Calculus Examples

$f (x) = 4 x \sqrt{100 - x^{2}}$

Step 1

Find the first derivative of the function.

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Step 1.1

Differentiate using the Constant Multiple Rule.

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Step 1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{100 - x^{2}}$ as ${(100 - x^{2})}^{\frac{1}{2}}$ .

$\frac{d}{d x} [4 x {(100 - x^{2})}^{\frac{1}{2}}]$

Step 1.1.2

Since $4$ is constant with respect to $x$ , the derivative of $4 x {(100 - x^{2})}^{\frac{1}{2}}$ with respect to $x$ is $4 \frac{d}{d x} [x {(100 - x^{2})}^{\frac{1}{2}}]$ .

$4 \frac{d}{d x} [x {(100 - x^{2})}^{\frac{1}{2}}]$

Step 1.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = {(100 - x^{2})}^{\frac{1}{2}}$ .

$4 (x \frac{d}{d x} [{(100 - x^{2})}^{\frac{1}{2}}] + {(100 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 1.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = 100 - x^{2}$ .

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Step 1.3.1

To apply the Chain Rule, set $u$ as $100 - x^{2}$ .

$4 (x (\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [100 - x^{2}]) + {(100 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 1.3.2

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

$4 (x (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [100 - x^{2}]) + {(100 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 1.3.3

Replace all occurrences of $u$ with $100 - x^{2}$ .

$4 (x (\frac{1}{2} {(100 - x^{2})}^{\frac{1}{2} - 1} \frac{d}{d x} [100 - x^{2}]) + {(100 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 1.4

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

$4 (x (\frac{1}{2} {(100 - x^{2})}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [100 - x^{2}]) + {(100 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 1.5

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

$4 (x (\frac{1}{2} {(100 - x^{2})}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [100 - x^{2}]) + {(100 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 1.6

Combine the numerators over the common denominator.

$4 (x (\frac{1}{2} {(100 - x^{2})}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [100 - x^{2}]) + {(100 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 1.7

Simplify the numerator.

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Step 1.7.1

Multiply $- 1$ by $2$ .

$4 (x (\frac{1}{2} {(100 - x^{2})}^{\frac{1 - 2}{2}} \frac{d}{d x} [100 - x^{2}]) + {(100 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 1.7.2

Subtract $2$ from $1$ .

$4 (x (\frac{1}{2} {(100 - x^{2})}^{\frac{- 1}{2}} \frac{d}{d x} [100 - x^{2}]) + {(100 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 1.8

Combine fractions.

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Step 1.8.1

Move the negative in front of the fraction.

$4 (x (\frac{1}{2} {(100 - x^{2})}^{- \frac{1}{2}} \frac{d}{d x} [100 - x^{2}]) + {(100 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 1.8.2

Combine $\frac{1}{2}$ and ${(100 - x^{2})}^{- \frac{1}{2}}$ .

$4 (x (\frac{{(100 - x^{2})}^{- \frac{1}{2}}}{2} \frac{d}{d x} [100 - x^{2}]) + {(100 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 1.8.3

Move ${(100 - x^{2})}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$4 (x (\frac{1}{2 {(100 - x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [100 - x^{2}]) + {(100 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 1.8.4

Combine $\frac{1}{2 {(100 - x^{2})}^{\frac{1}{2}}}$ and $x$ .

$4 (\frac{x}{2 {(100 - x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [100 - x^{2}] + {(100 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 1.9

By the Sum Rule, the derivative of $100 - x^{2}$ with respect to $x$ is $\frac{d}{d x} [100] + \frac{d}{d x} [- x^{2}]$ .

$4 (\frac{x}{2 {(100 - x^{2})}^{\frac{1}{2}}} (\frac{d}{d x} [100] + \frac{d}{d x} [- x^{2}]) + {(100 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 1.10

Since $100$ is constant with respect to $x$ , the derivative of $100$ with respect to $x$ is $0$ .

$4 (\frac{x}{2 {(100 - x^{2})}^{\frac{1}{2}}} (0 + \frac{d}{d x} [- x^{2}]) + {(100 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 1.11

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

$4 (\frac{x}{2 {(100 - x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [- x^{2}] + {(100 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 1.12

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

$4 (\frac{x}{2 {(100 - x^{2})}^{\frac{1}{2}}} (- \frac{d}{d x} [x^{2}]) + {(100 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 1.13

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

$4 (\frac{x}{2 {(100 - x^{2})}^{\frac{1}{2}}} (- (2 x)) + {(100 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 1.14

Combine fractions.

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Step 1.14.1

Multiply $2$ by $- 1$ .

$4 (\frac{x}{2 {(100 - x^{2})}^{\frac{1}{2}}} (- 2 x) + {(100 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 1.14.2

Combine $- 2$ and $\frac{x}{2 {(100 - x^{2})}^{\frac{1}{2}}}$ .

$4 (\frac{- 2 x}{2 {(100 - x^{2})}^{\frac{1}{2}}} x + {(100 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 1.14.3

Combine $\frac{- 2 x}{2 {(100 - x^{2})}^{\frac{1}{2}}}$ and $x$ .

$4 (\frac{- 2 x \cdot x}{2 {(100 - x^{2})}^{\frac{1}{2}}} + {(100 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 1.15

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

$4 (\frac{- 2 (x^{1} x)}{2 {(100 - x^{2})}^{\frac{1}{2}}} + {(100 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 1.16

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

$4 (\frac{- 2 (x^{1} x^{1})}{2 {(100 - x^{2})}^{\frac{1}{2}}} + {(100 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 1.17

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

$4 (\frac{- 2 x^{1 + 1}}{2 {(100 - x^{2})}^{\frac{1}{2}}} + {(100 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 1.18

Add $1$ and $1$ .

$4 (\frac{- 2 x^{2}}{2 {(100 - x^{2})}^{\frac{1}{2}}} + {(100 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 1.19

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

$4 (\frac{2 (- x^{2})}{2 {(100 - x^{2})}^{\frac{1}{2}}} + {(100 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 1.20

Cancel the common factors.

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Step 1.20.1

Factor $2$ out of $2 {(100 - x^{2})}^{\frac{1}{2}}$ .

$4 (\frac{2 (- x^{2})}{2 ({(100 - x^{2})}^{\frac{1}{2}})} + {(100 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 1.20.2

Cancel the common factor.

$4 (\frac{2 (- x^{2})}{2 {(100 - x^{2})}^{\frac{1}{2}}} + {(100 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 1.20.3

Rewrite the expression.

$4 (\frac{- x^{2}}{{(100 - x^{2})}^{\frac{1}{2}}} + {(100 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 1.21

Move the negative in front of the fraction.

$4 (- \frac{x^{2}}{{(100 - x^{2})}^{\frac{1}{2}}} + {(100 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 1.22

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

$4 (- \frac{x^{2}}{{(100 - x^{2})}^{\frac{1}{2}}} + {(100 - x^{2})}^{\frac{1}{2}} \cdot 1)$

Step 1.23

Multiply ${(100 - x^{2})}^{\frac{1}{2}}$ by $1$ .

$4 (- \frac{x^{2}}{{(100 - x^{2})}^{\frac{1}{2}}} + {(100 - x^{2})}^{\frac{1}{2}})$

Step 1.24

To write ${(100 - x^{2})}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{{(100 - x^{2})}^{\frac{1}{2}}}{{(100 - x^{2})}^{\frac{1}{2}}}$ .

$4 (- \frac{x^{2}}{{(100 - x^{2})}^{\frac{1}{2}}} + \frac{{(100 - x^{2})}^{\frac{1}{2}} {(100 - x^{2})}^{\frac{1}{2}}}{{(100 - x^{2})}^{\frac{1}{2}}})$

Step 1.25

Combine the numerators over the common denominator.

$4 \frac{- x^{2} + {(100 - x^{2})}^{\frac{1}{2}} {(100 - x^{2})}^{\frac{1}{2}}}{{(100 - x^{2})}^{\frac{1}{2}}}$

Step 1.26

Multiply ${(100 - x^{2})}^{\frac{1}{2}}$ by ${(100 - x^{2})}^{\frac{1}{2}}$ 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.

$4 \frac{- x^{2} + {(100 - x^{2})}^{\frac{1}{2} + \frac{1}{2}}}{{(100 - x^{2})}^{\frac{1}{2}}}$

Step 1.26.2

Combine the numerators over the common denominator.

$4 \frac{- x^{2} + {(100 - x^{2})}^{\frac{1 + 1}{2}}}{{(100 - x^{2})}^{\frac{1}{2}}}$

Step 1.26.3

Add $1$ and $1$ .

$4 \frac{- x^{2} + {(100 - x^{2})}^{\frac{2}{2}}}{{(100 - x^{2})}^{\frac{1}{2}}}$

Step 1.26.4

Divide $2$ by $2$ .

$4 \frac{- x^{2} + {(100 - x^{2})}^{1}}{{(100 - x^{2})}^{\frac{1}{2}}}$

Step 1.27

Simplify ${(100 - x^{2})}^{1}$ .

$4 \frac{- x^{2} + 100 - x^{2}}{{(100 - x^{2})}^{\frac{1}{2}}}$

Step 1.28

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

$4 \frac{- 2 x^{2} + 100}{{(100 - x^{2})}^{\frac{1}{2}}}$

Step 1.29

Combine $4$ and $\frac{- 2 x^{2} + 100}{{(100 - x^{2})}^{\frac{1}{2}}}$ .

$\frac{4 (- 2 x^{2} + 100)}{{(100 - x^{2})}^{\frac{1}{2}}}$

Step 1.30

Simplify.

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Step 1.30.1

Apply the distributive property.

$\frac{4 (- 2 x^{2}) + 4 \cdot 100}{{(100 - x^{2})}^{\frac{1}{2}}}$

Step 1.30.2

Simplify each term.

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Step 1.30.2.1

Multiply $- 2$ by $4$ .

$\frac{- 8 x^{2} + 4 \cdot 100}{{(100 - x^{2})}^{\frac{1}{2}}}$

Step 1.30.2.2

Multiply $4$ by $100$ .

$\frac{- 8 x^{2} + 400}{{(100 - x^{2})}^{\frac{1}{2}}}$

Step 1.30.3

Reorder terms.

$\frac{- 8 x^{2} + 400}{{(- x^{2} + 100)}^{\frac{1}{2}}}$

Step 1.30.4

Factor $8$ out of $- 8 x^{2} + 400$ .

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Step 1.30.4.1

Factor $8$ out of $- 8 x^{2}$ .

$\frac{8 (- x^{2}) + 400}{{(- x^{2} + 100)}^{\frac{1}{2}}}$

Step 1.30.4.2

Factor $8$ out of $400$ .

$\frac{8 (- x^{2}) + 8 (50)}{{(- x^{2} + 100)}^{\frac{1}{2}}}$

Step 1.30.4.3

Factor $8$ out of $8 (- x^{2}) + 8 (50)$ .

$\frac{8 (- x^{2} + 50)}{{(- x^{2} + 100)}^{\frac{1}{2}}}$

Step 1.30.5

Factor $- 1$ out of $- x^{2}$ .

$\frac{8 (- (x^{2}) + 50)}{{(- x^{2} + 100)}^{\frac{1}{2}}}$

Step 1.30.6

Rewrite $50$ as $- 1 (- 50)$ .

$\frac{8 (- (x^{2}) - 1 (- 50))}{{(- x^{2} + 100)}^{\frac{1}{2}}}$

Step 1.30.7

Factor $- 1$ out of $- (x^{2}) - 1 (- 50)$ .

$\frac{8 (- (x^{2} - 50))}{{(- x^{2} + 100)}^{\frac{1}{2}}}$

Step 1.30.8

Rewrite $- (x^{2} - 50)$ as $- 1 (x^{2} - 50)$ .

$\frac{8 (- 1 (x^{2} - 50))}{{(- x^{2} + 100)}^{\frac{1}{2}}}$

Step 1.30.9

Move the negative in front of the fraction.

$- \frac{8 (x^{2} - 50)}{{(- x^{2} + 100)}^{\frac{1}{2}}}$

Step 2

Find the second derivative of the function.

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Step 2.1

Since $- 8$ is constant with respect to $x$ , the derivative of $- \frac{8 (x^{2} - 50)}{{(- x^{2} + 100)}^{\frac{1}{2}}}$ with respect to $x$ is $- 8 \frac{d}{d x} [\frac{x^{2} - 50}{{(- x^{2} + 100)}^{\frac{1}{2}}}]$ .

$f'' (x) = - 8 \frac{d}{d x} \frac{x^{2} - 50}{{(- x^{2} + 100)}^{\frac{1}{2}}}$

Step 2.2

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x^{2} - 50$ and $g (x) = {(- x^{2} + 100)}^{\frac{1}{2}}$ .

$f'' (x) = - 8 \frac{{(- x^{2} + 100)}^{\frac{1}{2}} \frac{d}{d x} (x^{2} - 50) - (x^{2} - 50) \frac{d}{d x} {(- x^{2} + 100)}^{\frac{1}{2}}}{{({(- x^{2} + 100)}^{\frac{1}{2}})}^{2}}$

Step 2.3

Multiply the exponents in ${({(- x^{2} + 100)}^{\frac{1}{2}})}^{2}$ .

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Step 2.3.1

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

$f'' (x) = - 8 \frac{{(- x^{2} + 100)}^{\frac{1}{2}} \frac{d}{d x} (x^{2} - 50) - (x^{2} - 50) \frac{d}{d x} {(- x^{2} + 100)}^{\frac{1}{2}}}{{(- x^{2} + 100)}^{\frac{1}{2} \cdot 2}}$

Step 2.3.2

Cancel the common factor of $2$ .

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Step 2.3.2.1

Cancel the common factor.

$f'' (x) = - 8 \frac{{(- x^{2} + 100)}^{\frac{1}{2}} \frac{d}{d x} (x^{2} - 50) - (x^{2} - 50) \frac{d}{d x} {(- x^{2} + 100)}^{\frac{1}{2}}}{{(- x^{2} + 100)}^{\frac{1}{2} \cdot 2}}$

Step 2.3.2.2

Rewrite the expression.

$f'' (x) = - 8 \frac{{(- x^{2} + 100)}^{\frac{1}{2}} \frac{d}{d x} (x^{2} - 50) - (x^{2} - 50) \frac{d}{d x} {(- x^{2} + 100)}^{\frac{1}{2}}}{- x^{2} + 100}$

Step 2.4

Simplify.

$f'' (x) = - 8 \frac{{(- x^{2} + 100)}^{\frac{1}{2}} \frac{d}{d x} (x^{2} - 50) - (x^{2} - 50) \frac{d}{d x} {(- x^{2} + 100)}^{\frac{1}{2}}}{- x^{2} + 100}$

Step 2.5

Differentiate.

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Step 2.5.1

By the Sum Rule, the derivative of $x^{2} - 50$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 50]$ .

$f'' (x) = - 8 \frac{{(- x^{2} + 100)}^{\frac{1}{2}} (\frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 50)) - (x^{2} - 50) \frac{d}{d x} {(- x^{2} + 100)}^{\frac{1}{2}}}{- x^{2} + 100}$

Step 2.5.2

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

$f'' (x) = - 8 \frac{{(- x^{2} + 100)}^{\frac{1}{2}} (2 x + \frac{d}{d x} (- 50)) - (x^{2} - 50) \frac{d}{d x} {(- x^{2} + 100)}^{\frac{1}{2}}}{- x^{2} + 100}$

Step 2.5.3

Since $- 50$ is constant with respect to $x$ , the derivative of $- 50$ with respect to $x$ is $0$ .

$f'' (x) = - 8 \frac{{(- x^{2} + 100)}^{\frac{1}{2}} (2 x + 0) - (x^{2} - 50) \frac{d}{d x} {(- x^{2} + 100)}^{\frac{1}{2}}}{- x^{2} + 100}$

Step 2.5.4

Simplify the expression.

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Step 2.5.4.1

Add $2 x$ and $0$ .

$f'' (x) = - 8 \frac{{(- x^{2} + 100)}^{\frac{1}{2}} (2 x) - (x^{2} - 50) \frac{d}{d x} {(- x^{2} + 100)}^{\frac{1}{2}}}{- x^{2} + 100}$

Step 2.5.4.2

Move $2$ to the left of ${(- x^{2} + 100)}^{\frac{1}{2}}$ .

$f'' (x) = - 8 \frac{2 \cdot ({(- x^{2} + 100)}^{\frac{1}{2}} x) - (x^{2} - 50) \frac{d}{d x} {(- x^{2} + 100)}^{\frac{1}{2}}}{- x^{2} + 100}$

Step 2.6

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = - x^{2} + 100$ .

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Step 2.6.1

To apply the Chain Rule, set $u$ as $- x^{2} + 100$ .

$f'' (x) = - 8 \frac{2 {(- x^{2} + 100)}^{\frac{1}{2}} x - (x^{2} - 50) (\frac{d}{d u} (u^{\frac{1}{2}}) \frac{d}{d x} (- x^{2} + 100))}{- x^{2} + 100}$

Step 2.6.2

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

$f'' (x) = - 8 \frac{2 {(- x^{2} + 100)}^{\frac{1}{2}} x - (x^{2} - 50) (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} (- x^{2} + 100))}{- x^{2} + 100}$

Step 2.6.3

Replace all occurrences of $u$ with $- x^{2} + 100$ .

$f'' (x) = - 8 \frac{2 {(- x^{2} + 100)}^{\frac{1}{2}} x - (x^{2} - 50) (\frac{1}{2} \cdot {(- x^{2} + 100)}^{\frac{1}{2} - 1} \frac{d}{d x} (- x^{2} + 100))}{- x^{2} + 100}$

Step 2.7

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

$f'' (x) = - 8 \frac{2 {(- x^{2} + 100)}^{\frac{1}{2}} x - (x^{2} - 50) (\frac{1}{2} \cdot {(- x^{2} + 100)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} (- x^{2} + 100))}{- x^{2} + 100}$

Step 2.8

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

$f'' (x) = - 8 \frac{2 {(- x^{2} + 100)}^{\frac{1}{2}} x - (x^{2} - 50) (\frac{1}{2} \cdot {(- x^{2} + 100)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} (- x^{2} + 100))}{- x^{2} + 100}$

Step 2.9

Combine the numerators over the common denominator.

$f'' (x) = - 8 \frac{2 {(- x^{2} + 100)}^{\frac{1}{2}} x - (x^{2} - 50) (\frac{1}{2} \cdot {(- x^{2} + 100)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} (- x^{2} + 100))}{- x^{2} + 100}$

Step 2.10

Simplify the numerator.

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Step 2.10.1

Multiply $- 1$ by $2$ .

$f'' (x) = - 8 \frac{2 {(- x^{2} + 100)}^{\frac{1}{2}} x - (x^{2} - 50) (\frac{1}{2} \cdot {(- x^{2} + 100)}^{\frac{1 - 2}{2}} \frac{d}{d x} (- x^{2} + 100))}{- x^{2} + 100}$

Step 2.10.2

Subtract $2$ from $1$ .

$f'' (x) = - 8 \frac{2 {(- x^{2} + 100)}^{\frac{1}{2}} x - (x^{2} - 50) (\frac{1}{2} \cdot {(- x^{2} + 100)}^{\frac{- 1}{2}} \frac{d}{d x} (- x^{2} + 100))}{- x^{2} + 100}$

Step 2.11

Combine fractions.

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Step 2.11.1

Move the negative in front of the fraction.

$f'' (x) = - 8 \frac{2 {(- x^{2} + 100)}^{\frac{1}{2}} x - (x^{2} - 50) (\frac{1}{2} \cdot {(- x^{2} + 100)}^{- \frac{1}{2}} \frac{d}{d x} (- x^{2} + 100))}{- x^{2} + 100}$

Step 2.11.2

Combine $\frac{1}{2}$ and ${(- x^{2} + 100)}^{- \frac{1}{2}}$ .

$f'' (x) = - 8 \frac{2 {(- x^{2} + 100)}^{\frac{1}{2}} x - (x^{2} - 50) (\frac{{(- x^{2} + 100)}^{- \frac{1}{2}}}{2} \cdot \frac{d}{d x} (- x^{2} + 100))}{- x^{2} + 100}$

Step 2.11.3

Move ${(- x^{2} + 100)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (x) = - 8 \frac{2 {(- x^{2} + 100)}^{\frac{1}{2}} x - (x^{2} - 50) (\frac{1}{2 {(- x^{2} + 100)}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- x^{2} + 100))}{- x^{2} + 100}$

Step 2.12

By the Sum Rule, the derivative of $- x^{2} + 100$ with respect to $x$ is $\frac{d}{d x} [- x^{2}] + \frac{d}{d x} [100]$ .

$f'' (x) = - 8 \frac{2 {(- x^{2} + 100)}^{\frac{1}{2}} x - (x^{2} - 50) (\frac{1}{2 {(- x^{2} + 100)}^{\frac{1}{2}}} \cdot (\frac{d}{d x} (- x^{2}) + \frac{d}{d x} (100)))}{- x^{2} + 100}$

Step 2.13

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

$f'' (x) = - 8 \frac{2 {(- x^{2} + 100)}^{\frac{1}{2}} x - (x^{2} - 50) (\frac{1}{2 {(- x^{2} + 100)}^{\frac{1}{2}}} \cdot (- \frac{d}{d x} x^{2} + \frac{d}{d x} (100)))}{- x^{2} + 100}$

Step 2.14

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

$f'' (x) = - 8 \frac{2 {(- x^{2} + 100)}^{\frac{1}{2}} x - (x^{2} - 50) (\frac{1}{2 {(- x^{2} + 100)}^{\frac{1}{2}}} \cdot (- (2 x) + \frac{d}{d x} (100)))}{- x^{2} + 100}$

Step 2.15

Multiply $2$ by $- 1$ .

$f'' (x) = - 8 \frac{2 {(- x^{2} + 100)}^{\frac{1}{2}} x - (x^{2} - 50) (\frac{1}{2 {(- x^{2} + 100)}^{\frac{1}{2}}} \cdot (- 2 x + \frac{d}{d x} (100)))}{- x^{2} + 100}$

Step 2.16

Since $100$ is constant with respect to $x$ , the derivative of $100$ with respect to $x$ is $0$ .

$f'' (x) = - 8 \frac{2 {(- x^{2} + 100)}^{\frac{1}{2}} x - (x^{2} - 50) (\frac{1}{2 {(- x^{2} + 100)}^{\frac{1}{2}}} \cdot (- 2 x + 0))}{- x^{2} + 100}$

Step 2.17

Simplify terms.

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Step 2.17.1

Add $- 2 x$ and $0$ .

$f'' (x) = - 8 \frac{2 {(- x^{2} + 100)}^{\frac{1}{2}} x - (x^{2} - 50) (\frac{1}{2 {(- x^{2} + 100)}^{\frac{1}{2}}} \cdot (- 2 x))}{- x^{2} + 100}$

Step 2.17.2

Combine $- 2$ and $\frac{1}{2 {(- x^{2} + 100)}^{\frac{1}{2}}}$ .

$f'' (x) = - 8 \frac{2 {(- x^{2} + 100)}^{\frac{1}{2}} x - (x^{2} - 50) (\frac{- 2}{2 {(- x^{2} + 100)}^{\frac{1}{2}}} \cdot x)}{- x^{2} + 100}$

Step 2.17.3

Combine $\frac{- 2}{2 {(- x^{2} + 100)}^{\frac{1}{2}}}$ and $x$ .

$f'' (x) = - 8 \frac{2 {(- x^{2} + 100)}^{\frac{1}{2}} x - (x^{2} - 50) \frac{- 2 x}{2 {(- x^{2} + 100)}^{\frac{1}{2}}}}{- x^{2} + 100}$

Step 2.17.4

Factor $2$ out of $- 2 x$ .

$f'' (x) = - 8 \frac{2 {(- x^{2} + 100)}^{\frac{1}{2}} x - (x^{2} - 50) \frac{2 (- x)}{2 {(- x^{2} + 100)}^{\frac{1}{2}}}}{- x^{2} + 100}$

Step 2.18

Cancel the common factors.

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Step 2.18.1

Factor $2$ out of $2 {(- x^{2} + 100)}^{\frac{1}{2}}$ .

$f'' (x) = - 8 \frac{2 {(- x^{2} + 100)}^{\frac{1}{2}} x - (x^{2} - 50) \frac{2 (- x)}{2 ({(- x^{2} + 100)}^{\frac{1}{2}})}}{- x^{2} + 100}$

Step 2.18.2

Cancel the common factor.

$f'' (x) = - 8 \frac{2 {(- x^{2} + 100)}^{\frac{1}{2}} x - (x^{2} - 50) \frac{2 (- x)}{2 {(- x^{2} + 100)}^{\frac{1}{2}}}}{- x^{2} + 100}$

Step 2.18.3

Rewrite the expression.

$f'' (x) = - 8 \frac{2 {(- x^{2} + 100)}^{\frac{1}{2}} x - (x^{2} - 50) \frac{- x}{{(- x^{2} + 100)}^{\frac{1}{2}}}}{- x^{2} + 100}$

Step 2.19

Simplify the expression.

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Step 2.19.1

Move the negative in front of the fraction.

$f'' (x) = - 8 \frac{2 {(- x^{2} + 100)}^{\frac{1}{2}} x - (x^{2} - 50) (- \frac{x}{{(- x^{2} + 100)}^{\frac{1}{2}}})}{- x^{2} + 100}$

Step 2.19.2

Multiply $- 1$ by $- 1$ .

$f'' (x) = - 8 \frac{2 {(- x^{2} + 100)}^{\frac{1}{2}} x + 1 (x^{2} - 50) (\frac{x}{{(- x^{2} + 100)}^{\frac{1}{2}}})}{- x^{2} + 100}$

Step 2.19.3

Multiply $x^{2} - 50$ by $1$ .

$f'' (x) = - 8 \frac{2 {(- x^{2} + 100)}^{\frac{1}{2}} x + (x^{2} - 50) (\frac{x}{{(- x^{2} + 100)}^{\frac{1}{2}}})}{- x^{2} + 100}$

Step 2.20

Combine $- 8$ and $\frac{2 {(- x^{2} + 100)}^{\frac{1}{2}} x + (x^{2} - 50) \frac{x}{{(- x^{2} + 100)}^{\frac{1}{2}}}}{- x^{2} + 100}$ .

$f'' (x) = \frac{- 8 (2 {(- x^{2} + 100)}^{\frac{1}{2}} x + (x^{2} - 50) (\frac{x}{{(- x^{2} + 100)}^{\frac{1}{2}}}))}{- x^{2} + 100}$

Step 2.21

Move the negative in front of the fraction.

$f'' (x) = - \frac{8 (2 {(- x^{2} + 100)}^{\frac{1}{2}} x + (x^{2} - 50) (\frac{x}{{(- x^{2} + 100)}^{\frac{1}{2}}}))}{- x^{2} + 100}$

Step 2.22

Simplify.

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Step 2.22.1

Apply the distributive property.

$f'' (x) = - \frac{8 (2 {(- x^{2} + 100)}^{\frac{1}{2}} x) + 8 ((x^{2} - 50) (\frac{x}{{(- x^{2} + 100)}^{\frac{1}{2}}}))}{- x^{2} + 100}$

Step 2.22.2

Simplify the numerator.

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Step 2.22.2.1

Multiply $2$ by $8$ .

$f'' (x) = - \frac{16 ({(- x^{2} + 100)}^{\frac{1}{2}} x) + 8 ((x^{2} - 50) (\frac{x}{{(- x^{2} + 100)}^{\frac{1}{2}}}))}{- x^{2} + 100}$

Step 2.22.2.2

Multiply $x^{2} - 50$ by $\frac{x}{{(- x^{2} + 100)}^{\frac{1}{2}}}$ .

$f'' (x) = - \frac{16 {(- x^{2} + 100)}^{\frac{1}{2}} x + 8 (\frac{(x^{2} - 50) x}{{(- x^{2} + 100)}^{\frac{1}{2}}})}{- x^{2} + 100}$

Step 2.22.2.3

Combine $8$ and $\frac{(x^{2} - 50) x}{{(- x^{2} + 100)}^{\frac{1}{2}}}$ .

$f'' (x) = - \frac{16 {(- x^{2} + 100)}^{\frac{1}{2}} x + \frac{8 ((x^{2} - 50) x)}{{(- x^{2} + 100)}^{\frac{1}{2}}}}{- x^{2} + 100}$

Step 2.22.2.4

To write $16 {(- x^{2} + 100)}^{\frac{1}{2}} x$ as a fraction with a common denominator, multiply by $\frac{{(- x^{2} + 100)}^{\frac{1}{2}}}{{(- x^{2} + 100)}^{\frac{1}{2}}}$ .

$f'' (x) = - \frac{\frac{16 {(- x^{2} + 100)}^{\frac{1}{2}} x {(- x^{2} + 100)}^{\frac{1}{2}}}{{(- x^{2} + 100)}^{\frac{1}{2}}} + \frac{8 (x^{2} - 50) x}{{(- x^{2} + 100)}^{\frac{1}{2}}}}{- x^{2} + 100}$

Step 2.22.2.5

Combine the numerators over the common denominator.

$f'' (x) = - \frac{\frac{16 {(- x^{2} + 100)}^{\frac{1}{2}} x {(- x^{2} + 100)}^{\frac{1}{2}} + 8 (x^{2} - 50) x}{{(- x^{2} + 100)}^{\frac{1}{2}}}}{- x^{2} + 100}$

Step 2.22.2.6

Rewrite $\frac{16 {(- x^{2} + 100)}^{\frac{1}{2}} x {(- x^{2} + 100)}^{\frac{1}{2}} + 8 (x^{2} - 50) x}{{(- x^{2} + 100)}^{\frac{1}{2}}}$ in a factored form.

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Step 2.22.2.6.1

Factor $8 x$ out of $16 {(- x^{2} + 100)}^{\frac{1}{2}} x {(- x^{2} + 100)}^{\frac{1}{2}} + 8 (x^{2} - 50) x$ .

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Step 2.22.2.6.1.1

Factor $8 x$ out of $16 {(- x^{2} + 100)}^{\frac{1}{2}} x {(- x^{2} + 100)}^{\frac{1}{2}}$ .

$f'' (x) = - \frac{\frac{8 x (2 {(- x^{2} + 100)}^{\frac{1}{2}} {(- x^{2} + 100)}^{\frac{1}{2}}) + 8 (x^{2} - 50) x}{{(- x^{2} + 100)}^{\frac{1}{2}}}}{- x^{2} + 100}$

Step 2.22.2.6.1.2

Factor $8 x$ out of $8 (x^{2} - 50) x$ .

$f'' (x) = - \frac{\frac{8 x (2 {(- x^{2} + 100)}^{\frac{1}{2}} {(- x^{2} + 100)}^{\frac{1}{2}}) + 8 x (x^{2} - 50)}{{(- x^{2} + 100)}^{\frac{1}{2}}}}{- x^{2} + 100}$

Step 2.22.2.6.1.3

Factor $8 x$ out of $8 x (2 {(- x^{2} + 100)}^{\frac{1}{2}} {(- x^{2} + 100)}^{\frac{1}{2}}) + 8 x (x^{2} - 50)$ .

$f'' (x) = - \frac{\frac{8 x (2 {(- x^{2} + 100)}^{\frac{1}{2}} {(- x^{2} + 100)}^{\frac{1}{2}} + x^{2} - 50)}{{(- x^{2} + 100)}^{\frac{1}{2}}}}{- x^{2} + 100}$

Step 2.22.2.6.2

Combine exponents.

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Step 2.22.2.6.2.1

Multiply ${(- x^{2} + 100)}^{\frac{1}{2}}$ by ${(- x^{2} + 100)}^{\frac{1}{2}}$ by adding the exponents.

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Step 2.22.2.6.2.1.1

Move ${(- x^{2} + 100)}^{\frac{1}{2}}$ .

$f'' (x) = - \frac{\frac{8 x (2 ({(- x^{2} + 100)}^{\frac{1}{2}} {(- x^{2} + 100)}^{\frac{1}{2}}) + x^{2} - 50)}{{(- x^{2} + 100)}^{\frac{1}{2}}}}{- x^{2} + 100}$

Step 2.22.2.6.2.1.2

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

$f'' (x) = - \frac{\frac{8 x (2 {(- x^{2} + 100)}^{\frac{1}{2} + \frac{1}{2}} + x^{2} - 50)}{{(- x^{2} + 100)}^{\frac{1}{2}}}}{- x^{2} + 100}$

Step 2.22.2.6.2.1.3

Combine the numerators over the common denominator.

$f'' (x) = - \frac{\frac{8 x (2 {(- x^{2} + 100)}^{\frac{1 + 1}{2}} + x^{2} - 50)}{{(- x^{2} + 100)}^{\frac{1}{2}}}}{- x^{2} + 100}$

Step 2.22.2.6.2.1.4

Add $1$ and $1$ .

$f'' (x) = - \frac{\frac{8 x (2 {(- x^{2} + 100)}^{\frac{2}{2}} + x^{2} - 50)}{{(- x^{2} + 100)}^{\frac{1}{2}}}}{- x^{2} + 100}$

Step 2.22.2.6.2.1.5

Divide $2$ by $2$ .

$f'' (x) = - \frac{\frac{8 x (2 (- x^{2} + 100) + x^{2} - 50)}{{(- x^{2} + 100)}^{\frac{1}{2}}}}{- x^{2} + 100}$

Step 2.22.2.6.2.2

Simplify $2 {(- x^{2} + 100)}^{1}$ .

$f'' (x) = - \frac{\frac{8 x (2 (- x^{2} + 100) + x^{2} - 50)}{{(- x^{2} + 100)}^{\frac{1}{2}}}}{- x^{2} + 100}$

Step 2.22.2.7

Simplify the numerator.

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Step 2.22.2.7.1

Apply the distributive property.

$f'' (x) = - \frac{\frac{8 x (2 (- x^{2}) + 2 \cdot 100 + x^{2} - 50)}{{(- x^{2} + 100)}^{\frac{1}{2}}}}{- x^{2} + 100}$

Step 2.22.2.7.2

Multiply $- 1$ by $2$ .

$f'' (x) = - \frac{\frac{8 x (- 2 x^{2} + 2 \cdot 100 + x^{2} - 50)}{{(- x^{2} + 100)}^{\frac{1}{2}}}}{- x^{2} + 100}$

Step 2.22.2.7.3

Multiply $2$ by $100$ .

$f'' (x) = - \frac{\frac{8 x (- 2 x^{2} + 200 + x^{2} - 50)}{{(- x^{2} + 100)}^{\frac{1}{2}}}}{- x^{2} + 100}$

Step 2.22.2.7.4

Add $- 2 x^{2}$ and $x^{2}$ .

$f'' (x) = - \frac{\frac{8 x (- x^{2} + 200 - 50)}{{(- x^{2} + 100)}^{\frac{1}{2}}}}{- x^{2} + 100}$

Step 2.22.2.7.5

Subtract $50$ from $200$ .

$f'' (x) = - \frac{\frac{8 x (- x^{2} + 150)}{{(- x^{2} + 100)}^{\frac{1}{2}}}}{- x^{2} + 100}$

Step 2.22.3

Combine terms.

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Step 2.22.3.1

Rewrite $\frac{\frac{8 x (- x^{2} + 150)}{{(- x^{2} + 100)}^{\frac{1}{2}}}}{- x^{2} + 100}$ as a product.

$f'' (x) = - (\frac{8 x (- x^{2} + 150)}{{(- x^{2} + 100)}^{\frac{1}{2}}} \cdot \frac{1}{- x^{2} + 100})$

Step 2.22.3.2

Multiply $\frac{8 x (- x^{2} + 150)}{{(- x^{2} + 100)}^{\frac{1}{2}}}$ by $\frac{1}{- x^{2} + 100}$ .

$f'' (x) = - \frac{8 x (- x^{2} + 150)}{{(- x^{2} + 100)}^{\frac{1}{2}} (- x^{2} + 100)}$

Step 2.22.3.3

Multiply ${(- x^{2} + 100)}^{\frac{1}{2}}$ by $- x^{2} + 100$ by adding the exponents.

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Step 2.22.3.3.1

Multiply ${(- x^{2} + 100)}^{\frac{1}{2}}$ by $- x^{2} + 100$ .

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Step 2.22.3.3.1.1

Raise $- x^{2} + 100$ to the power of $1$ .

$f'' (x) = - \frac{8 x (- x^{2} + 150)}{{(- x^{2} + 100)}^{\frac{1}{2}} (- x^{2} + 100)}$

Step 2.22.3.3.1.2

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

$f'' (x) = - \frac{8 x (- x^{2} + 150)}{{(- x^{2} + 100)}^{\frac{1}{2} + 1}}$

Step 2.22.3.3.2

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

$f'' (x) = - \frac{8 x (- x^{2} + 150)}{{(- x^{2} + 100)}^{\frac{1}{2} + \frac{2}{2}}}$

Step 2.22.3.3.3

Combine the numerators over the common denominator.

$f'' (x) = - \frac{8 x (- x^{2} + 150)}{{(- x^{2} + 100)}^{\frac{1 + 2}{2}}}$

Step 2.22.3.3.4

Add $1$ and $2$ .

$f'' (x) = - \frac{8 x (- x^{2} + 150)}{{(- x^{2} + 100)}^{\frac{3}{2}}}$

Step 2.22.4

Factor $- 1$ out of $- x^{2}$ .

$f'' (x) = - \frac{8 x (- (x^{2}) + 150)}{{(- x^{2} + 100)}^{\frac{3}{2}}}$

Step 2.22.5

Rewrite $150$ as $- 1 (- 150)$ .

$f'' (x) = - \frac{8 x (- (x^{2}) - 1 \cdot - 150)}{{(- x^{2} + 100)}^{\frac{3}{2}}}$

Step 2.22.6

Factor $- 1$ out of $- (x^{2}) - 1 (- 150)$ .

$f'' (x) = - \frac{8 x (- (x^{2} - 150))}{{(- x^{2} + 100)}^{\frac{3}{2}}}$

Step 2.22.7

Rewrite $- (x^{2} - 150)$ as $- 1 (x^{2} - 150)$ .

$f'' (x) = - \frac{8 x (- 1 (x^{2} - 150))}{{(- x^{2} + 100)}^{\frac{3}{2}}}$

Step 2.22.8

Move the negative in front of the fraction.

$f'' (x) = \frac{(8 x) (x^{2} - 150)}{{(- x^{2} + 100)}^{\frac{3}{2}}}$

Step 2.22.9

Multiply $- 1$ by $- 1$ .

$f'' (x) = 1 (\frac{(8 x) (x^{2} - 150)}{{(- x^{2} + 100)}^{\frac{3}{2}}})$

Step 2.22.10

Multiply $\frac{(8 x) (x^{2} - 150)}{{(- x^{2} + 100)}^{\frac{3}{2}}}$ by $1$ .

$f'' (x) = \frac{(8 x) (x^{2} - 150)}{{(- x^{2} + 100)}^{\frac{3}{2}}}$

Step 3

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

$- \frac{8 (x^{2} - 50)}{{(- x^{2} + 100)}^{\frac{1}{2}}} = 0$

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

Differentiate using the Constant Multiple Rule.

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Step 4.1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{100 - x^{2}}$ as ${(100 - x^{2})}^{\frac{1}{2}}$ .

$\frac{d}{d x} [4 x {(100 - x^{2})}^{\frac{1}{2}}]$

Step 4.1.1.2

Since $4$ is constant with respect to $x$ , the derivative of $4 x {(100 - x^{2})}^{\frac{1}{2}}$ with respect to $x$ is $4 \frac{d}{d x} [x {(100 - x^{2})}^{\frac{1}{2}}]$ .

$4 \frac{d}{d x} [x {(100 - x^{2})}^{\frac{1}{2}}]$

Step 4.1.2

$4 (x \frac{d}{d x} [{(100 - x^{2})}^{\frac{1}{2}}] + {(100 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 4.1.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = 100 - x^{2}$ .

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Step 4.1.3.1

To apply the Chain Rule, set $u$ as $100 - x^{2}$ .

$4 (x (\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [100 - x^{2}]) + {(100 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

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 $n = \frac{1}{2}$ .

$4 (x (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [100 - x^{2}]) + {(100 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 4.1.3.3

Replace all occurrences of $u$ with $100 - x^{2}$ .

$4 (x (\frac{1}{2} {(100 - x^{2})}^{\frac{1}{2} - 1} \frac{d}{d x} [100 - x^{2}]) + {(100 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 4.1.4

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

$4 (x (\frac{1}{2} {(100 - x^{2})}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [100 - x^{2}]) + {(100 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 4.1.5

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

$4 (x (\frac{1}{2} {(100 - x^{2})}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [100 - x^{2}]) + {(100 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 4.1.6

Combine the numerators over the common denominator.

$4 (x (\frac{1}{2} {(100 - x^{2})}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [100 - x^{2}]) + {(100 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 4.1.7

Simplify the numerator.

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Step 4.1.7.1

Multiply $- 1$ by $2$ .

$4 (x (\frac{1}{2} {(100 - x^{2})}^{\frac{1 - 2}{2}} \frac{d}{d x} [100 - x^{2}]) + {(100 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 4.1.7.2

Subtract $2$ from $1$ .

$4 (x (\frac{1}{2} {(100 - x^{2})}^{\frac{- 1}{2}} \frac{d}{d x} [100 - x^{2}]) + {(100 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 4.1.8

Combine fractions.

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Step 4.1.8.1

Move the negative in front of the fraction.

$4 (x (\frac{1}{2} {(100 - x^{2})}^{- \frac{1}{2}} \frac{d}{d x} [100 - x^{2}]) + {(100 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 4.1.8.2

Combine $\frac{1}{2}$ and ${(100 - x^{2})}^{- \frac{1}{2}}$ .

$4 (x (\frac{{(100 - x^{2})}^{- \frac{1}{2}}}{2} \frac{d}{d x} [100 - x^{2}]) + {(100 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 4.1.8.3

Move ${(100 - x^{2})}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$4 (x (\frac{1}{2 {(100 - x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [100 - x^{2}]) + {(100 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 4.1.8.4

Combine $\frac{1}{2 {(100 - x^{2})}^{\frac{1}{2}}}$ and $x$ .

$4 (\frac{x}{2 {(100 - x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [100 - x^{2}] + {(100 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 4.1.9

By the Sum Rule, the derivative of $100 - x^{2}$ with respect to $x$ is $\frac{d}{d x} [100] + \frac{d}{d x} [- x^{2}]$ .

$4 (\frac{x}{2 {(100 - x^{2})}^{\frac{1}{2}}} (\frac{d}{d x} [100] + \frac{d}{d x} [- x^{2}]) + {(100 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 4.1.10

Since $100$ is constant with respect to $x$ , the derivative of $100$ with respect to $x$ is $0$ .

$4 (\frac{x}{2 {(100 - x^{2})}^{\frac{1}{2}}} (0 + \frac{d}{d x} [- x^{2}]) + {(100 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 4.1.11

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

$4 (\frac{x}{2 {(100 - x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [- x^{2}] + {(100 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 4.1.12

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

$4 (\frac{x}{2 {(100 - x^{2})}^{\frac{1}{2}}} (- \frac{d}{d x} [x^{2}]) + {(100 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 4.1.13

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

$4 (\frac{x}{2 {(100 - x^{2})}^{\frac{1}{2}}} (- (2 x)) + {(100 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 4.1.14

Combine fractions.

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Step 4.1.14.1

Multiply $2$ by $- 1$ .

$4 (\frac{x}{2 {(100 - x^{2})}^{\frac{1}{2}}} (- 2 x) + {(100 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 4.1.14.2

Combine $- 2$ and $\frac{x}{2 {(100 - x^{2})}^{\frac{1}{2}}}$ .

$4 (\frac{- 2 x}{2 {(100 - x^{2})}^{\frac{1}{2}}} x + {(100 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 4.1.14.3

Combine $\frac{- 2 x}{2 {(100 - x^{2})}^{\frac{1}{2}}}$ and $x$ .

$4 (\frac{- 2 x \cdot x}{2 {(100 - x^{2})}^{\frac{1}{2}}} + {(100 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 4.1.15

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

$4 (\frac{- 2 (x^{1} x)}{2 {(100 - x^{2})}^{\frac{1}{2}}} + {(100 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 4.1.16

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

$4 (\frac{- 2 (x^{1} x^{1})}{2 {(100 - x^{2})}^{\frac{1}{2}}} + {(100 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 4.1.17

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

$4 (\frac{- 2 x^{1 + 1}}{2 {(100 - x^{2})}^{\frac{1}{2}}} + {(100 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 4.1.18

Add $1$ and $1$ .

$4 (\frac{- 2 x^{2}}{2 {(100 - x^{2})}^{\frac{1}{2}}} + {(100 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 4.1.19

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

$4 (\frac{2 (- x^{2})}{2 {(100 - x^{2})}^{\frac{1}{2}}} + {(100 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 4.1.20

Cancel the common factors.

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Step 4.1.20.1

Factor $2$ out of $2 {(100 - x^{2})}^{\frac{1}{2}}$ .

$4 (\frac{2 (- x^{2})}{2 ({(100 - x^{2})}^{\frac{1}{2}})} + {(100 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 4.1.20.2

Cancel the common factor.

$4 (\frac{2 (- x^{2})}{2 {(100 - x^{2})}^{\frac{1}{2}}} + {(100 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 4.1.20.3

Rewrite the expression.

$4 (\frac{- x^{2}}{{(100 - x^{2})}^{\frac{1}{2}}} + {(100 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 4.1.21

Move the negative in front of the fraction.

$4 (- \frac{x^{2}}{{(100 - x^{2})}^{\frac{1}{2}}} + {(100 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 4.1.22

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

$4 (- \frac{x^{2}}{{(100 - x^{2})}^{\frac{1}{2}}} + {(100 - x^{2})}^{\frac{1}{2}} \cdot 1)$

Step 4.1.23

Multiply ${(100 - x^{2})}^{\frac{1}{2}}$ by $1$ .

$4 (- \frac{x^{2}}{{(100 - x^{2})}^{\frac{1}{2}}} + {(100 - x^{2})}^{\frac{1}{2}})$

Step 4.1.24

To write ${(100 - x^{2})}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{{(100 - x^{2})}^{\frac{1}{2}}}{{(100 - x^{2})}^{\frac{1}{2}}}$ .

$4 (- \frac{x^{2}}{{(100 - x^{2})}^{\frac{1}{2}}} + \frac{{(100 - x^{2})}^{\frac{1}{2}} {(100 - x^{2})}^{\frac{1}{2}}}{{(100 - x^{2})}^{\frac{1}{2}}})$

Step 4.1.25

Combine the numerators over the common denominator.

$4 \frac{- x^{2} + {(100 - x^{2})}^{\frac{1}{2}} {(100 - x^{2})}^{\frac{1}{2}}}{{(100 - x^{2})}^{\frac{1}{2}}}$

Step 4.1.26

Multiply ${(100 - x^{2})}^{\frac{1}{2}}$ by ${(100 - x^{2})}^{\frac{1}{2}}$ 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.

$4 \frac{- x^{2} + {(100 - x^{2})}^{\frac{1}{2} + \frac{1}{2}}}{{(100 - x^{2})}^{\frac{1}{2}}}$

Step 4.1.26.2

Combine the numerators over the common denominator.

$4 \frac{- x^{2} + {(100 - x^{2})}^{\frac{1 + 1}{2}}}{{(100 - x^{2})}^{\frac{1}{2}}}$

Step 4.1.26.3

Add $1$ and $1$ .

$4 \frac{- x^{2} + {(100 - x^{2})}^{\frac{2}{2}}}{{(100 - x^{2})}^{\frac{1}{2}}}$

Step 4.1.26.4

Divide $2$ by $2$ .

$4 \frac{- x^{2} + {(100 - x^{2})}^{1}}{{(100 - x^{2})}^{\frac{1}{2}}}$

Step 4.1.27

Simplify ${(100 - x^{2})}^{1}$ .

$4 \frac{- x^{2} + 100 - x^{2}}{{(100 - x^{2})}^{\frac{1}{2}}}$

Step 4.1.28

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

$4 \frac{- 2 x^{2} + 100}{{(100 - x^{2})}^{\frac{1}{2}}}$

Step 4.1.29

Combine $4$ and $\frac{- 2 x^{2} + 100}{{(100 - x^{2})}^{\frac{1}{2}}}$ .

$\frac{4 (- 2 x^{2} + 100)}{{(100 - x^{2})}^{\frac{1}{2}}}$

Step 4.1.30

Simplify.

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Step 4.1.30.1

Apply the distributive property.

$\frac{4 (- 2 x^{2}) + 4 \cdot 100}{{(100 - x^{2})}^{\frac{1}{2}}}$

Step 4.1.30.2

Simplify each term.

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Step 4.1.30.2.1

Multiply $- 2$ by $4$ .

$\frac{- 8 x^{2} + 4 \cdot 100}{{(100 - x^{2})}^{\frac{1}{2}}}$

Step 4.1.30.2.2

Multiply $4$ by $100$ .

$\frac{- 8 x^{2} + 400}{{(100 - x^{2})}^{\frac{1}{2}}}$

Step 4.1.30.3

Reorder terms.

$\frac{- 8 x^{2} + 400}{{(- x^{2} + 100)}^{\frac{1}{2}}}$

Step 4.1.30.4

Factor $8$ out of $- 8 x^{2} + 400$ .

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Step 4.1.30.4.1

Factor $8$ out of $- 8 x^{2}$ .

$\frac{8 (- x^{2}) + 400}{{(- x^{2} + 100)}^{\frac{1}{2}}}$

Step 4.1.30.4.2

Factor $8$ out of $400$ .

$\frac{8 (- x^{2}) + 8 (50)}{{(- x^{2} + 100)}^{\frac{1}{2}}}$

Step 4.1.30.4.3

Factor $8$ out of $8 (- x^{2}) + 8 (50)$ .

$\frac{8 (- x^{2} + 50)}{{(- x^{2} + 100)}^{\frac{1}{2}}}$

Step 4.1.30.5

Factor $- 1$ out of $- x^{2}$ .

$\frac{8 (- (x^{2}) + 50)}{{(- x^{2} + 100)}^{\frac{1}{2}}}$

Step 4.1.30.6

Rewrite $50$ as $- 1 (- 50)$ .

$\frac{8 (- (x^{2}) - 1 (- 50))}{{(- x^{2} + 100)}^{\frac{1}{2}}}$

Step 4.1.30.7

Factor $- 1$ out of $- (x^{2}) - 1 (- 50)$ .

$\frac{8 (- (x^{2} - 50))}{{(- x^{2} + 100)}^{\frac{1}{2}}}$

Step 4.1.30.8

Rewrite $- (x^{2} - 50)$ as $- 1 (x^{2} - 50)$ .

$\frac{8 (- 1 (x^{2} - 50))}{{(- x^{2} + 100)}^{\frac{1}{2}}}$

Step 4.1.30.9

Move the negative in front of the fraction.

$f' (x) = - \frac{8 (x^{2} - 50)}{{(- x^{2} + 100)}^{\frac{1}{2}}}$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{8 (x^{2} - 50)}{{(- x^{2} + 100)}^{\frac{1}{2}}}$ .

$- \frac{8 (x^{2} - 50)}{{(- x^{2} + 100)}^{\frac{1}{2}}}$

Step 5

Set the first derivative equal to $0$ then solve the equation $- \frac{8 (x^{2} - 50)}{{(- x^{2} + 100)}^{\frac{1}{2}}} = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$- \frac{8 (x^{2} - 50)}{{(- x^{2} + 100)}^{\frac{1}{2}}} = 0$

Step 5.2

Set the numerator equal to zero.

$8 (x^{2} - 50) = 0$

Step 5.3

Solve the equation for $x$ .

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Step 5.3.1

Divide each term in $8 (x^{2} - 50) = 0$ by $8$ and simplify.

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Step 5.3.1.1

Divide each term in $8 (x^{2} - 50) = 0$ by $8$ .

$\frac{8 (x^{2} - 50)}{8} = \frac{0}{8}$

Step 5.3.1.2

Simplify the left side.

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Step 5.3.1.2.1

Cancel the common factor of $8$ .

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Step 5.3.1.2.1.1

Cancel the common factor.

$\frac{8 (x^{2} - 50)}{8} = \frac{0}{8}$

Step 5.3.1.2.1.2

Divide $x^{2} - 50$ by $1$ .

$x^{2} - 50 = \frac{0}{8}$

Step 5.3.1.3

Simplify the right side.

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Step 5.3.1.3.1

Divide $0$ by $8$ .

$x^{2} - 50 = 0$

Step 5.3.2

Add $50$ to both sides of the equation.

$x^{2} = 50$

Step 5.3.3

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

$x = \pm \sqrt{50}$

Step 5.3.4

Simplify $\pm \sqrt{50}$ .

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Step 5.3.4.1

Rewrite $50$ as $5^{2} \cdot 2$ .

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Step 5.3.4.1.1

Factor $25$ out of $50$ .

$x = \pm \sqrt{25 (2)}$

Step 5.3.4.1.2

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

$x = \pm \sqrt{5^{2} \cdot 2}$

Step 5.3.4.2

Pull terms out from under the radical.

$x = \pm 5 \sqrt{2}$

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.

$x = 5 \sqrt{2}$

Step 5.3.5.2

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

$x = - 5 \sqrt{2}$

Step 5.3.5.3

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

$x = 5 \sqrt{2}, - 5 \sqrt{2}$

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.

$- \frac{8 (x^{2} - 50)}{\sqrt{{(- x^{2} + 100)}^{1}}}$

Step 6.1.2

Anything raised to $1$ is the base itself.

$- \frac{8 (x^{2} - 50)}{\sqrt{- x^{2} + 100}}$

Step 6.2

Set the denominator in $\frac{8 (x^{2} - 50)}{\sqrt{- x^{2} + 100}}$ equal to $0$ to find where the expression is undefined.

$\sqrt{- x^{2} + 100} = 0$

Step 6.3

Solve for $x$ .

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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.

${\sqrt{- x^{2} + 100}}^{2} = 0^{2}$

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 $\sqrt{- x^{2} + 100}$ as ${(- x^{2} + 100)}^{\frac{1}{2}}$ .

${({(- x^{2} + 100)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 6.3.2.2

Simplify the left side.

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Step 6.3.2.2.1

Simplify ${({(- x^{2} + 100)}^{\frac{1}{2}})}^{2}$ .

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Step 6.3.2.2.1.1

Multiply the exponents in ${({(- x^{2} + 100)}^{\frac{1}{2}})}^{2}$ .

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Step 6.3.2.2.1.1.1

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

${(- x^{2} + 100)}^{\frac{1}{2} \cdot 2} = 0^{2}$

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.

${(- x^{2} + 100)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 6.3.2.2.1.1.2.2

Rewrite the expression.

${(- x^{2} + 100)}^{1} = 0^{2}$

Step 6.3.2.2.1.2

Simplify.

$- x^{2} + 100 = 0^{2}$

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 $0$ .

$- x^{2} + 100 = 0$

Step 6.3.3

Solve for $x$ .

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Step 6.3.3.1

Subtract $100$ from both sides of the equation.

$- x^{2} = - 100$

Step 6.3.3.2

Divide each term in $- x^{2} = - 100$ by $- 1$ and simplify.

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Step 6.3.3.2.1

Divide each term in $- x^{2} = - 100$ by $- 1$ .

$\frac{- x^{2}}{- 1} = \frac{- 100}{- 1}$

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.

$\frac{x^{2}}{1} = \frac{- 100}{- 1}$

Step 6.3.3.2.2.2

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

$x^{2} = \frac{- 100}{- 1}$

Step 6.3.3.2.3

Simplify the right side.

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Step 6.3.3.2.3.1

Divide $- 100$ by $- 1$ .

$x^{2} = 100$

Step 6.3.3.3

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

$x = \pm \sqrt{100}$

Step 6.3.3.4

Simplify $\pm \sqrt{100}$ .

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Step 6.3.3.4.1

Rewrite $100$ as $10^{2}$ .

$x = \pm \sqrt{10^{2}}$

Step 6.3.3.4.2

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

$x = \pm 10$

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.

$x = 10$

Step 6.3.3.5.2

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

$x = - 10$

Step 6.3.3.5.3

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

$x = 10, - 10$

Step 6.4

Set the radicand in $\sqrt{- x^{2} + 100}$ less than $0$ to find where the expression is undefined.

$- x^{2} + 100 < 0$

Step 6.5

Solve for $x$ .

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Step 6.5.1

Subtract $100$ from both sides of the inequality.

$- x^{2} < - 100$

Step 6.5.2

Divide each term in $- x^{2} < - 100$ by $- 1$ and simplify.

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Step 6.5.2.1

Divide each term in $- x^{2} < - 100$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x^{2}}{- 1} > \frac{- 100}{- 1}$

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.

$\frac{x^{2}}{1} > \frac{- 100}{- 1}$

Step 6.5.2.2.2

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

$x^{2} > \frac{- 100}{- 1}$

Step 6.5.2.3

Simplify the right side.

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Step 6.5.2.3.1

Divide $- 100$ by $- 1$ .

$x^{2} > 100$

Step 6.5.3

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

$\sqrt{x^{2}} > \sqrt{100}$

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.

$| x | > \sqrt{100}$

Step 6.5.4.2

Simplify the right side.

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Step 6.5.4.2.1

Simplify $\sqrt{100}$ .

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Step 6.5.4.2.1.1

Rewrite $100$ as $10^{2}$ .

$| x | > \sqrt{10^{2}}$

Step 6.5.4.2.1.2

Pull terms out from under the radical.

$| x | > | 10 |$

Step 6.5.4.2.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $10$ is $10$ .

$| x | > 10$

Step 6.5.5

Write $| x | > 10$ 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.

$x \geq 0$

Step 6.5.5.2

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

$x > 10$

Step 6.5.5.3

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

$x < 0$

Step 6.5.5.4

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

$- x > 10$

Step 6.5.5.5

Write as a piecewise.

${\begin{cases} x > 10 & x \geq 0 \\ - x > 10 & x < 0 \end{cases}$

Step 6.5.6

Find the intersection of $x > 10$ and $x \geq 0$ .

$x > 10$

Step 6.5.7

Divide each term in $- x > 10$ by $- 1$ and simplify.

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Step 6.5.7.1

Divide each term in $- x > 10$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} < \frac{10}{- 1}$

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.

$\frac{x}{1} < \frac{10}{- 1}$

Step 6.5.7.2.2

Divide $x$ by $1$ .

$x < \frac{10}{- 1}$

Step 6.5.7.3

Simplify the right side.

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Step 6.5.7.3.1

Divide $10$ by $- 1$ .

$x < - 10$

Step 6.5.8

Find the union of the solutions.

$x < - 10$ or $x > 10$

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 $0$ .

$x \leq - 10, x \geq 10$

$(- \infty, - 10] \cup [10, \infty)$

$x \leq - 10, x \geq 10$

$(- \infty, - 10] \cup [10, \infty)$

Step 7

Critical points to evaluate.

$x = 5 \sqrt{2}, - 5 \sqrt{2}, - 10, 10$

Step 8

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

$\frac{(8 (5 \sqrt{2})) ({(5 \sqrt{2})}^{2} - 150)}{{(- {(5 \sqrt{2})}^{2} + 100)}^{\frac{3}{2}}}$

Step 9

Evaluate the second derivative.

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Step 9.1

Multiply $8$ by $5$ .

$\frac{40 \sqrt{2} ({(5 \sqrt{2})}^{2} - 150)}{{(- {(5 \sqrt{2})}^{2} + 100)}^{\frac{3}{2}}}$

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

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

$\frac{40 \sqrt{2} ({(5 \sqrt{2})}^{2} - 150)}{{(- (5^{2} {\sqrt{2}}^{2}) + 100)}^{\frac{3}{2}}}$

Step 9.2.1.2

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

$\frac{40 \sqrt{2} ({(5 \sqrt{2})}^{2} - 150)}{{(- (25 {\sqrt{2}}^{2}) + 100)}^{\frac{3}{2}}}$

Step 9.2.1.3

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

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Step 9.2.1.3.1

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

$\frac{40 \sqrt{2} ({(5 \sqrt{2})}^{2} - 150)}{{(- (25 {(2^{\frac{1}{2}})}^{2}) + 100)}^{\frac{3}{2}}}$

Step 9.2.1.3.2

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

$\frac{40 \sqrt{2} ({(5 \sqrt{2})}^{2} - 150)}{{(- (25 \cdot 2^{\frac{1}{2} \cdot 2}) + 100)}^{\frac{3}{2}}}$

Step 9.2.1.3.3

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

$\frac{40 \sqrt{2} ({(5 \sqrt{2})}^{2} - 150)}{{(- (25 \cdot 2^{\frac{2}{2}}) + 100)}^{\frac{3}{2}}}$

Step 9.2.1.3.4

Cancel the common factor of $2$ .

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Step 9.2.1.3.4.1

Cancel the common factor.

$\frac{40 \sqrt{2} ({(5 \sqrt{2})}^{2} - 150)}{{(- (25 \cdot 2^{\frac{2}{2}}) + 100)}^{\frac{3}{2}}}$

Step 9.2.1.3.4.2

Rewrite the expression.

$\frac{40 \sqrt{2} ({(5 \sqrt{2})}^{2} - 150)}{{(- (25 \cdot 2^{1}) + 100)}^{\frac{3}{2}}}$

Step 9.2.1.3.5

Evaluate the exponent.

$\frac{40 \sqrt{2} ({(5 \sqrt{2})}^{2} - 150)}{{(- (25 \cdot 2) + 100)}^{\frac{3}{2}}}$

Step 9.2.1.4

Multiply $- (25 \cdot 2)$ .

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Step 9.2.1.4.1

Multiply $25$ by $2$ .

$\frac{40 \sqrt{2} ({(5 \sqrt{2})}^{2} - 150)}{{(- 1 \cdot 50 + 100)}^{\frac{3}{2}}}$

Step 9.2.1.4.2

Multiply $- 1$ by $50$ .

$\frac{40 \sqrt{2} ({(5 \sqrt{2})}^{2} - 150)}{{(- 50 + 100)}^{\frac{3}{2}}}$

Step 9.2.2

Add $- 50$ and $100$ .

$\frac{40 \sqrt{2} ({(5 \sqrt{2})}^{2} - 150)}{50^{\frac{3}{2}}}$

Step 9.3

Simplify the numerator.

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Step 9.3.1

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

$\frac{40 \sqrt{2} (5^{2} {\sqrt{2}}^{2} - 150)}{50^{\frac{3}{2}}}$

Step 9.3.2

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

$\frac{40 \sqrt{2} (25 {\sqrt{2}}^{2} - 150)}{50^{\frac{3}{2}}}$

Step 9.3.3

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

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Step 9.3.3.1

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

$\frac{40 \sqrt{2} (25 {(2^{\frac{1}{2}})}^{2} - 150)}{50^{\frac{3}{2}}}$

Step 9.3.3.2

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

$\frac{40 \sqrt{2} (25 \cdot 2^{\frac{1}{2} \cdot 2} - 150)}{50^{\frac{3}{2}}}$

Step 9.3.3.3

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

$\frac{40 \sqrt{2} (25 \cdot 2^{\frac{2}{2}} - 150)}{50^{\frac{3}{2}}}$

Step 9.3.3.4

Cancel the common factor of $2$ .

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Step 9.3.3.4.1

Cancel the common factor.

$\frac{40 \sqrt{2} (25 \cdot 2^{\frac{2}{2}} - 150)}{50^{\frac{3}{2}}}$

Step 9.3.3.4.2

Rewrite the expression.

$\frac{40 \sqrt{2} (25 \cdot 2^{1} - 150)}{50^{\frac{3}{2}}}$

Step 9.3.3.5

Evaluate the exponent.

$\frac{40 \sqrt{2} (25 \cdot 2 - 150)}{50^{\frac{3}{2}}}$

Step 9.3.4

Multiply $25$ by $2$ .

$\frac{40 \sqrt{2} (50 - 150)}{50^{\frac{3}{2}}}$

Step 9.3.5

Subtract $150$ from $50$ .

$\frac{40 \sqrt{2} \cdot - 100}{50^{\frac{3}{2}}}$

Step 9.3.6

Multiply $- 100$ by $40$ .

$\frac{- 4000 \sqrt{2}}{50^{\frac{3}{2}}}$

Step 9.4

Move the negative in front of the fraction.

$- \frac{4000 \sqrt{2}}{50^{\frac{3}{2}}}$

Step 10

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

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

Step 11

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

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Step 11.1

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

$f (5 \sqrt{2}) = 4 (5 \sqrt{2}) \sqrt{100 - {(5 \sqrt{2})}^{2}}$

Step 11.2

Simplify the result.

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Step 11.2.1

Simplify the expression.

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Step 11.2.1.1

Multiply $5$ by $4$ .

$f (5 \sqrt{2}) = 20 \sqrt{2} \sqrt{100 - {(5 \sqrt{2})}^{2}}$

Step 11.2.1.2

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

$f (5 \sqrt{2}) = 20 \sqrt{2} \sqrt{100 - (5^{2} {\sqrt{2}}^{2})}$

Step 11.2.1.3

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

$f (5 \sqrt{2}) = 20 \sqrt{2} \sqrt{100 - (25 {\sqrt{2}}^{2})}$

Step 11.2.2

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

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Step 11.2.2.1

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

$f (5 \sqrt{2}) = 20 \sqrt{2} \sqrt{100 - (25 {(2^{\frac{1}{2}})}^{2})}$

Step 11.2.2.2

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

$f (5 \sqrt{2}) = 20 \sqrt{2} \sqrt{100 - (25 \cdot 2^{\frac{1}{2} \cdot 2})}$

Step 11.2.2.3

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

$f (5 \sqrt{2}) = 20 \sqrt{2} \sqrt{100 - (25 \cdot 2^{\frac{2}{2}})}$

Step 11.2.2.4

Cancel the common factor of $2$ .

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Step 11.2.2.4.1

Cancel the common factor.

$f (5 \sqrt{2}) = 20 \sqrt{2} \sqrt{100 - (25 \cdot 2^{\frac{2}{2}})}$

Step 11.2.2.4.2

Rewrite the expression.

$f (5 \sqrt{2}) = 20 \sqrt{2} \sqrt{100 - (25 \cdot 2)}$

Step 11.2.2.5

Evaluate the exponent.

$f (5 \sqrt{2}) = 20 \sqrt{2} \sqrt{100 - (25 \cdot 2)}$

Step 11.2.3

Multiply $- (25 \cdot 2)$ .

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Step 11.2.3.1

Multiply $25$ by $2$ .

$f (5 \sqrt{2}) = 20 \sqrt{2} \sqrt{100 - 1 \cdot 50}$

Step 11.2.3.2

Multiply $- 1$ by $50$ .

$f (5 \sqrt{2}) = 20 \sqrt{2} \sqrt{100 - 50}$

Step 11.2.4

Subtract $50$ from $100$ .

$f (5 \sqrt{2}) = 20 \sqrt{2} \sqrt{50}$

Step 11.2.5

Rewrite $50$ as $5^{2} \cdot 2$ .

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Step 11.2.5.1

Factor $25$ out of $50$ .

$f (5 \sqrt{2}) = 20 \sqrt{2} \sqrt{25 (2)}$

Step 11.2.5.2

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

$f (5 \sqrt{2}) = 20 \sqrt{2} \sqrt{5^{2} \cdot 2}$

Step 11.2.6

Pull terms out from under the radical.

$f (5 \sqrt{2}) = 20 \sqrt{2} (5 \sqrt{2})$

Step 11.2.7

Multiply $20 \sqrt{2} (5 \sqrt{2})$ .

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Step 11.2.7.1

Multiply $5$ by $20$ .

$f (5 \sqrt{2}) = 100 \sqrt{2} \sqrt{2}$

Step 11.2.7.2

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

$f (5 \sqrt{2}) = 100 (\sqrt{2} \sqrt{2})$

Step 11.2.7.3

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

$f (5 \sqrt{2}) = 100 (\sqrt{2} \sqrt{2})$

Step 11.2.7.4

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

$f (5 \sqrt{2}) = 100 {\sqrt{2}}^{1 + 1}$

Step 11.2.7.5

Add $1$ and $1$ .

$f (5 \sqrt{2}) = 100 {\sqrt{2}}^{2}$

Step 11.2.8

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

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Step 11.2.8.1

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

$f (5 \sqrt{2}) = 100 {(2^{\frac{1}{2}})}^{2}$

Step 11.2.8.2

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

$f (5 \sqrt{2}) = 100 \cdot 2^{\frac{1}{2} \cdot 2}$

Step 11.2.8.3

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

$f (5 \sqrt{2}) = 100 \cdot 2^{\frac{2}{2}}$

Step 11.2.8.4

Cancel the common factor of $2$ .

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Step 11.2.8.4.1

Cancel the common factor.

$f (5 \sqrt{2}) = 100 \cdot 2^{\frac{2}{2}}$

Step 11.2.8.4.2

Rewrite the expression.

$f (5 \sqrt{2}) = 100 \cdot 2$

Step 11.2.8.5

Evaluate the exponent.

$f (5 \sqrt{2}) = 100 \cdot 2$

Step 11.2.9

Multiply $100$ by $2$ .

$f (5 \sqrt{2}) = 200$

Step 11.2.10

The final answer is $200$ .

$y = 200$

Step 12

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

$\frac{(8 (- 5 \sqrt{2})) ({(- 5 \sqrt{2})}^{2} - 150)}{{(- {(- 5 \sqrt{2})}^{2} + 100)}^{\frac{3}{2}}}$

Step 13

Evaluate the second derivative.

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Step 13.1

Multiply $- 5$ by $8$ .

$\frac{- 40 \sqrt{2} ({(- 5 \sqrt{2})}^{2} - 150)}{{(- {(- 5 \sqrt{2})}^{2} + 100)}^{\frac{3}{2}}}$

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

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

$\frac{- 40 \sqrt{2} ({(- 5 \sqrt{2})}^{2} - 150)}{{(- ({(- 5)}^{2} {\sqrt{2}}^{2}) + 100)}^{\frac{3}{2}}}$

Step 13.2.1.2

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

$\frac{- 40 \sqrt{2} ({(- 5 \sqrt{2})}^{2} - 150)}{{(- (25 {\sqrt{2}}^{2}) + 100)}^{\frac{3}{2}}}$

Step 13.2.1.3

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

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Step 13.2.1.3.1

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

$\frac{- 40 \sqrt{2} ({(- 5 \sqrt{2})}^{2} - 150)}{{(- (25 {(2^{\frac{1}{2}})}^{2}) + 100)}^{\frac{3}{2}}}$

Step 13.2.1.3.2

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

$\frac{- 40 \sqrt{2} ({(- 5 \sqrt{2})}^{2} - 150)}{{(- (25 \cdot 2^{\frac{1}{2} \cdot 2}) + 100)}^{\frac{3}{2}}}$

Step 13.2.1.3.3

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

$\frac{- 40 \sqrt{2} ({(- 5 \sqrt{2})}^{2} - 150)}{{(- (25 \cdot 2^{\frac{2}{2}}) + 100)}^{\frac{3}{2}}}$

Step 13.2.1.3.4

Cancel the common factor of $2$ .

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Step 13.2.1.3.4.1

Cancel the common factor.

$\frac{- 40 \sqrt{2} ({(- 5 \sqrt{2})}^{2} - 150)}{{(- (25 \cdot 2^{\frac{2}{2}}) + 100)}^{\frac{3}{2}}}$

Step 13.2.1.3.4.2

Rewrite the expression.

$\frac{- 40 \sqrt{2} ({(- 5 \sqrt{2})}^{2} - 150)}{{(- (25 \cdot 2^{1}) + 100)}^{\frac{3}{2}}}$

Step 13.2.1.3.5

Evaluate the exponent.

$\frac{- 40 \sqrt{2} ({(- 5 \sqrt{2})}^{2} - 150)}{{(- (25 \cdot 2) + 100)}^{\frac{3}{2}}}$

Step 13.2.1.4

Multiply $- (25 \cdot 2)$ .

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Step 13.2.1.4.1

Multiply $25$ by $2$ .

$\frac{- 40 \sqrt{2} ({(- 5 \sqrt{2})}^{2} - 150)}{{(- 1 \cdot 50 + 100)}^{\frac{3}{2}}}$

Step 13.2.1.4.2

Multiply $- 1$ by $50$ .

$\frac{- 40 \sqrt{2} ({(- 5 \sqrt{2})}^{2} - 150)}{{(- 50 + 100)}^{\frac{3}{2}}}$

Step 13.2.2

Add $- 50$ and $100$ .

$\frac{- 40 \sqrt{2} ({(- 5 \sqrt{2})}^{2} - 150)}{50^{\frac{3}{2}}}$

Step 13.3

Simplify the numerator.

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Step 13.3.1

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

$\frac{- 40 \sqrt{2} ({(- 5)}^{2} {\sqrt{2}}^{2} - 150)}{50^{\frac{3}{2}}}$

Step 13.3.2

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

$\frac{- 40 \sqrt{2} (25 {\sqrt{2}}^{2} - 150)}{50^{\frac{3}{2}}}$

Step 13.3.3

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

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Step 13.3.3.1

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

$\frac{- 40 \sqrt{2} (25 {(2^{\frac{1}{2}})}^{2} - 150)}{50^{\frac{3}{2}}}$

Step 13.3.3.2

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

$\frac{- 40 \sqrt{2} (25 \cdot 2^{\frac{1}{2} \cdot 2} - 150)}{50^{\frac{3}{2}}}$

Step 13.3.3.3

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

$\frac{- 40 \sqrt{2} (25 \cdot 2^{\frac{2}{2}} - 150)}{50^{\frac{3}{2}}}$

Step 13.3.3.4

Cancel the common factor of $2$ .

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Step 13.3.3.4.1

Cancel the common factor.

$\frac{- 40 \sqrt{2} (25 \cdot 2^{\frac{2}{2}} - 150)}{50^{\frac{3}{2}}}$

Step 13.3.3.4.2

Rewrite the expression.

$\frac{- 40 \sqrt{2} (25 \cdot 2^{1} - 150)}{50^{\frac{3}{2}}}$

Step 13.3.3.5

Evaluate the exponent.

$\frac{- 40 \sqrt{2} (25 \cdot 2 - 150)}{50^{\frac{3}{2}}}$

Step 13.3.4

Multiply $25$ by $2$ .

$\frac{- 40 \sqrt{2} (50 - 150)}{50^{\frac{3}{2}}}$

Step 13.3.5

Subtract $150$ from $50$ .

$\frac{- 40 \sqrt{2} \cdot - 100}{50^{\frac{3}{2}}}$

Step 13.3.6

Multiply $- 100$ by $- 40$ .

$\frac{4000 \sqrt{2}}{50^{\frac{3}{2}}}$

Step 14

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

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

Step 15

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

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Step 15.1

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

$f (- 5 \sqrt{2}) = 4 (- 5 \sqrt{2}) \sqrt{100 - {(- 5 \sqrt{2})}^{2}}$

Step 15.2

Simplify the result.

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Step 15.2.1

Simplify the expression.

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Step 15.2.1.1

Multiply $- 5$ by $4$ .

$f (- 5 \sqrt{2}) = - 20 \sqrt{2} \sqrt{100 - {(- 5 \sqrt{2})}^{2}}$

Step 15.2.1.2

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

$f (- 5 \sqrt{2}) = - 20 \sqrt{2} \sqrt{100 - ({(- 5)}^{2} {\sqrt{2}}^{2})}$

Step 15.2.1.3

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

$f (- 5 \sqrt{2}) = - 20 \sqrt{2} \sqrt{100 - (25 {\sqrt{2}}^{2})}$

Step 15.2.2

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

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Step 15.2.2.1

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

$f (- 5 \sqrt{2}) = - 20 \sqrt{2} \sqrt{100 - (25 {(2^{\frac{1}{2}})}^{2})}$

Step 15.2.2.2

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

$f (- 5 \sqrt{2}) = - 20 \sqrt{2} \sqrt{100 - (25 \cdot 2^{\frac{1}{2} \cdot 2})}$

Step 15.2.2.3

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

$f (- 5 \sqrt{2}) = - 20 \sqrt{2} \sqrt{100 - (25 \cdot 2^{\frac{2}{2}})}$

Step 15.2.2.4

Cancel the common factor of $2$ .

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Step 15.2.2.4.1

Cancel the common factor.

$f (- 5 \sqrt{2}) = - 20 \sqrt{2} \sqrt{100 - (25 \cdot 2^{\frac{2}{2}})}$

Step 15.2.2.4.2

Rewrite the expression.

$f (- 5 \sqrt{2}) = - 20 \sqrt{2} \sqrt{100 - (25 \cdot 2)}$

Step 15.2.2.5

Evaluate the exponent.

$f (- 5 \sqrt{2}) = - 20 \sqrt{2} \sqrt{100 - (25 \cdot 2)}$

Step 15.2.3

Multiply $- (25 \cdot 2)$ .

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Step 15.2.3.1

Multiply $25$ by $2$ .

$f (- 5 \sqrt{2}) = - 20 \sqrt{2} \sqrt{100 - 1 \cdot 50}$

Step 15.2.3.2

Multiply $- 1$ by $50$ .

$f (- 5 \sqrt{2}) = - 20 \sqrt{2} \sqrt{100 - 50}$

Step 15.2.4

Subtract $50$ from $100$ .

$f (- 5 \sqrt{2}) = - 20 \sqrt{2} \sqrt{50}$

Step 15.2.5

Rewrite $50$ as $5^{2} \cdot 2$ .

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Step 15.2.5.1

Factor $25$ out of $50$ .

$f (- 5 \sqrt{2}) = - 20 \sqrt{2} \sqrt{25 (2)}$

Step 15.2.5.2

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

$f (- 5 \sqrt{2}) = - 20 \sqrt{2} \sqrt{5^{2} \cdot 2}$

Step 15.2.6

Pull terms out from under the radical.

$f (- 5 \sqrt{2}) = - 20 \sqrt{2} (5 \sqrt{2})$

Step 15.2.7

Multiply $- 20 \sqrt{2} (5 \sqrt{2})$ .

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Step 15.2.7.1

Multiply $5$ by $- 20$ .

$f (- 5 \sqrt{2}) = - 100 \sqrt{2} \sqrt{2}$

Step 15.2.7.2

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

$f (- 5 \sqrt{2}) = - 100 (\sqrt{2} \sqrt{2})$

Step 15.2.7.3

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

$f (- 5 \sqrt{2}) = - 100 (\sqrt{2} \sqrt{2})$

Step 15.2.7.4

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

$f (- 5 \sqrt{2}) = - 100 {\sqrt{2}}^{1 + 1}$

Step 15.2.7.5

Add $1$ and $1$ .

$f (- 5 \sqrt{2}) = - 100 {\sqrt{2}}^{2}$

Step 15.2.8

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

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Step 15.2.8.1

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

$f (- 5 \sqrt{2}) = - 100 {(2^{\frac{1}{2}})}^{2}$

Step 15.2.8.2

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

$f (- 5 \sqrt{2}) = - 100 \cdot 2^{\frac{1}{2} \cdot 2}$

Step 15.2.8.3

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

$f (- 5 \sqrt{2}) = - 100 \cdot 2^{\frac{2}{2}}$

Step 15.2.8.4

Cancel the common factor of $2$ .

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Step 15.2.8.4.1

Cancel the common factor.

$f (- 5 \sqrt{2}) = - 100 \cdot 2^{\frac{2}{2}}$

Step 15.2.8.4.2

Rewrite the expression.

$f (- 5 \sqrt{2}) = - 100 \cdot 2$

Step 15.2.8.5

Evaluate the exponent.

$f (- 5 \sqrt{2}) = - 100 \cdot 2$

Step 15.2.9

Multiply $- 100$ by $2$ .

$f (- 5 \sqrt{2}) = - 200$

Step 15.2.10

The final answer is $- 200$ .

$y = - 200$

Step 16

Evaluate the second derivative at $x = - 10$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{(8 (- 10)) ({(- 10)}^{2} - 150)}{{(- {(- 10)}^{2} + 100)}^{\frac{3}{2}}}$

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 $- 10$ to the power of $2$ .

$\frac{(8 (- 10)) ({(- 10)}^{2} - 150)}{{(- 1 \cdot 100 + 100)}^{\frac{3}{2}}}$

Step 17.1.2

Multiply $- 1$ by $100$ .

$\frac{(8 (- 10)) ({(- 10)}^{2} - 150)}{{(- 100 + 100)}^{\frac{3}{2}}}$

Step 17.2

Reduce the expression by cancelling the common factors.

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Step 17.2.1

Add $- 100$ and $100$ .

$\frac{(8 (- 10)) ({(- 10)}^{2} - 150)}{0^{\frac{3}{2}}}$

Step 17.2.2

Simplify the expression.

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Step 17.2.2.1

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

$\frac{(8 (- 10)) ({(- 10)}^{2} - 150)}{{(0^{2})}^{\frac{3}{2}}}$

Step 17.2.2.2

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

$\frac{(8 (- 10)) ({(- 10)}^{2} - 150)}{0^{2 (\frac{3}{2})}}$

Step 17.2.3

Cancel the common factor of $2$ .

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Step 17.2.3.1

Cancel the common factor.

$\frac{(8 (- 10)) ({(- 10)}^{2} - 150)}{0^{2 (\frac{3}{2})}}$

Step 17.2.3.2

Rewrite the expression.

$\frac{(8 (- 10)) ({(- 10)}^{2} - 150)}{0^{3}}$

Step 17.2.4

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

$\frac{(8 (- 10)) ({(- 10)}^{2} - 150)}{0}$

Step 17.2.5

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

Undefined

$\frac{(8 (- 10)) ({(- 10)}^{2} - 150)}{0}$

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