $1$ .

$f'' (t) = \frac{{(- t^{2} + 25)}^{\frac{1}{2}} (- 4 t) - (- 2 t^{2} + 25) (\frac{1}{2} \cdot {(- t^{2} + 25)}^{\frac{- 1}{2}} \frac{d}{d t} (- t^{2} + 25))}{- t^{2} + 25}$

Step 2.10

Combine fractions.

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Step 2.10.1

Move the negative in front of the fraction.

$f'' (t) = \frac{{(- t^{2} + 25)}^{\frac{1}{2}} (- 4 t) - (- 2 t^{2} + 25) (\frac{1}{2} \cdot {(- t^{2} + 25)}^{- \frac{1}{2}} \frac{d}{d t} (- t^{2} + 25))}{- t^{2} + 25}$

Step 2.10.2

Combine $\frac{1}{2}$ and ${(- t^{2} + 25)}^{- \frac{1}{2}}$ .

$f'' (t) = \frac{{(- t^{2} + 25)}^{\frac{1}{2}} (- 4 t) - (- 2 t^{2} + 25) (\frac{{(- t^{2} + 25)}^{- \frac{1}{2}}}{2} \cdot \frac{d}{d t} (- t^{2} + 25))}{- t^{2} + 25}$

Step 2.10.3

Move ${(- t^{2} + 25)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (t) = \frac{{(- t^{2} + 25)}^{\frac{1}{2}} (- 4 t) - (- 2 t^{2} + 25) (\frac{1}{2 {(- t^{2} + 25)}^{\frac{1}{2}}} \cdot \frac{d}{d t} (- t^{2} + 25))}{- t^{2} + 25}$

Step 2.11

By the Sum Rule, the derivative of $- t^{2} + 25$ with respect to $t$ is $\frac{d}{d t} [- t^{2}] + \frac{d}{d t} [25]$ .

$f'' (t) = \frac{{(- t^{2} + 25)}^{\frac{1}{2}} (- 4 t) - (- 2 t^{2} + 25) (\frac{1}{2 {(- t^{2} + 25)}^{\frac{1}{2}}} \cdot (\frac{d}{d t} (- t^{2}) + \frac{d}{d t} (25)))}{- t^{2} + 25}$

Step 2.12

Since $- 1$ is constant with respect to $t$ , the derivative of $- t^{2}$ with respect to $t$ is $- \frac{d}{d t} [t^{2}]$ .

$f'' (t) = \frac{{(- t^{2} + 25)}^{\frac{1}{2}} (- 4 t) - (- 2 t^{2} + 25) (\frac{1}{2 {(- t^{2} + 25)}^{\frac{1}{2}}} \cdot (- \frac{d}{d t} t^{2} + \frac{d}{d t} (25)))}{- t^{2} + 25}$

Step 2.13

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 2$ .

$f'' (t) = \frac{{(- t^{2} + 25)}^{\frac{1}{2}} (- 4 t) - (- 2 t^{2} + 25) (\frac{1}{2 {(- t^{2} + 25)}^{\frac{1}{2}}} \cdot (- (2 t) + \frac{d}{d t} (25)))}{- t^{2} + 25}$

Step 2.14

Multiply $2$ by $- 1$ .

$f'' (t) = \frac{{(- t^{2} + 25)}^{\frac{1}{2}} (- 4 t) - (- 2 t^{2} + 25) (\frac{1}{2 {(- t^{2} + 25)}^{\frac{1}{2}}} \cdot (- 2 t + \frac{d}{d t} (25)))}{- t^{2} + 25}$

Step 2.15

Since $25$ is constant with respect to $t$ , the derivative of $25$ with respect to $t$ is $0$ .

$f'' (t) = \frac{{(- t^{2} + 25)}^{\frac{1}{2}} (- 4 t) - (- 2 t^{2} + 25) (\frac{1}{2 {(- t^{2} + 25)}^{\frac{1}{2}}} \cdot (- 2 t + 0))}{- t^{2} + 25}$

Step 2.16

Simplify terms.

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Step 2.16.1

Add $- 2 t$ and $0$ .

$f'' (t) = \frac{{(- t^{2} + 25)}^{\frac{1}{2}} (- 4 t) - (- 2 t^{2} + 25) (\frac{1}{2 {(- t^{2} + 25)}^{\frac{1}{2}}} \cdot (- 2 t))}{- t^{2} + 25}$

Step 2.16.2

Combine $- 2$ and $\frac{1}{2 {(- t^{2} + 25)}^{\frac{1}{2}}}$ .

$f'' (t) = \frac{{(- t^{2} + 25)}^{\frac{1}{2}} (- 4 t) - (- 2 t^{2} + 25) (\frac{- 2}{2 {(- t^{2} + 25)}^{\frac{1}{2}}} \cdot t)}{- t^{2} + 25}$

Step 2.16.3

Combine $\frac{- 2}{2 {(- t^{2} + 25)}^{\frac{1}{2}}}$ and $t$ .

$f'' (t) = \frac{{(- t^{2} + 25)}^{\frac{1}{2}} (- 4 t) - (- 2 t^{2} + 25) \frac{- 2 t}{2 {(- t^{2} + 25)}^{\frac{1}{2}}}}{- t^{2} + 25}$

Step 2.16.4

Factor $2$ out of $- 2 t$ .

$f'' (t) = \frac{{(- t^{2} + 25)}^{\frac{1}{2}} (- 4 t) - (- 2 t^{2} + 25) \frac{2 (- t)}{2 {(- t^{2} + 25)}^{\frac{1}{2}}}}{- t^{2} + 25}$

Step 2.17

Cancel the common factors.

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Step 2.17.1

Factor $2$ out of $2 {(- t^{2} + 25)}^{\frac{1}{2}}$ .

$f'' (t) = \frac{{(- t^{2} + 25)}^{\frac{1}{2}} (- 4 t) - (- 2 t^{2} + 25) \frac{2 (- t)}{2 ({(- t^{2} + 25)}^{\frac{1}{2}})}}{- t^{2} + 25}$

Step 2.17.2

Cancel the common factor.

$f'' (t) = \frac{{(- t^{2} + 25)}^{\frac{1}{2}} (- 4 t) - (- 2 t^{2} + 25) \frac{2 (- t)}{2 {(- t^{2} + 25)}^{\frac{1}{2}}}}{- t^{2} + 25}$

Step 2.17.3

Rewrite the expression.

$f'' (t) = \frac{{(- t^{2} + 25)}^{\frac{1}{2}} (- 4 t) - (- 2 t^{2} + 25) \frac{- t}{{(- t^{2} + 25)}^{\frac{1}{2}}}}{- t^{2} + 25}$

Step 2.18

Move the negative in front of the fraction.

$f'' (t) = \frac{{(- t^{2} + 25)}^{\frac{1}{2}} (- 4 t) - (- 2 t^{2} + 25) (- \frac{t}{{(- t^{2} + 25)}^{\frac{1}{2}}})}{- t^{2} + 25}$

Step 2.19

Multiply $- 1$ by $- 1$ .

$f'' (t) = \frac{{(- t^{2} + 25)}^{\frac{1}{2}} (- 4 t) + 1 (- 2 t^{2} + 25) (\frac{t}{{(- t^{2} + 25)}^{\frac{1}{2}}})}{- t^{2} + 25}$

Step 2.20

Multiply $- 2 t^{2} + 25$ by $1$ .

$f'' (t) = \frac{{(- t^{2} + 25)}^{\frac{1}{2}} (- 4 t) + (- 2 t^{2} + 25) (\frac{t}{{(- t^{2} + 25)}^{\frac{1}{2}}})}{- t^{2} + 25}$

Step 2.21

Simplify.

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Step 2.21.1

Simplify the numerator.

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Step 2.21.1.1

Rewrite using the commutative property of multiplication.

$f'' (t) = \frac{- 4 {(- t^{2} + 25)}^{\frac{1}{2}} t + (- 2 t^{2} + 25) (\frac{t}{{(- t^{2} + 25)}^{\frac{1}{2}}})}{- t^{2} + 25}$

Step 2.21.1.2

Multiply $- 2 t^{2} + 25$ by $\frac{t}{{(- t^{2} + 25)}^{\frac{1}{2}}}$ .

$f'' (t) = \frac{- 4 {(- t^{2} + 25)}^{\frac{1}{2}} t + \frac{(- 2 t^{2} + 25) t}{{(- t^{2} + 25)}^{\frac{1}{2}}}}{- t^{2} + 25}$

Step 2.21.1.3

To write $- 4 {(- t^{2} + 25)}^{\frac{1}{2}} t$ as a fraction with a common denominator, multiply by $\frac{{(- t^{2} + 25)}^{\frac{1}{2}}}{{(- t^{2} + 25)}^{\frac{1}{2}}}$ .

$f'' (t) = \frac{- 4 {(- t^{2} + 25)}^{\frac{1}{2}} t \cdot \frac{{(- t^{2} + 25)}^{\frac{1}{2}}}{{(- t^{2} + 25)}^{\frac{1}{2}}} + \frac{(- 2 t^{2} + 25) t}{{(- t^{2} + 25)}^{\frac{1}{2}}}}{- t^{2} + 25}$

Step 2.21.1.4

Combine $- 4 {(- t^{2} + 25)}^{\frac{1}{2}} t$ and $\frac{{(- t^{2} + 25)}^{\frac{1}{2}}}{{(- t^{2} + 25)}^{\frac{1}{2}}}$ .

$f'' (t) = \frac{\frac{- 4 {(- t^{2} + 25)}^{\frac{1}{2}} t {(- t^{2} + 25)}^{\frac{1}{2}}}{{(- t^{2} + 25)}^{\frac{1}{2}}} + \frac{(- 2 t^{2} + 25) t}{{(- t^{2} + 25)}^{\frac{1}{2}}}}{- t^{2} + 25}$

Step 2.21.1.5

Combine the numerators over the common denominator.

$f'' (t) = \frac{\frac{- 4 {(- t^{2} + 25)}^{\frac{1}{2}} t {(- t^{2} + 25)}^{\frac{1}{2}} + (- 2 t^{2} + 25) t}{{(- t^{2} + 25)}^{\frac{1}{2}}}}{- t^{2} + 25}$

Step 2.21.1.6

Rewrite $\frac{- 4 {(- t^{2} + 25)}^{\frac{1}{2}} t {(- t^{2} + 25)}^{\frac{1}{2}} + (- 2 t^{2} + 25) t}{{(- t^{2} + 25)}^{\frac{1}{2}}}$ in a factored form.

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Step 2.21.1.6.1

Factor $t$ out of $- 4 {(- t^{2} + 25)}^{\frac{1}{2}} t {(- t^{2} + 25)}^{\frac{1}{2}} + (- 2 t^{2} + 25) t$ .

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Step 2.21.1.6.1.1

Factor $t$ out of $- 4 {(- t^{2} + 25)}^{\frac{1}{2}} t {(- t^{2} + 25)}^{\frac{1}{2}}$ .

$f'' (t) = \frac{\frac{t (- 4 {(- t^{2} + 25)}^{\frac{1}{2}} {(- t^{2} + 25)}^{\frac{1}{2}}) + (- 2 t^{2} + 25) t}{{(- t^{2} + 25)}^{\frac{1}{2}}}}{- t^{2} + 25}$

Step 2.21.1.6.1.2

Factor $t$ out of $(- 2 t^{2} + 25) t$ .

$f'' (t) = \frac{\frac{t (- 4 {(- t^{2} + 25)}^{\frac{1}{2}} {(- t^{2} + 25)}^{\frac{1}{2}}) + t (- 2 t^{2} + 25)}{{(- t^{2} + 25)}^{\frac{1}{2}}}}{- t^{2} + 25}$

Step 2.21.1.6.1.3

Factor $t$ out of $t (- 4 {(- t^{2} + 25)}^{\frac{1}{2}} {(- t^{2} + 25)}^{\frac{1}{2}}) + t (- 2 t^{2} + 25)$ .

$f'' (t) = \frac{\frac{t (- 4 {(- t^{2} + 25)}^{\frac{1}{2}} {(- t^{2} + 25)}^{\frac{1}{2}} - 2 t^{2} + 25)}{{(- t^{2} + 25)}^{\frac{1}{2}}}}{- t^{2} + 25}$

Step 2.21.1.6.2

Multiply ${(- t^{2} + 25)}^{\frac{1}{2}}$ by ${(- t^{2} + 25)}^{\frac{1}{2}}$ by adding the exponents.

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Step 2.21.1.6.2.1

Move ${(- t^{2} + 25)}^{\frac{1}{2}}$ .

$f'' (t) = \frac{\frac{t (- 4 ({(- t^{2} + 25)}^{\frac{1}{2}} {(- t^{2} + 25)}^{\frac{1}{2}}) - 2 t^{2} + 25)}{{(- t^{2} + 25)}^{\frac{1}{2}}}}{- t^{2} + 25}$

Step 2.21.1.6.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (t) = \frac{\frac{t (- 4 {(- t^{2} + 25)}^{\frac{1}{2} + \frac{1}{2}} - 2 t^{2} + 25)}{{(- t^{2} + 25)}^{\frac{1}{2}}}}{- t^{2} + 25}$

Step 2.21.1.6.2.3

Combine the numerators over the common denominator.

$f'' (t) = \frac{\frac{t (- 4 {(- t^{2} + 25)}^{\frac{1 + 1}{2}} - 2 t^{2} + 25)}{{(- t^{2} + 25)}^{\frac{1}{2}}}}{- t^{2} + 25}$

Step 2.21.1.6.2.4

Add $1$ and $1$ .

$f'' (t) = \frac{\frac{t (- 4 {(- t^{2} + 25)}^{\frac{2}{2}} - 2 t^{2} + 25)}{{(- t^{2} + 25)}^{\frac{1}{2}}}}{- t^{2} + 25}$

Step 2.21.1.6.2.5

Divide $2$ by $2$ .

$f'' (t) = \frac{\frac{t (- 4 (- t^{2} + 25) - 2 t^{2} + 25)}{{(- t^{2} + 25)}^{\frac{1}{2}}}}{- t^{2} + 25}$

Step 2.21.1.6.3

Simplify $- 4 {(- t^{2} + 25)}^{1}$ .

$f'' (t) = \frac{\frac{t (- 4 (- t^{2} + 25) - 2 t^{2} + 25)}{{(- t^{2} + 25)}^{\frac{1}{2}}}}{- t^{2} + 25}$

Step 2.21.1.6.4

Apply the distributive property.

$f'' (t) = \frac{\frac{t (- 4 (- t^{2}) - 4 \cdot 25 - 2 t^{2} + 25)}{{(- t^{2} + 25)}^{\frac{1}{2}}}}{- t^{2} + 25}$

Step 2.21.1.6.5

Multiply $- 1$ by $- 4$ .

$f'' (t) = \frac{\frac{t (4 t^{2} - 4 \cdot 25 - 2 t^{2} + 25)}{{(- t^{2} + 25)}^{\frac{1}{2}}}}{- t^{2} + 25}$

Step 2.21.1.6.6

Multiply $- 4$ by $25$ .

$f'' (t) = \frac{\frac{t (4 t^{2} - 100 - 2 t^{2} + 25)}{{(- t^{2} + 25)}^{\frac{1}{2}}}}{- t^{2} + 25}$

Step 2.21.1.6.7

Subtract $2 t^{2}$ from $4 t^{2}$ .

$f'' (t) = \frac{\frac{t (2 t^{2} - 100 + 25)}{{(- t^{2} + 25)}^{\frac{1}{2}}}}{- t^{2} + 25}$

Step 2.21.1.6.8

Add $- 100$ and $25$ .

$f'' (t) = \frac{\frac{t (2 t^{2} - 75)}{{(- t^{2} + 25)}^{\frac{1}{2}}}}{- t^{2} + 25}$

Step 2.21.2

Combine terms.

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Step 2.21.2.1

Rewrite $\frac{\frac{t (2 t^{2} - 75)}{{(- t^{2} + 25)}^{\frac{1}{2}}}}{- t^{2} + 25}$ as a product.

$f'' (t) = \frac{t (2 t^{2} - 75)}{{(- t^{2} + 25)}^{\frac{1}{2}}} \cdot \frac{1}{- t^{2} + 25}$

Step 2.21.2.2

Multiply $\frac{t (2 t^{2} - 75)}{{(- t^{2} + 25)}^{\frac{1}{2}}}$ by $\frac{1}{- t^{2} + 25}$ .

$f'' (t) = \frac{t (2 t^{2} - 75)}{{(- t^{2} + 25)}^{\frac{1}{2}} (- t^{2} + 25)}$

Step 2.21.2.3

Multiply ${(- t^{2} + 25)}^{\frac{1}{2}}$ by $- t^{2} + 25$ by adding the exponents.

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Step 2.21.2.3.1

Multiply ${(- t^{2} + 25)}^{\frac{1}{2}}$ by $- t^{2} + 25$ .

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Step 2.21.2.3.1.1

Raise $- t^{2} + 25$ to the power of $1$ .

$f'' (t) = \frac{t (2 t^{2} - 75)}{{(- t^{2} + 25)}^{\frac{1}{2}} (- t^{2} + 25)}$

Step 2.21.2.3.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (t) = \frac{t (2 t^{2} - 75)}{{(- t^{2} + 25)}^{\frac{1}{2} + 1}}$

Step 2.21.2.3.2

Write $1$ as a fraction with a common denominator.

$f'' (t) = \frac{t (2 t^{2} - 75)}{{(- t^{2} + 25)}^{\frac{1}{2} + \frac{2}{2}}}$

Step 2.21.2.3.3

Combine the numerators over the common denominator.

$f'' (t) = \frac{t (2 t^{2} - 75)}{{(- t^{2} + 25)}^{\frac{1 + 2}{2}}}$

Step 2.21.2.3.4

Add $1$ and $2$ .

$f'' (t) = \frac{t (2 t^{2} - 75)}{{(- t^{2} + 25)}^{\frac{3}{2}}}$

Step 3

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$\frac{- 2 t^{2} + 25}{{(- t^{2} + 25)}^{\frac{1}{2}}} = 0$

Step 4

Find the first derivative.

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Step 4.1

Find the first derivative.

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Step 4.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{25 - t^{2}}$ as ${(25 - t^{2})}^{\frac{1}{2}}$ .

$f' (t) = \frac{d}{d t} (t {(25 - t^{2})}^{\frac{1}{2}})$

Step 4.1.2

$f' (t) = t \frac{d}{d t} ({(25 - t^{2})}^{\frac{1}{2}}) + {(25 - t^{2})}^{\frac{1}{2}} \frac{d}{d t} (t)$

Step 4.1.3

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = t^{\frac{1}{2}}$ and $g (t) = 25 - t^{2}$ .

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Step 4.1.3.1

To apply the Chain Rule, set $u$ as $25 - t^{2}$ .

$f' (t) = t (\frac{d}{d u} (u^{\frac{1}{2}}) \frac{d}{d t} (25 - t^{2})) + {(25 - t^{2})}^{\frac{1}{2}} \frac{d}{d t} (t)$

Step 4.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{1}{2}$ .

$f' (t) = t (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d t} (25 - t^{2})) + {(25 - t^{2})}^{\frac{1}{2}} \frac{d}{d t} (t)$

Step 4.1.3.3

Replace all occurrences of $u$ with $25 - t^{2}$ .

$f' (t) = t (\frac{1}{2} \cdot {(25 - t^{2})}^{\frac{1}{2} - 1} \frac{d}{d t} (25 - t^{2})) + {(25 - t^{2})}^{\frac{1}{2}} \frac{d}{d t} (t)$

Step 4.1.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f' (t) = t (\frac{1}{2} \cdot {(25 - t^{2})}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d t} (25 - t^{2})) + {(25 - t^{2})}^{\frac{1}{2}} \frac{d}{d t} (t)$

Step 4.1.5

Combine $- 1$ and $\frac{2}{2}$ .

$f' (t) = t (\frac{1}{2} \cdot {(25 - t^{2})}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d t} (25 - t^{2})) + {(25 - t^{2})}^{\frac{1}{2}} \frac{d}{d t} (t)$

Step 4.1.6

Combine the numerators over the common denominator.

$f' (t) = t (\frac{1}{2} \cdot {(25 - t^{2})}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d t} (25 - t^{2})) + {(25 - t^{2})}^{\frac{1}{2}} \frac{d}{d t} (t)$

Step 4.1.7

Simplify the numerator.

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Step 4.1.7.1

Multiply $- 1$ by $2$ .

$f' (t) = t (\frac{1}{2} \cdot {(25 - t^{2})}^{\frac{1 - 2}{2}} \frac{d}{d t} (25 - t^{2})) + {(25 - t^{2})}^{\frac{1}{2}} \frac{d}{d t} (t)$

Step 4.1.7.2

Subtract $2$ from $1$ .

$f' (t) = t (\frac{1}{2} \cdot {(25 - t^{2})}^{\frac{- 1}{2}} \frac{d}{d t} (25 - t^{2})) + {(25 - t^{2})}^{\frac{1}{2}} \frac{d}{d t} (t)$

Step 4.1.8

Combine fractions.

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Step 4.1.8.1

Move the negative in front of the fraction.

$f' (t) = t (\frac{1}{2} \cdot {(25 - t^{2})}^{- \frac{1}{2}} \frac{d}{d t} (25 - t^{2})) + {(25 - t^{2})}^{\frac{1}{2}} \frac{d}{d t} (t)$

Step 4.1.8.2

Combine $\frac{1}{2}$ and ${(25 - t^{2})}^{- \frac{1}{2}}$ .

$f' (t) = t (\frac{{(25 - t^{2})}^{- \frac{1}{2}}}{2} \cdot \frac{d}{d t} (25 - t^{2})) + {(25 - t^{2})}^{\frac{1}{2}} \frac{d}{d t} (t)$

Step 4.1.8.3

Move ${(25 - t^{2})}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f' (t) = t (\frac{1}{2 {(25 - t^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d t} (25 - t^{2})) + {(25 - t^{2})}^{\frac{1}{2}} \frac{d}{d t} (t)$

Step 4.1.8.4

Combine $\frac{1}{2 {(25 - t^{2})}^{\frac{1}{2}}}$ and $t$ .

$f' (t) = \frac{t}{2 {(25 - t^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d t} (25 - t^{2}) + {(25 - t^{2})}^{\frac{1}{2}} \frac{d}{d t} (t)$

Step 4.1.9

By the Sum Rule, the derivative of $25 - t^{2}$ with respect to $t$ is $\frac{d}{d t} [25] + \frac{d}{d t} [- t^{2}]$ .

$f' (t) = \frac{t}{2 {(25 - t^{2})}^{\frac{1}{2}}} \cdot (\frac{d}{d t} (25) + \frac{d}{d t} (- t^{2})) + {(25 - t^{2})}^{\frac{1}{2}} \frac{d}{d t} (t)$

Step 4.1.10

Since $25$ is constant with respect to $t$ , the derivative of $25$ with respect to $t$ is $0$ .

$f' (t) = \frac{t}{2 {(25 - t^{2})}^{\frac{1}{2}}} \cdot (0 + \frac{d}{d t} (- t^{2})) + {(25 - t^{2})}^{\frac{1}{2}} \frac{d}{d t} (t)$

Step 4.1.11

Add $0$ and $\frac{d}{d t} [- t^{2}]$ .

$f' (t) = \frac{t}{2 {(25 - t^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d t} (- t^{2}) + {(25 - t^{2})}^{\frac{1}{2}} \frac{d}{d t} (t)$

Step 4.1.12

Since $- 1$ is constant with respect to $t$ , the derivative of $- t^{2}$ with respect to $t$ is $- \frac{d}{d t} [t^{2}]$ .

$f' (t) = \frac{t}{2 {(25 - t^{2})}^{\frac{1}{2}}} \cdot (- \frac{d}{d t} t^{2}) + {(25 - t^{2})}^{\frac{1}{2}} \frac{d}{d t} (t)$

Step 4.1.13

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 2$ .

$f' (t) = \frac{t}{2 {(25 - t^{2})}^{\frac{1}{2}}} \cdot (- (2 t)) + {(25 - t^{2})}^{\frac{1}{2}} \frac{d}{d t} (t)$

Step 4.1.14

Combine fractions.

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Step 4.1.14.1

Multiply $2$ by $- 1$ .

$f' (t) = \frac{t}{2 {(25 - t^{2})}^{\frac{1}{2}}} \cdot (- 2 t) + {(25 - t^{2})}^{\frac{1}{2}} \frac{d}{d t} (t)$

Step 4.1.14.2

Combine $- 2$ and $\frac{t}{2 {(25 - t^{2})}^{\frac{1}{2}}}$ .

$f' (t) = \frac{- 2 t}{2 {(25 - t^{2})}^{\frac{1}{2}}} \cdot t + {(25 - t^{2})}^{\frac{1}{2}} \frac{d}{d t} (t)$

Step 4.1.14.3

Combine $\frac{- 2 t}{2 {(25 - t^{2})}^{\frac{1}{2}}}$ and $t$ .

$f' (t) = \frac{- 2 t \cdot t}{2 {(25 - t^{2})}^{\frac{1}{2}}} + {(25 - t^{2})}^{\frac{1}{2}} \frac{d}{d t} (t)$

Step 4.1.15

Raise $t$ to the power of $1$ .

$f' (t) = \frac{- 2 (t \cdot t)}{2 {(25 - t^{2})}^{\frac{1}{2}}} + {(25 - t^{2})}^{\frac{1}{2}} \frac{d}{d t} (t)$

Step 4.1.16

Raise $t$ to the power of $1$ .

$f' (t) = \frac{- 2 (t \cdot t)}{2 {(25 - t^{2})}^{\frac{1}{2}}} + {(25 - t^{2})}^{\frac{1}{2}} \frac{d}{d t} (t)$

Step 4.1.17

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f' (t) = \frac{- 2 t^{1 + 1}}{2 {(25 - t^{2})}^{\frac{1}{2}}} + {(25 - t^{2})}^{\frac{1}{2}} \frac{d}{d t} (t)$

Step 4.1.18

Add $1$ and $1$ .

$f' (t) = \frac{- 2 t^{2}}{2 {(25 - t^{2})}^{\frac{1}{2}}} + {(25 - t^{2})}^{\frac{1}{2}} \frac{d}{d t} (t)$

Step 4.1.19

Factor $2$ out of $- 2 t^{2}$ .

$f' (t) = \frac{2 (- t^{2})}{2 {(25 - t^{2})}^{\frac{1}{2}}} + {(25 - t^{2})}^{\frac{1}{2}} \frac{d}{d t} (t)$

Step 4.1.20

Cancel the common factors.

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Step 4.1.20.1

Factor $2$ out of $2 {(25 - t^{2})}^{\frac{1}{2}}$ .

$f' (t) = \frac{2 (- t^{2})}{2 ({(25 - t^{2})}^{\frac{1}{2}})} + {(25 - t^{2})}^{\frac{1}{2}} \frac{d}{d t} (t)$

Step 4.1.20.2

Cancel the common factor.

$f' (t) = \frac{2 (- t^{2})}{2 {(25 - t^{2})}^{\frac{1}{2}}} + {(25 - t^{2})}^{\frac{1}{2}} \frac{d}{d t} (t)$

Step 4.1.20.3

Rewrite the expression.

$f' (t) = \frac{- t^{2}}{{(25 - t^{2})}^{\frac{1}{2}}} + {(25 - t^{2})}^{\frac{1}{2}} \frac{d}{d t} (t)$

Step 4.1.21

Move the negative in front of the fraction.

$f' (t) = - \frac{t^{2}}{{(25 - t^{2})}^{\frac{1}{2}}} + {(25 - t^{2})}^{\frac{1}{2}} \frac{d}{d t} (t)$

Step 4.1.22

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 1$ .

$f' (t) = - \frac{t^{2}}{{(25 - t^{2})}^{\frac{1}{2}}} + {(25 - t^{2})}^{\frac{1}{2}} \cdot 1$

Step 4.1.23

Multiply ${(25 - t^{2})}^{\frac{1}{2}}$ by $1$ .

$f' (t) = - \frac{t^{2}}{{(25 - t^{2})}^{\frac{1}{2}}} + {(25 - t^{2})}^{\frac{1}{2}}$

Step 4.1.24

To write ${(25 - t^{2})}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{{(25 - t^{2})}^{\frac{1}{2}}}{{(25 - t^{2})}^{\frac{1}{2}}}$ .

$f' (t) = - \frac{t^{2}}{{(25 - t^{2})}^{\frac{1}{2}}} + \frac{{(25 - t^{2})}^{\frac{1}{2}} {(25 - t^{2})}^{\frac{1}{2}}}{{(25 - t^{2})}^{\frac{1}{2}}}$

Step 4.1.25

Combine the numerators over the common denominator.

$f' (t) = \frac{- t^{2} + {(25 - t^{2})}^{\frac{1}{2}} {(25 - t^{2})}^{\frac{1}{2}}}{{(25 - t^{2})}^{\frac{1}{2}}}$

Step 4.1.26

Multiply ${(25 - t^{2})}^{\frac{1}{2}}$ by ${(25 - t^{2})}^{\frac{1}{2}}$ by adding the exponents.

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Step 4.1.26.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f' (t) = \frac{- t^{2} + {(25 - t^{2})}^{\frac{1}{2} + \frac{1}{2}}}{{(25 - t^{2})}^{\frac{1}{2}}}$

Step 4.1.26.2

Combine the numerators over the common denominator.

$f' (t) = \frac{- t^{2} + {(25 - t^{2})}^{\frac{1 + 1}{2}}}{{(25 - t^{2})}^{\frac{1}{2}}}$

Step 4.1.26.3

Add $1$ and $1$ .

$f' (t) = \frac{- t^{2} + {(25 - t^{2})}^{\frac{2}{2}}}{{(25 - t^{2})}^{\frac{1}{2}}}$

Step 4.1.26.4

Divide $2$ by $2$ .

$f' (t) = \frac{- t^{2} + 25 - t^{2}}{{(25 - t^{2})}^{\frac{1}{2}}}$

Step 4.1.27

Simplify ${(25 - t^{2})}^{1}$ .

$f' (t) = \frac{- t^{2} + 25 - t^{2}}{{(25 - t^{2})}^{\frac{1}{2}}}$

Step 4.1.28

Subtract $t^{2}$ from $- t^{2}$ .

$f' (t) = \frac{- 2 t^{2} + 25}{{(25 - t^{2})}^{\frac{1}{2}}}$

Step 4.1.29

Reorder terms.

$f' (t) = \frac{- 2 t^{2} + 25}{{(- t^{2} + 25)}^{\frac{1}{2}}}$

Step 4.2

The first derivative of $f (t)$ with respect to $t$ is $\frac{- 2 t^{2} + 25}{{(- t^{2} + 25)}^{\frac{1}{2}}}$ .

$\frac{- 2 t^{2} + 25}{{(- t^{2} + 25)}^{\frac{1}{2}}}$

Step 5

Set the first derivative equal to $0$ then solve the equation $\frac{- 2 t^{2} + 25}{{(- t^{2} + 25)}^{\frac{1}{2}}} = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$\frac{- 2 t^{2} + 25}{{(- t^{2} + 25)}^{\frac{1}{2}}} = 0$

Step 5.2

Set the numerator equal to zero.

$- 2 t^{2} + 25 = 0$

Step 5.3

Solve the equation for $t$ .

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Step 5.3.1

Subtract $25$ from both sides of the equation.

$- 2 t^{2} = - 25$

Step 5.3.2

Divide each term in $- 2 t^{2} = - 25$ by $- 2$ and simplify.

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Step 5.3.2.1

Divide each term in $- 2 t^{2} = - 25$ by $- 2$ .

$\frac{- 2 t^{2}}{- 2} = \frac{- 25}{- 2}$

Step 5.3.2.2

Simplify the left side.

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Step 5.3.2.2.1

Cancel the common factor of $- 2$ .

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Step 5.3.2.2.1.1

Cancel the common factor.

$\frac{- 2 t^{2}}{- 2} = \frac{- 25}{- 2}$

Step 5.3.2.2.1.2

Divide $t^{2}$ by $1$ .

$t^{2} = \frac{- 25}{- 2}$

Step 5.3.2.3

Simplify the right side.

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Step 5.3.2.3.1

Dividing two negative values results in a positive value.

$t^{2} = \frac{25}{2}$

Step 5.3.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$t = \pm \sqrt{\frac{25}{2}}$

Step 5.3.4

Simplify $\pm \sqrt{\frac{25}{2}}$ .

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Step 5.3.4.1

Rewrite $\sqrt{\frac{25}{2}}$ as $\frac{\sqrt{25}}{\sqrt{2}}$ .

$t = \pm \frac{\sqrt{25}}{\sqrt{2}}$

Step 5.3.4.2

Simplify the numerator.

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Step 5.3.4.2.1

Rewrite $25$ as $5^{2}$ .

$t = \pm \frac{\sqrt{5^{2}}}{\sqrt{2}}$

Step 5.3.4.2.2

Pull terms out from under the radical, assuming positive real numbers.

$t = \pm \frac{5}{\sqrt{2}}$

Step 5.3.4.3

Multiply $\frac{5}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$t = \pm \frac{5}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 5.3.4.4

Combine and simplify the denominator.

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Step 5.3.4.4.1

Multiply $\frac{5}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$t = \pm \frac{5 \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 5.3.4.4.2

Raise $\sqrt{2}$ to the power of $1$ .

$t = \pm \frac{5 \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 5.3.4.4.3

Raise $\sqrt{2}$ to the power of $1$ .

$t = \pm \frac{5 \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 5.3.4.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$t = \pm \frac{5 \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 5.3.4.4.5

Add $1$ and $1$ .

$t = \pm \frac{5 \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 5.3.4.4.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 5.3.4.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$t = \pm \frac{5 \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 5.3.4.4.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$t = \pm \frac{5 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 5.3.4.4.6.3

Combine $\frac{1}{2}$ and $2$ .

$t = \pm \frac{5 \sqrt{2}}{2^{\frac{2}{2}}}$

Step 5.3.4.4.6.4

Cancel the common factor of $2$ .

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Step 5.3.4.4.6.4.1

Cancel the common factor.

$t = \pm \frac{5 \sqrt{2}}{2^{\frac{2}{2}}}$

Step 5.3.4.4.6.4.2

Rewrite the expression.

$t = \pm \frac{5 \sqrt{2}}{2^{1}}$

Step 5.3.4.4.6.5

Evaluate the exponent.

$t = \pm \frac{5 \sqrt{2}}{2}$

Step 5.3.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 5.3.5.1

First, use the positive value of the $\pm$ to find the first solution.

$t = \frac{5 \sqrt{2}}{2}$

Step 5.3.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$t = - \frac{5 \sqrt{2}}{2}$

Step 5.3.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$t = \frac{5 \sqrt{2}}{2}, - \frac{5 \sqrt{2}}{2}$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

Convert expressions with fractional exponents to radicals.

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Step 6.1.1

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$\frac{- 2 t^{2} + 25}{\sqrt{{(- t^{2} + 25)}^{1}}}$

Step 6.1.2

Anything raised to $1$ is the base itself.

$\frac{- 2 t^{2} + 25}{\sqrt{- t^{2} + 25}}$

Step 6.2

Set the denominator in $\frac{- 2 t^{2} + 25}{\sqrt{- t^{2} + 25}}$ equal to $0$ to find where the expression is undefined.

$\sqrt{- t^{2} + 25} = 0$

Step 6.3

Solve for $t$ .

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Step 6.3.1

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{- t^{2} + 25}}^{2} = 0^{2}$

Step 6.3.2

Simplify each side of the equation.

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Step 6.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{- t^{2} + 25}$ as ${(- t^{2} + 25)}^{\frac{1}{2}}$ .

${({(- t^{2} + 25)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 6.3.2.2

Simplify the left side.

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Step 6.3.2.2.1

Simplify ${({(- t^{2} + 25)}^{\frac{1}{2}})}^{2}$ .

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Step 6.3.2.2.1.1

Multiply the exponents in ${({(- t^{2} + 25)}^{\frac{1}{2}})}^{2}$ .

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Step 6.3.2.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(- t^{2} + 25)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 6.3.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 6.3.2.2.1.1.2.1

Cancel the common factor.

${(- t^{2} + 25)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 6.3.2.2.1.1.2.2

Rewrite the expression.

${(- t^{2} + 25)}^{1} = 0^{2}$

Step 6.3.2.2.1.2

Simplify.

$- t^{2} + 25 = 0^{2}$

Step 6.3.2.3

Simplify the right side.

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Step 6.3.2.3.1

Raising $0$ to any positive power yields $0$ .

$- t^{2} + 25 = 0$

Step 6.3.3

Solve for $t$ .

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Step 6.3.3.1

Subtract $25$ from both sides of the equation.

$- t^{2} = - 25$

Step 6.3.3.2

Divide each term in $- t^{2} = - 25$ by $- 1$ and simplify.

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Step 6.3.3.2.1

Divide each term in $- t^{2} = - 25$ by $- 1$ .

$\frac{- t^{2}}{- 1} = \frac{- 25}{- 1}$

Step 6.3.3.2.2

Simplify the left side.

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Step 6.3.3.2.2.1

Dividing two negative values results in a positive value.

$\frac{t^{2}}{1} = \frac{- 25}{- 1}$

Step 6.3.3.2.2.2

Divide $t^{2}$ by $1$ .

$t^{2} = \frac{- 25}{- 1}$

Step 6.3.3.2.3

Simplify the right side.

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Step 6.3.3.2.3.1

Divide $- 25$ by $- 1$ .

$t^{2} = 25$

Step 6.3.3.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$t = \pm \sqrt{25}$

Step 6.3.3.4

Simplify $\pm \sqrt{25}$ .

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Step 6.3.3.4.1

Rewrite $25$ as $5^{2}$ .

$t = \pm \sqrt{5^{2}}$

Step 6.3.3.4.2

Pull terms out from under the radical, assuming positive real numbers.

$t = \pm 5$

Step 6.3.3.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 6.3.3.5.1

First, use the positive value of the $\pm$ to find the first solution.

$t = 5$

Step 6.3.3.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$t = - 5$

Step 6.3.3.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$t = 5, - 5$

Step 6.4

Set the radicand in $\sqrt{- t^{2} + 25}$ less than $0$ to find where the expression is undefined.

$- t^{2} + 25 < 0$

Step 6.5

Solve for $t$ .

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Step 6.5.1

Subtract $25$ from both sides of the inequality.

$- t^{2} < - 25$

Step 6.5.2

Divide each term in $- t^{2} < - 25$ by $- 1$ and simplify.

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Step 6.5.2.1

Divide each term in $- t^{2} < - 25$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- t^{2}}{- 1} > \frac{- 25}{- 1}$

Step 6.5.2.2

Simplify the left side.

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Step 6.5.2.2.1

Dividing two negative values results in a positive value.

$\frac{t^{2}}{1} > \frac{- 25}{- 1}$

Step 6.5.2.2.2

Divide $t^{2}$ by $1$ .

$t^{2} > \frac{- 25}{- 1}$

Step 6.5.2.3

Simplify the right side.

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Step 6.5.2.3.1

Divide $- 25$ by $- 1$ .

$t^{2} > 25$

Step 6.5.3

Take the specified root of both sides of the inequality to eliminate the exponent on the left side.

$\sqrt{t^{2}} > \sqrt{25}$

Step 6.5.4

Simplify the equation.

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Step 6.5.4.1

Simplify the left side.

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Step 6.5.4.1.1

Pull terms out from under the radical.

$| t | > \sqrt{25}$

Step 6.5.4.2

Simplify the right side.

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Step 6.5.4.2.1

Simplify $\sqrt{25}$ .

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Step 6.5.4.2.1.1

Rewrite $25$ as $5^{2}$ .

$| t | > \sqrt{5^{2}}$

Step 6.5.4.2.1.2

Pull terms out from under the radical.

$| t | > | 5 |$

Step 6.5.4.2.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $5$ is $5$ .

$| t | > 5$

Step 6.5.5

Write $| t | > 5$ as a piecewise.

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Step 6.5.5.1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$t \geq 0$

Step 6.5.5.2

In the piece where $t$ is non-negative, remove the absolute value.

$t > 5$

Step 6.5.5.3

To find the interval for the second piece, find where the inside of the absolute value is negative.

$t < 0$

Step 6.5.5.4

In the piece where $t$ is negative, remove the absolute value and multiply by $- 1$ .

$- t > 5$

Step 6.5.5.5

Write as a piecewise.

${\begin{cases} t > 5 & t \geq 0 \\ - t > 5 & t < 0 \end{cases}$

Step 6.5.6

Find the intersection of $t > 5$ and $t \geq 0$ .

$t > 5$

Step 6.5.7

Divide each term in $- t > 5$ by $- 1$ and simplify.

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Step 6.5.7.1

Divide each term in $- t > 5$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- t}{- 1} < \frac{5}{- 1}$

Step 6.5.7.2

Simplify the left side.

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Step 6.5.7.2.1

Dividing two negative values results in a positive value.

$\frac{t}{1} < \frac{5}{- 1}$

Step 6.5.7.2.2

Divide $t$ by $1$ .

$t < \frac{5}{- 1}$

Step 6.5.7.3

Simplify the right side.

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Step 6.5.7.3.1

Divide $5$ by $- 1$ .

$t < - 5$

Step 6.5.8

Find the union of the solutions.

$t < - 5$ or $t > 5$

Step 6.6

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$t \leq - 5, t \geq 5$

$(- \infty, - 5] \cup [5, \infty)$

$t \leq - 5, t \geq 5$

$(- \infty, - 5] \cup [5, \infty)$

Step 7

Critical points to evaluate.

$t = \frac{5 \sqrt{2}}{2}, - \frac{5 \sqrt{2}}{2}, - 5, 5$

Step 8

Evaluate the second derivative at $t = \frac{5 \sqrt{2}}{2}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{(\frac{5 \sqrt{2}}{2}) (2 {(\frac{5 \sqrt{2}}{2})}^{2} - 75)}{{(- {(\frac{5 \sqrt{2}}{2})}^{2} + 25)}^{\frac{3}{2}}}$

Step 9

Evaluate the second derivative.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 9.1.1.1

Apply the product rule to $\frac{5 \sqrt{2}}{2}$ .

$\frac{\frac{5 \sqrt{2}}{2} (2 \frac{{(5 \sqrt{2})}^{2}}{2^{2}} - 75)}{{(- {(\frac{5 \sqrt{2}}{2})}^{2} + 25)}^{\frac{3}{2}}}$

Step 9.1.1.2

Apply the product rule to $5 \sqrt{2}$ .

$\frac{\frac{5 \sqrt{2}}{2} (2 \frac{5^{2} {\sqrt{2}}^{2}}{2^{2}} - 75)}{{(- {(\frac{5 \sqrt{2}}{2})}^{2} + 25)}^{\frac{3}{2}}}$

Step 9.1.2

Simplify the numerator.

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Step 9.1.2.1

Raise $5$ to the power of $2$ .

$\frac{\frac{5 \sqrt{2}}{2} (2 \frac{25 {\sqrt{2}}^{2}}{2^{2}} - 75)}{{(- {(\frac{5 \sqrt{2}}{2})}^{2} + 25)}^{\frac{3}{2}}}$

Step 9.1.2.2

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 9.1.2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\frac{\frac{5 \sqrt{2}}{2} (2 \frac{25 {(2^{\frac{1}{2}})}^{2}}{2^{2}} - 75)}{{(- {(\frac{5 \sqrt{2}}{2})}^{2} + 25)}^{\frac{3}{2}}}$

Step 9.1.2.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\frac{5 \sqrt{2}}{2} (2 \frac{25 \cdot 2^{\frac{1}{2} \cdot 2}}{2^{2}} - 75)}{{(- {(\frac{5 \sqrt{2}}{2})}^{2} + 25)}^{\frac{3}{2}}}$

Step 9.1.2.2.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{\frac{5 \sqrt{2}}{2} (2 \frac{25 \cdot 2^{\frac{2}{2}}}{2^{2}} - 75)}{{(- {(\frac{5 \sqrt{2}}{2})}^{2} + 25)}^{\frac{3}{2}}}$

Step 9.1.2.2.4

Cancel the common factor of $2$ .

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Step 9.1.2.2.4.1

Cancel the common factor.

$\frac{\frac{5 \sqrt{2}}{2} (2 \frac{25 \cdot 2^{\frac{2}{2}}}{2^{2}} - 75)}{{(- {(\frac{5 \sqrt{2}}{2})}^{2} + 25)}^{\frac{3}{2}}}$

Step 9.1.2.2.4.2

Rewrite the expression.

$\frac{\frac{5 \sqrt{2}}{2} (2 \frac{25 \cdot 2^{1}}{2^{2}} - 75)}{{(- {(\frac{5 \sqrt{2}}{2})}^{2} + 25)}^{\frac{3}{2}}}$

Step 9.1.2.2.5

Evaluate the exponent.

$\frac{\frac{5 \sqrt{2}}{2} (2 \frac{25 \cdot 2}{2^{2}} - 75)}{{(- {(\frac{5 \sqrt{2}}{2})}^{2} + 25)}^{\frac{3}{2}}}$

Step 9.1.3

Raise $2$ to the power of $2$ .

$\frac{\frac{5 \sqrt{2}}{2} (2 \frac{25 \cdot 2}{4} - 75)}{{(- {(\frac{5 \sqrt{2}}{2})}^{2} + 25)}^{\frac{3}{2}}}$

Step 9.1.4

Multiply $25$ by $2$ .

$\frac{\frac{5 \sqrt{2}}{2} (2 (\frac{50}{4}) - 75)}{{(- {(\frac{5 \sqrt{2}}{2})}^{2} + 25)}^{\frac{3}{2}}}$

Step 9.1.5

Cancel the common factor of $2$ .

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Step 9.1.5.1

Factor $2$ out of $4$ .

$\frac{\frac{5 \sqrt{2}}{2} (2 \frac{50}{2 (2)} - 75)}{{(- {(\frac{5 \sqrt{2}}{2})}^{2} + 25)}^{\frac{3}{2}}}$

Step 9.1.5.2

Cancel the common factor.

$\frac{\frac{5 \sqrt{2}}{2} (2 \frac{50}{2 \cdot 2} - 75)}{{(- {(\frac{5 \sqrt{2}}{2})}^{2} + 25)}^{\frac{3}{2}}}$

Step 9.1.5.3

Rewrite the expression.

$\frac{\frac{5 \sqrt{2}}{2} (\frac{50}{2} - 75)}{{(- {(\frac{5 \sqrt{2}}{2})}^{2} + 25)}^{\frac{3}{2}}}$

Step 9.1.6

Divide $50$ by $2$ .

$\frac{\frac{5 \sqrt{2}}{2} (25 - 75)}{{(- {(\frac{5 \sqrt{2}}{2})}^{2} + 25)}^{\frac{3}{2}}}$

Step 9.1.7

Subtract $75$ from $25$ .

$\frac{\frac{5 \sqrt{2}}{2} \cdot - 50}{{(- {(\frac{5 \sqrt{2}}{2})}^{2} + 25)}^{\frac{3}{2}}}$

Step 9.2

Simplify the denominator.

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Step 9.2.1

Simplify each term.

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Step 9.2.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 9.2.1.1.1

Apply the product rule to $\frac{5 \sqrt{2}}{2}$ .

$\frac{\frac{5 \sqrt{2}}{2} \cdot - 50}{{(- \frac{{(5 \sqrt{2})}^{2}}{2^{2}} + 25)}^{\frac{3}{2}}}$

Step 9.2.1.1.2

Apply the product rule to $5 \sqrt{2}$ .

$\frac{\frac{5 \sqrt{2}}{2} \cdot - 50}{{(- \frac{5^{2} {\sqrt{2}}^{2}}{2^{2}} + 25)}^{\frac{3}{2}}}$

Step 9.2.1.2

Simplify the numerator.

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Step 9.2.1.2.1

Raise $5$ to the power of $2$ .

$\frac{\frac{5 \sqrt{2}}{2} \cdot - 50}{{(- \frac{25 {\sqrt{2}}^{2}}{2^{2}} + 25)}^{\frac{3}{2}}}$

Step 9.2.1.2.2

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 9.2.1.2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\frac{\frac{5 \sqrt{2}}{2} \cdot - 50}{{(- \frac{25 {(2^{\frac{1}{2}})}^{2}}{2^{2}} + 25)}^{\frac{3}{2}}}$

Step 9.2.1.2.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\frac{5 \sqrt{2}}{2} \cdot - 50}{{(- \frac{25 \cdot 2^{\frac{1}{2} \cdot 2}}{2^{2}} + 25)}^{\frac{3}{2}}}$

Step 9.2.1.2.2.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{\frac{5 \sqrt{2}}{2} \cdot - 50}{{(- \frac{25 \cdot 2^{\frac{2}{2}}}{2^{2}} + 25)}^{\frac{3}{2}}}$

Step 9.2.1.2.2.4

Cancel the common factor of $2$ .

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Step 9.2.1.2.2.4.1

Cancel the common factor.

$\frac{\frac{5 \sqrt{2}}{2} \cdot - 50}{{(- \frac{25 \cdot 2^{\frac{2}{2}}}{2^{2}} + 25)}^{\frac{3}{2}}}$

Step 9.2.1.2.2.4.2

Rewrite the expression.

$\frac{\frac{5 \sqrt{2}}{2} \cdot - 50}{{(- \frac{25 \cdot 2^{1}}{2^{2}} + 25)}^{\frac{3}{2}}}$

Step 9.2.1.2.2.5

Evaluate the exponent.

$\frac{\frac{5 \sqrt{2}}{2} \cdot - 50}{{(- \frac{25 \cdot 2}{2^{2}} + 25)}^{\frac{3}{2}}}$

Step 9.2.1.3

Raise $2$ to the power of $2$ .

$\frac{\frac{5 \sqrt{2}}{2} \cdot - 50}{{(- \frac{25 \cdot 2}{4} + 25)}^{\frac{3}{2}}}$

Step 9.2.1.4

Multiply $25$ by $2$ .

$\frac{\frac{5 \sqrt{2}}{2} \cdot - 50}{{(- \frac{50}{4} + 25)}^{\frac{3}{2}}}$

Step 9.2.1.5

Cancel the common factor of $50$ and $4$ .

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Step 9.2.1.5.1

Factor $2$ out of $50$ .

$\frac{\frac{5 \sqrt{2}}{2} \cdot - 50}{{(- \frac{2 (25)}{4} + 25)}^{\frac{3}{2}}}$

Step 9.2.1.5.2

Cancel the common factors.

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Step 9.2.1.5.2.1

Factor $2$ out of $4$ .

$\frac{\frac{5 \sqrt{2}}{2} \cdot - 50}{{(- \frac{2 \cdot 25}{2 \cdot 2} + 25)}^{\frac{3}{2}}}$

Step 9.2.1.5.2.2

Cancel the common factor.

$\frac{\frac{5 \sqrt{2}}{2} \cdot - 50}{{(- \frac{2 \cdot 25}{2 \cdot 2} + 25)}^{\frac{3}{2}}}$

Step 9.2.1.5.2.3

Rewrite the expression.

$\frac{\frac{5 \sqrt{2}}{2} \cdot - 50}{{(- \frac{25}{2} + 25)}^{\frac{3}{2}}}$

Step 9.2.2

To write $25$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{\frac{5 \sqrt{2}}{2} \cdot - 50}{{(- \frac{25}{2} + 25 \cdot \frac{2}{2})}^{\frac{3}{2}}}$

Step 9.2.3

Combine $25$ and $\frac{2}{2}$ .

$\frac{\frac{5 \sqrt{2}}{2} \cdot - 50}{{(- \frac{25}{2} + \frac{25 \cdot 2}{2})}^{\frac{3}{2}}}$

Step 9.2.4

Combine the numerators over the common denominator.

$\frac{\frac{5 \sqrt{2}}{2} \cdot - 50}{{(\frac{- 25 + 25 \cdot 2}{2})}^{\frac{3}{2}}}$

Step 9.2.5

Simplify the numerator.

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Step 9.2.5.1

Multiply $25$ by $2$ .

$\frac{\frac{5 \sqrt{2}}{2} \cdot - 50}{{(\frac{- 25 + 50}{2})}^{\frac{3}{2}}}$

Step 9.2.5.2

Add $- 25$ and $50$ .

$\frac{\frac{5 \sqrt{2}}{2} \cdot - 50}{{(\frac{25}{2})}^{\frac{3}{2}}}$

Step 9.2.6

Apply the product rule to $\frac{25}{2}$ .

$\frac{\frac{5 \sqrt{2}}{2} \cdot - 50}{\frac{25^{\frac{3}{2}}}{2^{\frac{3}{2}}}}$

Step 9.2.7

Simplify the numerator.

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Step 9.2.7.1

Rewrite $25$ as $5^{2}$ .

$\frac{\frac{5 \sqrt{2}}{2} \cdot - 50}{\frac{{(5^{2})}^{\frac{3}{2}}}{2^{\frac{3}{2}}}}$

Step 9.2.7.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\frac{5 \sqrt{2}}{2} \cdot - 50}{\frac{5^{2 (\frac{3}{2})}}{2^{\frac{3}{2}}}}$

Step 9.2.7.3

Cancel the common factor of $2$ .

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Step 9.2.7.3.1

Cancel the common factor.

$\frac{\frac{5 \sqrt{2}}{2} \cdot - 50}{\frac{5^{2 (\frac{3}{2})}}{2^{\frac{3}{2}}}}$

Step 9.2.7.3.2

Rewrite the expression.

$\frac{\frac{5 \sqrt{2}}{2} \cdot - 50}{\frac{5^{3}}{2^{\frac{3}{2}}}}$

Step 9.2.7.4

Raise $5$ to the power of $3$ .

$\frac{\frac{5 \sqrt{2}}{2} \cdot - 50}{\frac{125}{2^{\frac{3}{2}}}}$

Step 9.3

Combine fractions.

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Step 9.3.1

Combine $\frac{5 \sqrt{2}}{2}$ and $- 50$ .

$\frac{\frac{5 \sqrt{2} \cdot - 50}{2}}{\frac{125}{2^{\frac{3}{2}}}}$

Step 9.3.2

Multiply $- 50$ by $5$ .

$\frac{\frac{- 250 \sqrt{2}}{2}}{\frac{125}{2^{\frac{3}{2}}}}$

Step 9.4

Reduce the expression by cancelling the common factors.

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Step 9.4.1

Reduce the expression $\frac{- 250 \sqrt{2}}{2}$ by cancelling the common factors.

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Step 9.4.1.1

Factor $2$ out of $- 250 \sqrt{2}$ .

$\frac{\frac{2 (- 125 \sqrt{2})}{2}}{\frac{125}{2^{\frac{3}{2}}}}$

Step 9.4.1.2

Factor $2$ out of $2$ .

$\frac{\frac{2 (- 125 \sqrt{2})}{2 (1)}}{\frac{125}{2^{\frac{3}{2}}}}$

Step 9.4.1.3

Cancel the common factor.

$\frac{\frac{2 (- 125 \sqrt{2})}{2 \cdot 1}}{\frac{125}{2^{\frac{3}{2}}}}$

Step 9.4.1.4

Rewrite the expression.

$\frac{\frac{- 125 \sqrt{2}}{1}}{\frac{125}{2^{\frac{3}{2}}}}$

Step 9.4.2

Divide $- 125 \sqrt{2}$ by $1$ .

$\frac{- 125 \sqrt{2}}{\frac{125}{2^{\frac{3}{2}}}}$

Step 9.5

Multiply the numerator by the reciprocal of the denominator.

$- 125 \sqrt{2} \frac{2^{\frac{3}{2}}}{125}$

Step 9.6

Cancel the common factor of $125$ .

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Step 9.6.1

Factor $125$ out of $- 125 \sqrt{2}$ .

$125 (- \sqrt{2}) \frac{2^{\frac{3}{2}}}{125}$

Step 9.6.2

Cancel the common factor.

$125 (- \sqrt{2}) \frac{2^{\frac{3}{2}}}{125}$

Step 9.6.3

Rewrite the expression.

$- \sqrt{2} \cdot 2^{\frac{3}{2}}$

Step 9.7

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$- (2^{\frac{3}{2}} \cdot 2^{\frac{1}{2}})$

Step 9.8

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 2^{\frac{3}{2} + \frac{1}{2}}$

Step 9.9

Simplify the expression.

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Step 9.9.1

Combine the numerators over the common denominator.

$- 2^{\frac{3 + 1}{2}}$

Step 9.9.2

Add $3$ and $1$ .

$- 2^{\frac{4}{2}}$

Step 9.10

Cancel the common factor of $4$ and $2$ .

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Step 9.10.1

Factor $2$ out of $4$ .

$- 2^{\frac{2 \cdot 2}{2}}$

Step 9.10.2

Cancel the common factors.

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Step 9.10.2.1

Factor $2$ out of $2$ .

$- 2^{\frac{2 \cdot 2}{2 (1)}}$

Step 9.10.2.2

Cancel the common factor.

$- 2^{\frac{2 \cdot 2}{2 \cdot 1}}$

Step 9.10.2.3

Rewrite the expression.

$- 2^{\frac{2}{1}}$

Step 9.10.2.4

Divide $2$ by $1$ .

$- 2^{2}$

Step 9.11

Simplify the expression.

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Step 9.11.1

Raise $2$ to the power of $2$ .

$- 1 \cdot 4$

Step 9.11.2

Multiply $- 1$ by $4$ .

$- 4$

Step 10

$t = \frac{5 \sqrt{2}}{2}$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$t = \frac{5 \sqrt{2}}{2}$ is a local maximum

Step 11

Find the y-value when $t = \frac{5 \sqrt{2}}{2}$ .

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Step 11.1

Replace the variable $t$ with $\frac{5 \sqrt{2}}{2}$ in the expression.

$f (\frac{5 \sqrt{2}}{2}) = (\frac{5 \sqrt{2}}{2}) \sqrt{25 - {(\frac{5 \sqrt{2}}{2})}^{2}}$

Step 11.2

Simplify the result.

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Step 11.2.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 11.2.1.1

Apply the product rule to $\frac{5 \sqrt{2}}{2}$ .

$f (\frac{5 \sqrt{2}}{2}) = \frac{5 \sqrt{2}}{2} \cdot \sqrt{25 - \frac{{(5 \sqrt{2})}^{2}}{2^{2}}}$

Step 11.2.1.2

Apply the product rule to $5 \sqrt{2}$ .

$f (\frac{5 \sqrt{2}}{2}) = \frac{5 \sqrt{2}}{2} \cdot \sqrt{25 - \frac{5^{2} {\sqrt{2}}^{2}}{2^{2}}}$

Step 11.2.2

Simplify the numerator.

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Step 11.2.2.1

Raise $5$ to the power of $2$ .

$f (\frac{5 \sqrt{2}}{2}) = \frac{5 \sqrt{2}}{2} \cdot \sqrt{25 - \frac{25 {\sqrt{2}}^{2}}{2^{2}}}$

Step 11.2.2.2

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 11.2.2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$f (\frac{5 \sqrt{2}}{2}) = \frac{5 \sqrt{2}}{2} \cdot \sqrt{25 - \frac{25 {(2^{\frac{1}{2}})}^{2}}{2^{2}}}$

Step 11.2.2.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (\frac{5 \sqrt{2}}{2}) = \frac{5 \sqrt{2}}{2} \cdot \sqrt{25 - \frac{25 \cdot 2^{\frac{1}{2} \cdot 2}}{2^{2}}}$

Step 11.2.2.2.3

Combine $\frac{1}{2}$ and $2$ .

$f (\frac{5 \sqrt{2}}{2}) = \frac{5 \sqrt{2}}{2} \cdot \sqrt{25 - \frac{25 \cdot 2^{\frac{2}{2}}}{2^{2}}}$

Step 11.2.2.2.4

Cancel the common factor of $2$ .

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Step 11.2.2.2.4.1

Cancel the common factor.

$f (\frac{5 \sqrt{2}}{2}) = \frac{5 \sqrt{2}}{2} \cdot \sqrt{25 - \frac{25 \cdot 2^{\frac{2}{2}}}{2^{2}}}$

Step 11.2.2.2.4.2

Rewrite the expression.

$f (\frac{5 \sqrt{2}}{2}) = \frac{5 \sqrt{2}}{2} \cdot \sqrt{25 - \frac{25 \cdot 2}{2^{2}}}$

Step 11.2.2.2.5

Evaluate the exponent.

$f (\frac{5 \sqrt{2}}{2}) = \frac{5 \sqrt{2}}{2} \cdot \sqrt{25 - \frac{25 \cdot 2}{2^{2}}}$

Step 11.2.3

Reduce the expression by cancelling the common factors.

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Step 11.2.3.1

Raise $2$ to the power of $2$ .

$f (\frac{5 \sqrt{2}}{2}) = \frac{5 \sqrt{2}}{2} \cdot \sqrt{25 - \frac{25 \cdot 2}{4}}$

Step 11.2.3.2

Multiply $25$ by $2$ .

$f (\frac{5 \sqrt{2}}{2}) = \frac{5 \sqrt{2}}{2} \cdot \sqrt{25 - \frac{50}{4}}$

Step 11.2.3.3

Cancel the common factor of $50$ and $4$ .

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Step 11.2.3.3.1

Factor $2$ out of $50$ .

$f (\frac{5 \sqrt{2}}{2}) = \frac{5 \sqrt{2}}{2} \cdot \sqrt{25 - \frac{2 (25)}{4}}$

Step 11.2.3.3.2

Cancel the common factors.

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Step 11.2.3.3.2.1

Factor $2$ out of $4$ .

$f (\frac{5 \sqrt{2}}{2}) = \frac{5 \sqrt{2}}{2} \cdot \sqrt{25 - \frac{2 \cdot 25}{2 \cdot 2}}$

Step 11.2.3.3.2.2

Cancel the common factor.

$f (\frac{5 \sqrt{2}}{2}) = \frac{5 \sqrt{2}}{2} \cdot \sqrt{25 - \frac{2 \cdot 25}{2 \cdot 2}}$

Step 11.2.3.3.2.3

Rewrite the expression.

$f (\frac{5 \sqrt{2}}{2}) = \frac{5 \sqrt{2}}{2} \cdot \sqrt{25 - \frac{25}{2}}$

Step 11.2.4

To write $25$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (\frac{5 \sqrt{2}}{2}) = \frac{5 \sqrt{2}}{2} \cdot \sqrt{25 \cdot \frac{2}{2} - \frac{25}{2}}$

Step 11.2.5

Combine $25$ and $\frac{2}{2}$ .

$f (\frac{5 \sqrt{2}}{2}) = \frac{5 \sqrt{2}}{2} \cdot \sqrt{\frac{25 \cdot 2}{2} - \frac{25}{2}}$

Step 11.2.6

Combine the numerators over the common denominator.

$f (\frac{5 \sqrt{2}}{2}) = \frac{5 \sqrt{2}}{2} \cdot \sqrt{\frac{25 \cdot 2 - 25}{2}}$

Step 11.2.7

Simplify the numerator.

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Step 11.2.7.1

Multiply $25$ by $2$ .

$f (\frac{5 \sqrt{2}}{2}) = \frac{5 \sqrt{2}}{2} \cdot \sqrt{\frac{50 - 25}{2}}$

Step 11.2.7.2

Subtract $25$ from $50$ .

$f (\frac{5 \sqrt{2}}{2}) = \frac{5 \sqrt{2}}{2} \cdot \sqrt{\frac{25}{2}}$

Step 11.2.8

Rewrite $\sqrt{\frac{25}{2}}$ as $\frac{\sqrt{25}}{\sqrt{2}}$ .

$f (\frac{5 \sqrt{2}}{2}) = \frac{5 \sqrt{2}}{2} \cdot \frac{\sqrt{25}}{\sqrt{2}}$

Step 11.2.9

Simplify the numerator.

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Step 11.2.9.1

Rewrite $25$ as $5^{2}$ .

$f (\frac{5 \sqrt{2}}{2}) = \frac{5 \sqrt{2}}{2} \cdot \frac{\sqrt{5^{2}}}{\sqrt{2}}$

Step 11.2.9.2

Pull terms out from under the radical, assuming positive real numbers.

$f (\frac{5 \sqrt{2}}{2}) = \frac{5 \sqrt{2}}{2} \cdot \frac{5}{\sqrt{2}}$

Step 11.2.10

Simplify terms.

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Step 11.2.10.1

Cancel the common factor of $\sqrt{2}$ .

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Step 11.2.10.1.1

Factor $\sqrt{2}$ out of $5 \sqrt{2}$ .

$f (\frac{5 \sqrt{2}}{2}) = \frac{\sqrt{2} \cdot 5}{2} \cdot \frac{5}{\sqrt{2}}$

Step 11.2.10.1.2

Cancel the common factor.

$f (\frac{5 \sqrt{2}}{2}) = \frac{\sqrt{2} \cdot 5}{2} \cdot \frac{5}{\sqrt{2}}$

Step 11.2.10.1.3

Rewrite the expression.

$f (\frac{5 \sqrt{2}}{2}) = \frac{5}{2} \cdot 5$

Step 11.2.10.2

Combine $\frac{5}{2}$ and $5$ .

$f (\frac{5 \sqrt{2}}{2}) = \frac{5 \cdot 5}{2}$

Step 11.2.10.3

Multiply $5$ by $5$ .

$f (\frac{5 \sqrt{2}}{2}) = \frac{25}{2}$

Step 11.2.11

The final answer is $\frac{25}{2}$ .

$y = \frac{25}{2}$

Step 12

Evaluate the second derivative at $t = - \frac{5 \sqrt{2}}{2}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{(- \frac{5 \sqrt{2}}{2}) (2 {(- \frac{5 \sqrt{2}}{2})}^{2} - 75)}{{(- {(- \frac{5 \sqrt{2}}{2})}^{2} + 25)}^{\frac{3}{2}}}$

Step 13

Evaluate the second derivative.

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Step 13.1

Simplify the numerator.

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Step 13.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 13.1.1.1

Apply the product rule to $- \frac{5 \sqrt{2}}{2}$ .

$\frac{- \frac{5 \sqrt{2}}{2} (2 ({(- 1)}^{2} {(\frac{5 \sqrt{2}}{2})}^{2}) - 75)}{{(- {(- \frac{5 \sqrt{2}}{2})}^{2} + 25)}^{\frac{3}{2}}}$

Step 13.1.1.2

Apply the product rule to $\frac{5 \sqrt{2}}{2}$ .

$\frac{- \frac{5 \sqrt{2}}{2} (2 ({(- 1)}^{2} \frac{{(5 \sqrt{2})}^{2}}{2^{2}}) - 75)}{{(- {(- \frac{5 \sqrt{2}}{2})}^{2} + 25)}^{\frac{3}{2}}}$

Step 13.1.1.3

Apply the product rule to $5 \sqrt{2}$ .

$\frac{- \frac{5 \sqrt{2}}{2} (2 ({(- 1)}^{2} \frac{5^{2} {\sqrt{2}}^{2}}{2^{2}}) - 75)}{{(- {(- \frac{5 \sqrt{2}}{2})}^{2} + 25)}^{\frac{3}{2}}}$

Step 13.1.2

Raise $- 1$ to the power of $2$ .

$\frac{- \frac{5 \sqrt{2}}{2} (2 (1 \frac{5^{2} {\sqrt{2}}^{2}}{2^{2}}) - 75)}{{(- {(- \frac{5 \sqrt{2}}{2})}^{2} + 25)}^{\frac{3}{2}}}$

Step 13.1.3

Multiply $\frac{5^{2} {\sqrt{2}}^{2}}{2^{2}}$ by $1$ .

$\frac{- \frac{5 \sqrt{2}}{2} (2 \frac{5^{2} {\sqrt{2}}^{2}}{2^{2}} - 75)}{{(- {(- \frac{5 \sqrt{2}}{2})}^{2} + 25)}^{\frac{3}{2}}}$

Step 13.1.4

Simplify the numerator.

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Step 13.1.4.1

Raise $5$ to the power of $2$ .

$\frac{- \frac{5 \sqrt{2}}{2} (2 \frac{25 {\sqrt{2}}^{2}}{2^{2}} - 75)}{{(- {(- \frac{5 \sqrt{2}}{2})}^{2} + 25)}^{\frac{3}{2}}}$

Step 13.1.4.2

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 13.1.4.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\frac{- \frac{5 \sqrt{2}}{2} (2 \frac{25 {(2^{\frac{1}{2}})}^{2}}{2^{2}} - 75)}{{(- {(- \frac{5 \sqrt{2}}{2})}^{2} + 25)}^{\frac{3}{2}}}$

Step 13.1.4.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{- \frac{5 \sqrt{2}}{2} (2 \frac{25 \cdot 2^{\frac{1}{2} \cdot 2}}{2^{2}} - 75)}{{(- {(- \frac{5 \sqrt{2}}{2})}^{2} + 25)}^{\frac{3}{2}}}$

Step 13.1.4.2.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{- \frac{5 \sqrt{2}}{2} (2 \frac{25 \cdot 2^{\frac{2}{2}}}{2^{2}} - 75)}{{(- {(- \frac{5 \sqrt{2}}{2})}^{2} + 25)}^{\frac{3}{2}}}$

Step 13.1.4.2.4

Cancel the common factor of $2$ .

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Step 13.1.4.2.4.1

Cancel the common factor.

$\frac{- \frac{5 \sqrt{2}}{2} (2 \frac{25 \cdot 2^{\frac{2}{2}}}{2^{2}} - 75)}{{(- {(- \frac{5 \sqrt{2}}{2})}^{2} + 25)}^{\frac{3}{2}}}$

Step 13.1.4.2.4.2

Rewrite the expression.

$\frac{- \frac{5 \sqrt{2}}{2} (2 \frac{25 \cdot 2^{1}}{2^{2}} - 75)}{{(- {(- \frac{5 \sqrt{2}}{2})}^{2} + 25)}^{\frac{3}{2}}}$

Step 13.1.4.2.5

Evaluate the exponent.

$\frac{- \frac{5 \sqrt{2}}{2} (2 \frac{25 \cdot 2}{2^{2}} - 75)}{{(- {(- \frac{5 \sqrt{2}}{2})}^{2} + 25)}^{\frac{3}{2}}}$

Step 13.1.5

Raise $2$ to the power of $2$ .

$\frac{- \frac{5 \sqrt{2}}{2} (2 \frac{25 \cdot 2}{4} - 75)}{{(- {(- \frac{5 \sqrt{2}}{2})}^{2} + 25)}^{\frac{3}{2}}}$

Step 13.1.6

Multiply $25$ by $2$ .

$\frac{- \frac{5 \sqrt{2}}{2} (2 (\frac{50}{4}) - 75)}{{(- {(- \frac{5 \sqrt{2}}{2})}^{2} + 25)}^{\frac{3}{2}}}$

Step 13.1.7

Cancel the common factor of $2$ .

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Step 13.1.7.1

Factor $2$ out of $4$ .

$\frac{- \frac{5 \sqrt{2}}{2} (2 \frac{50}{2 (2)} - 75)}{{(- {(- \frac{5 \sqrt{2}}{2})}^{2} + 25)}^{\frac{3}{2}}}$

Step 13.1.7.2

Cancel the common factor.

$\frac{- \frac{5 \sqrt{2}}{2} (2 \frac{50}{2 \cdot 2} - 75)}{{(- {(- \frac{5 \sqrt{2}}{2})}^{2} + 25)}^{\frac{3}{2}}}$

Step 13.1.7.3

Rewrite the expression.

$\frac{- \frac{5 \sqrt{2}}{2} (\frac{50}{2} - 75)}{{(- {(- \frac{5 \sqrt{2}}{2})}^{2} + 25)}^{\frac{3}{2}}}$

Step 13.1.8

Divide $50$ by $2$ .

$\frac{- \frac{5 \sqrt{2}}{2} (25 - 75)}{{(- {(- \frac{5 \sqrt{2}}{2})}^{2} + 25)}^{\frac{3}{2}}}$

Step 13.1.9

Subtract $75$ from $25$ .

$\frac{- \frac{5 \sqrt{2}}{2} \cdot - 50}{{(- {(- \frac{5 \sqrt{2}}{2})}^{2} + 25)}^{\frac{3}{2}}}$

Step 13.1.10

Combine exponents.

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Step 13.1.10.1

Multiply $- 50$ by $- 1$ .

$\frac{50 \frac{5 \sqrt{2}}{2}}{{(- {(- \frac{5 \sqrt{2}}{2})}^{2} + 25)}^{\frac{3}{2}}}$

Step 13.1.10.2

Combine $50$ and $\frac{5 \sqrt{2}}{2}$ .

$\frac{\frac{50 (5 \sqrt{2})}{2}}{{(- {(- \frac{5 \sqrt{2}}{2})}^{2} + 25)}^{\frac{3}{2}}}$

Step 13.1.10.3

Multiply $5$ by $50$ .

$\frac{\frac{250 \sqrt{2}}{2}}{{(- {(- \frac{5 \sqrt{2}}{2})}^{2} + 25)}^{\frac{3}{2}}}$

Step 13.1.11

Cancel the common factor of $250$ and $2$ .

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Step 13.1.11.1

Factor $2$ out of $250 \sqrt{2}$ .

$\frac{\frac{2 (125 \sqrt{2})}{2}}{{(- {(- \frac{5 \sqrt{2}}{2})}^{2} + 25)}^{\frac{3}{2}}}$

Step 13.1.11.2

Cancel the common factors.

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Step 13.1.11.2.1

Factor $2$ out of $2$ .

$\frac{\frac{2 (125 \sqrt{2})}{2 (1)}}{{(- {(- \frac{5 \sqrt{2}}{2})}^{2} + 25)}^{\frac{3}{2}}}$

Step 13.1.11.2.2

Cancel the common factor.

$\frac{\frac{2 (125 \sqrt{2})}{2 \cdot 1}}{{(- {(- \frac{5 \sqrt{2}}{2})}^{2} + 25)}^{\frac{3}{2}}}$

Step 13.1.11.2.3

Rewrite the expression.

$\frac{\frac{125 \sqrt{2}}{1}}{{(- {(- \frac{5 \sqrt{2}}{2})}^{2} + 25)}^{\frac{3}{2}}}$

Step 13.1.11.2.4

Divide $125 \sqrt{2}$ by $1$ .

$\frac{125 \sqrt{2}}{{(- {(- \frac{5 \sqrt{2}}{2})}^{2} + 25)}^{\frac{3}{2}}}$

Step 13.2

Simplify the denominator.

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Step 13.2.1

Simplify each term.

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Step 13.2.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 13.2.1.1.1

Apply the product rule to $- \frac{5 \sqrt{2}}{2}$ .

$\frac{125 \sqrt{2}}{{(- ({(- 1)}^{2} {(\frac{5 \sqrt{2}}{2})}^{2}) + 25)}^{\frac{3}{2}}}$

Step 13.2.1.1.2

Apply the product rule to $\frac{5 \sqrt{2}}{2}$ .

$\frac{125 \sqrt{2}}{{(- ({(- 1)}^{2} \frac{{(5 \sqrt{2})}^{2}}{2^{2}}) + 25)}^{\frac{3}{2}}}$

Step 13.2.1.1.3

Apply the product rule to $5 \sqrt{2}$ .

$\frac{125 \sqrt{2}}{{(- ({(- 1)}^{2} \frac{5^{2} {\sqrt{2}}^{2}}{2^{2}}) + 25)}^{\frac{3}{2}}}$

Step 13.2.1.2

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

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Step 13.2.1.2.1

Move ${(- 1)}^{2}$ .

$\frac{125 \sqrt{2}}{{({(- 1)}^{2} \cdot - 1 \frac{5^{2} {\sqrt{2}}^{2}}{2^{2}} + 25)}^{\frac{3}{2}}}$

Step 13.2.1.2.2

Multiply ${(- 1)}^{2}$ by $- 1$ .

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Step 13.2.1.2.2.1

Raise $- 1$ to the power of $1$ .

$\frac{125 \sqrt{2}}{{({(- 1)}^{2} \cdot {(- 1)}^{1} \frac{5^{2} {\sqrt{2}}^{2}}{2^{2}} + 25)}^{\frac{3}{2}}}$

Step 13.2.1.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{125 \sqrt{2}}{{({(- 1)}^{2 + 1} \frac{5^{2} {\sqrt{2}}^{2}}{2^{2}} + 25)}^{\frac{3}{2}}}$

Step 13.2.1.2.3

Add $2$ and $1$ .

$\frac{125 \sqrt{2}}{{({(- 1)}^{3} \frac{5^{2} {\sqrt{2}}^{2}}{2^{2}} + 25)}^{\frac{3}{2}}}$

Step 13.2.1.3

Raise $- 1$ to the power of $3$ .

$\frac{125 \sqrt{2}}{{(- \frac{5^{2} {\sqrt{2}}^{2}}{2^{2}} + 25)}^{\frac{3}{2}}}$

Step 13.2.1.4

Simplify the numerator.

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Step 13.2.1.4.1

Raise $5$ to the power of $2$ .

$\frac{125 \sqrt{2}}{{(- \frac{25 {\sqrt{2}}^{2}}{2^{2}} + 25)}^{\frac{3}{2}}}$

Step 13.2.1.4.2

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 13.2.1.4.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\frac{125 \sqrt{2}}{{(- \frac{25 {(2^{\frac{1}{2}})}^{2}}{2^{2}} + 25)}^{\frac{3}{2}}}$

Step 13.2.1.4.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{125 \sqrt{2}}{{(- \frac{25 \cdot 2^{\frac{1}{2} \cdot 2}}{2^{2}} + 25)}^{\frac{3}{2}}}$

Step 13.2.1.4.2.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{125 \sqrt{2}}{{(- \frac{25 \cdot 2^{\frac{2}{2}}}{2^{2}} + 25)}^{\frac{3}{2}}}$

Step 13.2.1.4.2.4

Cancel the common factor of $2$ .

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Step 13.2.1.4.2.4.1

Cancel the common factor.

$\frac{125 \sqrt{2}}{{(- \frac{25 \cdot 2^{\frac{2}{2}}}{2^{2}} + 25)}^{\frac{3}{2}}}$

Step 13.2.1.4.2.4.2

Rewrite the expression.

$\frac{125 \sqrt{2}}{{(- \frac{25 \cdot 2^{1}}{2^{2}} + 25)}^{\frac{3}{2}}}$

Step 13.2.1.4.2.5

Evaluate the exponent.

$\frac{125 \sqrt{2}}{{(- \frac{25 \cdot 2}{2^{2}} + 25)}^{\frac{3}{2}}}$

Step 13.2.1.5

Raise $2$ to the power of $2$ .

$\frac{125 \sqrt{2}}{{(- \frac{25 \cdot 2}{4} + 25)}^{\frac{3}{2}}}$

Step 13.2.1.6

Multiply $25$ by $2$ .

$\frac{125 \sqrt{2}}{{(- \frac{50}{4} + 25)}^{\frac{3}{2}}}$

Step 13.2.1.7

Cancel the common factor of $50$ and $4$ .

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Step 13.2.1.7.1

Factor $2$ out of $50$ .

$\frac{125 \sqrt{2}}{{(- \frac{2 (25)}{4} + 25)}^{\frac{3}{2}}}$

Step 13.2.1.7.2

Cancel the common factors.

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Step 13.2.1.7.2.1

Factor $2$ out of $4$ .

$\frac{125 \sqrt{2}}{{(- \frac{2 \cdot 25}{2 \cdot 2} + 25)}^{\frac{3}{2}}}$

Step 13.2.1.7.2.2

Cancel the common factor.

$\frac{125 \sqrt{2}}{{(- \frac{2 \cdot 25}{2 \cdot 2} + 25)}^{\frac{3}{2}}}$

Step 13.2.1.7.2.3

Rewrite the expression.

$\frac{125 \sqrt{2}}{{(- \frac{25}{2} + 25)}^{\frac{3}{2}}}$

Step 13.2.2

To write $25$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{125 \sqrt{2}}{{(- \frac{25}{2} + 25 \cdot \frac{2}{2})}^{\frac{3}{2}}}$

Step 13.2.3

Combine $25$ and $\frac{2}{2}$ .

$\frac{125 \sqrt{2}}{{(- \frac{25}{2} + \frac{25 \cdot 2}{2})}^{\frac{3}{2}}}$

Step 13.2.4

Combine the numerators over the common denominator.

$\frac{125 \sqrt{2}}{{(\frac{- 25 + 25 \cdot 2}{2})}^{\frac{3}{2}}}$

Step 13.2.5

Simplify the numerator.

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Step 13.2.5.1

Multiply $25$ by $2$ .

$\frac{125 \sqrt{2}}{{(\frac{- 25 + 50}{2})}^{\frac{3}{2}}}$

Step 13.2.5.2

Add $- 25$ and $50$ .

$\frac{125 \sqrt{2}}{{(\frac{25}{2})}^{\frac{3}{2}}}$

Step 13.2.6

Apply the product rule to $\frac{25}{2}$ .

$\frac{125 \sqrt{2}}{\frac{25^{\frac{3}{2}}}{2^{\frac{3}{2}}}}$

Step 13.2.7

Simplify the numerator.

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Step 13.2.7.1

Rewrite $25$ as $5^{2}$ .

$\frac{125 \sqrt{2}}{\frac{{(5^{2})}^{\frac{3}{2}}}{2^{\frac{3}{2}}}}$

Step 13.2.7.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{125 \sqrt{2}}{\frac{5^{2 (\frac{3}{2})}}{2^{\frac{3}{2}}}}$

Step 13.2.7.3

Cancel the common factor of $2$ .

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Step 13.2.7.3.1

Cancel the common factor.

$\frac{125 \sqrt{2}}{\frac{5^{2 (\frac{3}{2})}}{2^{\frac{3}{2}}}}$

Step 13.2.7.3.2

Rewrite the expression.

$\frac{125 \sqrt{2}}{\frac{5^{3}}{2^{\frac{3}{2}}}}$

Step 13.2.7.4

Raise $5$ to the power of $3$ .

$\frac{125 \sqrt{2}}{\frac{125}{2^{\frac{3}{2}}}}$

Step 13.3

Multiply the numerator by the reciprocal of the denominator.

$125 \sqrt{2} \frac{2^{\frac{3}{2}}}{125}$

Step 13.4

Cancel the common factor of $125$ .

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Step 13.4.1

Factor $125$ out of $125 \sqrt{2}$ .

$125 (\sqrt{2}) \frac{2^{\frac{3}{2}}}{125}$

Step 13.4.2

Cancel the common factor.

$125 \sqrt{2} \frac{2^{\frac{3}{2}}}{125}$

Step 13.4.3

Rewrite the expression.

$\sqrt{2} \cdot 2^{\frac{3}{2}}$

Step 13.5

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$2^{\frac{1}{2}} \cdot 2^{\frac{3}{2}}$

Step 13.6

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2^{\frac{1}{2} + \frac{3}{2}}$

Step 13.7

Simplify the expression.

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Step 13.7.1

Combine the numerators over the common denominator.

$2^{\frac{1 + 3}{2}}$

Step 13.7.2

Add $1$ and $3$ .

$2^{\frac{4}{2}}$

Step 13.8

Cancel the common factor of $4$ and $2$ .

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Step 13.8.1

Factor $2$ out of $4$ .

$2^{\frac{2 \cdot 2}{2}}$

Step 13.8.2

Cancel the common factors.

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Step 13.8.2.1

Factor $2$ out of $2$ .

$2^{\frac{2 \cdot 2}{2 (1)}}$

Step 13.8.2.2

Cancel the common factor.

$2^{\frac{2 \cdot 2}{2 \cdot 1}}$

Step 13.8.2.3

Rewrite the expression.

$2^{\frac{2}{1}}$

Step 13.8.2.4

Divide $2$ by $1$ .

$2^{2}$

Step 13.9

Raise $2$ to the power of $2$ .

$4$

Step 14

$t = - \frac{5 \sqrt{2}}{2}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$t = - \frac{5 \sqrt{2}}{2}$ is a local minimum

Step 15

Find the y-value when $t = - \frac{5 \sqrt{2}}{2}$ .

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Step 15.1

Replace the variable $t$ with $- \frac{5 \sqrt{2}}{2}$ in the expression.

$f (- \frac{5 \sqrt{2}}{2}) = (- \frac{5 \sqrt{2}}{2}) \sqrt{25 - {(- \frac{5 \sqrt{2}}{2})}^{2}}$

Step 15.2

Simplify the result.

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Step 15.2.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 15.2.1.1

Apply the product rule to $- \frac{5 \sqrt{2}}{2}$ .

$f (- \frac{5 \sqrt{2}}{2}) = - \frac{5 \sqrt{2}}{2} \cdot \sqrt{25 - ({(- 1)}^{2} {(\frac{5 \sqrt{2}}{2})}^{2})}$

Step 15.2.1.2

Apply the product rule to $\frac{5 \sqrt{2}}{2}$ .

$f (- \frac{5 \sqrt{2}}{2}) = - \frac{5 \sqrt{2}}{2} \cdot \sqrt{25 - ({(- 1)}^{2} (\frac{{(5 \sqrt{2})}^{2}}{2^{2}}))}$

Step 15.2.1.3

Apply the product rule to $5 \sqrt{2}$ .

$f (- \frac{5 \sqrt{2}}{2}) = - \frac{5 \sqrt{2}}{2} \cdot \sqrt{25 - ({(- 1)}^{2} (\frac{5^{2} {\sqrt{2}}^{2}}{2^{2}}))}$

Step 15.2.2

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

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Step 15.2.2.1

Move ${(- 1)}^{2}$ .

$f (- \frac{5 \sqrt{2}}{2}) = - \frac{5 \sqrt{2}}{2} \cdot \sqrt{25 + {(- 1)}^{2} \cdot (- 1 \frac{5^{2} {\sqrt{2}}^{2}}{2^{2}})}$

Step 15.2.2.2

Multiply ${(- 1)}^{2}$ by $- 1$ .

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Step 15.2.2.2.1

Raise $- 1$ to the power of $1$ .

$f (- \frac{5 \sqrt{2}}{2}) = - \frac{5 \sqrt{2}}{2} \cdot \sqrt{25 + {(- 1)}^{2} \cdot ((- 1) (\frac{5^{2} {\sqrt{2}}^{2}}{2^{2}}))}$

Step 15.2.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (- \frac{5 \sqrt{2}}{2}) = - \frac{5 \sqrt{2}}{2} \cdot \sqrt{25 + {(- 1)}^{2 + 1} (\frac{5^{2} {\sqrt{2}}^{2}}{2^{2}})}$

Step 15.2.2.3

Add $2$ and $1$ .

$f (- \frac{5 \sqrt{2}}{2}) = - \frac{5 \sqrt{2}}{2} \cdot \sqrt{25 + {(- 1)}^{3} (\frac{5^{2} {\sqrt{2}}^{2}}{2^{2}})}$

Step 15.2.3

Raise $- 1$ to the power of $3$ .

$f (- \frac{5 \sqrt{2}}{2}) = - \frac{5 \sqrt{2}}{2} \cdot \sqrt{25 - \frac{5^{2} {\sqrt{2}}^{2}}{2^{2}}}$

Step 15.2.4

Simplify the numerator.

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Step 15.2.4.1

Raise $5$ to the power of $2$ .

$f (- \frac{5 \sqrt{2}}{2}) = - \frac{5 \sqrt{2}}{2} \cdot \sqrt{25 - \frac{25 {\sqrt{2}}^{2}}{2^{2}}}$

Step 15.2.4.2

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 15.2.4.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$f (- \frac{5 \sqrt{2}}{2}) = - \frac{5 \sqrt{2}}{2} \cdot \sqrt{25 - \frac{25 {(2^{\frac{1}{2}})}^{2}}{2^{2}}}$

Step 15.2.4.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (- \frac{5 \sqrt{2}}{2}) = - \frac{5 \sqrt{2}}{2} \cdot \sqrt{25 - \frac{25 \cdot 2^{\frac{1}{2} \cdot 2}}{2^{2}}}$

Step 15.2.4.2.3

Combine $\frac{1}{2}$ and $2$ .

$f (- \frac{5 \sqrt{2}}{2}) = - \frac{5 \sqrt{2}}{2} \cdot \sqrt{25 - \frac{25 \cdot 2^{\frac{2}{2}}}{2^{2}}}$

Step 15.2.4.2.4

Cancel the common factor of $2$ .

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Step 15.2.4.2.4.1

Cancel the common factor.

$f (- \frac{5 \sqrt{2}}{2}) = - \frac{5 \sqrt{2}}{2} \cdot \sqrt{25 - \frac{25 \cdot 2^{\frac{2}{2}}}{2^{2}}}$

Step 15.2.4.2.4.2

Rewrite the expression.

$f (- \frac{5 \sqrt{2}}{2}) = - \frac{5 \sqrt{2}}{2} \cdot \sqrt{25 - \frac{25 \cdot 2}{2^{2}}}$

Step 15.2.4.2.5

Evaluate the exponent.

$f (- \frac{5 \sqrt{2}}{2}) = - \frac{5 \sqrt{2}}{2} \cdot \sqrt{25 - \frac{25 \cdot 2}{2^{2}}}$

Step 15.2.5

Reduce the expression by cancelling the common factors.

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Step 15.2.5.1

Raise $2$ to the power of $2$ .

$f (- \frac{5 \sqrt{2}}{2}) = - \frac{5 \sqrt{2}}{2} \cdot \sqrt{25 - \frac{25 \cdot 2}{4}}$

Step 15.2.5.2

Multiply $25$ by $2$ .

$f (- \frac{5 \sqrt{2}}{2}) = - \frac{5 \sqrt{2}}{2} \cdot \sqrt{25 - \frac{50}{4}}$

Step 15.2.5.3

Cancel the common factor of $50$ and $4$ .

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Step 15.2.5.3.1

Factor $2$ out of $50$ .

$f (- \frac{5 \sqrt{2}}{2}) = - \frac{5 \sqrt{2}}{2} \cdot \sqrt{25 - \frac{2 (25)}{4}}$

Step 15.2.5.3.2

Cancel the common factors.

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Step 15.2.5.3.2.1

Factor $2$ out of $4$ .

$f (- \frac{5 \sqrt{2}}{2}) = - \frac{5 \sqrt{2}}{2} \cdot \sqrt{25 - \frac{2 \cdot 25}{2 \cdot 2}}$

Step 15.2.5.3.2.2

Cancel the common factor.

$f (- \frac{5 \sqrt{2}}{2}) = - \frac{5 \sqrt{2}}{2} \cdot \sqrt{25 - \frac{2 \cdot 25}{2 \cdot 2}}$

Step 15.2.5.3.2.3

Rewrite the expression.

$f (- \frac{5 \sqrt{2}}{2}) = - \frac{5 \sqrt{2}}{2} \cdot \sqrt{25 - \frac{25}{2}}$

Step 15.2.6

To write $25$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (- \frac{5 \sqrt{2}}{2}) = - \frac{5 \sqrt{2}}{2} \cdot \sqrt{25 \cdot \frac{2}{2} - \frac{25}{2}}$

Step 15.2.7

Combine $25$ and $\frac{2}{2}$ .

$f (- \frac{5 \sqrt{2}}{2}) = - \frac{5 \sqrt{2}}{2} \cdot \sqrt{\frac{25 \cdot 2}{2} - \frac{25}{2}}$

Step 15.2.8

Combine the numerators over the common denominator.

$f (- \frac{5 \sqrt{2}}{2}) = - \frac{5 \sqrt{2}}{2} \cdot \sqrt{\frac{25 \cdot 2 - 25}{2}}$

Step 15.2.9

Simplify the numerator.

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Step 15.2.9.1

Multiply $25$ by $2$ .

$f (- \frac{5 \sqrt{2}}{2}) = - \frac{5 \sqrt{2}}{2} \cdot \sqrt{\frac{50 - 25}{2}}$

Step 15.2.9.2

Subtract $25$ from $50$ .

$f (- \frac{5 \sqrt{2}}{2}) = - \frac{5 \sqrt{2}}{2} \cdot \sqrt{\frac{25}{2}}$

Step 15.2.10

Rewrite $\sqrt{\frac{25}{2}}$ as $\frac{\sqrt{25}}{\sqrt{2}}$ .

$f (- \frac{5 \sqrt{2}}{2}) = - \frac{5 \sqrt{2}}{2} \cdot \frac{\sqrt{25}}{\sqrt{2}}$

Step 15.2.11

Simplify the numerator.

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Step 15.2.11.1

Rewrite $25$ as $5^{2}$ .

$f (- \frac{5 \sqrt{2}}{2}) = - \frac{5 \sqrt{2}}{2} \cdot \frac{\sqrt{5^{2}}}{\sqrt{2}}$

Step 15.2.11.2

Pull terms out from under the radical, assuming positive real numbers.

$f (- \frac{5 \sqrt{2}}{2}) = - \frac{5 \sqrt{2}}{2} \cdot \frac{5}{\sqrt{2}}$

Step 15.2.12

Simplify terms.

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Step 15.2.12.1

Cancel the common factor of $\sqrt{2}$ .

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Step 15.2.12.1.1

Move the leading negative in $- \frac{5 \sqrt{2}}{2}$ into the numerator.

$f (- \frac{5 \sqrt{2}}{2}) = \frac{- 5 \sqrt{2}}{2} \cdot \frac{5}{\sqrt{2}}$

Step 15.2.12.1.2

Factor $\sqrt{2}$ out of $- 5 \sqrt{2}$ .

$f (- \frac{5 \sqrt{2}}{2}) = \frac{\sqrt{2} \cdot - 5}{2} \cdot \frac{5}{\sqrt{2}}$

Step 15.2.12.1.3

Cancel the common factor.

$f (- \frac{5 \sqrt{2}}{2}) = \frac{\sqrt{2} \cdot - 5}{2} \cdot \frac{5}{\sqrt{2}}$

Step 15.2.12.1.4

Rewrite the expression.

$f (- \frac{5 \sqrt{2}}{2}) = \frac{- 5}{2} \cdot 5$

Step 15.2.12.2

Combine $\frac{- 5}{2}$ and $5$ .

$f (- \frac{5 \sqrt{2}}{2}) = \frac{- 5 \cdot 5}{2}$

Step 15.2.12.3

Simplify the expression.

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Step 15.2.12.3.1

Multiply $- 5$ by $5$ .

$f (- \frac{5 \sqrt{2}}{2}) = \frac{- 25}{2}$

Step 15.2.12.3.2

Move the negative in front of the fraction.

$f (- \frac{5 \sqrt{2}}{2}) = - \frac{25}{2}$

Step 15.2.13

The final answer is $- \frac{25}{2}$ .

$y = - \frac{25}{2}$

Step 16

Evaluate the second derivative at $t = - 5$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{(- 5) (2 {(- 5)}^{2} - 75)}{{(- {(- 5)}^{2} + 25)}^{\frac{3}{2}}}$

Step 17

Evaluate the second derivative.

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Step 17.1

Simplify each term.

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Step 17.1.1

Raise $- 5$ to the power of $2$ .

$\frac{(- 5) (2 {(- 5)}^{2} - 75)}{{(- 1 \cdot 25 + 25)}^{\frac{3}{2}}}$

Step 17.1.2

Multiply $- 1$ by $25$ .

$\frac{(- 5) (2 {(- 5)}^{2} - 75)}{{(- 25 + 25)}^{\frac{3}{2}}}$

Step 17.2

Reduce the expression by cancelling the common factors.

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Step 17.2.1

Add $- 25$ and $25$ .

$\frac{(- 5) (2 {(- 5)}^{2} - 75)}{0^{\frac{3}{2}}}$

Step 17.2.2

Simplify the expression.

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Step 17.2.2.1

Rewrite $0$ as $0^{2}$ .

$\frac{(- 5) (2 {(- 5)}^{2} - 75)}{{(0^{2})}^{\frac{3}{2}}}$

Step 17.2.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{(- 5) (2 {(- 5)}^{2} - 75)}{0^{2 (\frac{3}{2})}}$

Step 17.2.3

Cancel the common factor of $2$ .

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Step 17.2.3.1

Cancel the common factor.

$\frac{(- 5) (2 {(- 5)}^{2} - 75)}{0^{2 (\frac{3}{2})}}$

Step 17.2.3.2

Rewrite the expression.

$\frac{(- 5) (2 {(- 5)}^{2} - 75)}{0^{3}}$

Step 17.2.4

Raising $0$ to any positive power yields $0$ .

$\frac{(- 5) (2 {(- 5)}^{2} - 75)}{0}$

Step 17.2.5

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{(- 5) (2 {(- 5)}^{2} - 75)}{0}$

Step 17.3

The expression contains a division by $0$ . The expression is undefined.

Undefined

Step 18

Since the first derivative test failed, there are no local extrema.

No Local Extrema

Step 19