Calculus Examples

$f (x) = 7 x \sqrt{3 - x}$

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 $\sqrt{3 - x}$ as ${(3 - x)}^{\frac{1}{2}}$ .

$\frac{d}{d x} [7 x {(3 - x)}^{\frac{1}{2}}]$

Step 1.1.2

Since $7$ is constant with respect to $x$ , the derivative of $7 x {(3 - x)}^{\frac{1}{2}}$ with respect to $x$ is $7 \frac{d}{d x} [x {(3 - x)}^{\frac{1}{2}}]$ .

$7 \frac{d}{d x} [x {(3 - x)}^{\frac{1}{2}}]$

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 $g (x) = {(3 - x)}^{\frac{1}{2}}$ .

$7 (x \frac{d}{d x} [{(3 - x)}^{\frac{1}{2}}] + {(3 - x)}^{\frac{1}{2}} \frac{d}{d x} [x])$

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 $g (x) = 3 - x$ .

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Step 1.3.1

To apply the Chain Rule, set $u$ as $3 - x$ .

$7 (x (\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [3 - x]) + {(3 - x)}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 1.3.2

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

$7 (x (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [3 - x]) + {(3 - x)}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 1.3.3

Replace all occurrences of $u$ with $3 - x$ .

$7 (x (\frac{1}{2} {(3 - x)}^{\frac{1}{2} - 1} \frac{d}{d x} [3 - x]) + {(3 - x)}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 1.4

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

$7 (x (\frac{1}{2} {(3 - x)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [3 - x]) + {(3 - x)}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 1.5

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

$7 (x (\frac{1}{2} {(3 - x)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [3 - x]) + {(3 - x)}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 1.6

Combine the numerators over the common denominator.

$7 (x (\frac{1}{2} {(3 - x)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [3 - x]) + {(3 - x)}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 1.7

Simplify the numerator.

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Step 1.7.1

Multiply $- 1$ by $2$ .

$7 (x (\frac{1}{2} {(3 - x)}^{\frac{1 - 2}{2}} \frac{d}{d x} [3 - x]) + {(3 - x)}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 1.7.2

Subtract $2$ from $1$ .

$7 (x (\frac{1}{2} {(3 - x)}^{\frac{- 1}{2}} \frac{d}{d x} [3 - x]) + {(3 - x)}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 1.8

Combine fractions.

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Step 1.8.1

Move the negative in front of the fraction.

$7 (x (\frac{1}{2} {(3 - x)}^{- \frac{1}{2}} \frac{d}{d x} [3 - x]) + {(3 - x)}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 1.8.2

Combine $\frac{1}{2}$ and ${(3 - x)}^{- \frac{1}{2}}$ .

$7 (x (\frac{{(3 - x)}^{- \frac{1}{2}}}{2} \frac{d}{d x} [3 - x]) + {(3 - x)}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 1.8.3

Move ${(3 - x)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$7 (x (\frac{1}{2 {(3 - x)}^{\frac{1}{2}}} \frac{d}{d x} [3 - x]) + {(3 - x)}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 1.8.4

Combine $\frac{1}{2 {(3 - x)}^{\frac{1}{2}}}$ and $x$ .

$7 (\frac{x}{2 {(3 - x)}^{\frac{1}{2}}} \frac{d}{d x} [3 - x] + {(3 - x)}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 1.9

By the Sum Rule, the derivative of $3 - x$ with respect to $x$ is $\frac{d}{d x} [3] + \frac{d}{d x} [- x]$ .

$7 (\frac{x}{2 {(3 - x)}^{\frac{1}{2}}} (\frac{d}{d x} [3] + \frac{d}{d x} [- x]) + {(3 - x)}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 1.10

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$7 (\frac{x}{2 {(3 - x)}^{\frac{1}{2}}} (0 + \frac{d}{d x} [- x]) + {(3 - x)}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 1.11

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

$7 (\frac{x}{2 {(3 - x)}^{\frac{1}{2}}} \frac{d}{d x} [- x] + {(3 - x)}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 1.12

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

$7 (\frac{x}{2 {(3 - x)}^{\frac{1}{2}}} (- \frac{d}{d x} [x]) + {(3 - x)}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 1.13

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

$7 (\frac{x}{2 {(3 - x)}^{\frac{1}{2}}} (- 1 \cdot 1) + {(3 - x)}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 1.14

Combine fractions.

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Step 1.14.1

Multiply $- 1$ by $1$ .

$7 (\frac{x}{2 {(3 - x)}^{\frac{1}{2}}} \cdot - 1 + {(3 - x)}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 1.14.2

Combine $\frac{x}{2 {(3 - x)}^{\frac{1}{2}}}$ and $- 1$ .

$7 (\frac{x \cdot - 1}{2 {(3 - x)}^{\frac{1}{2}}} + {(3 - x)}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 1.14.3

Simplify the expression.

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Step 1.14.3.1

Move $- 1$ to the left of $x$ .

$7 (\frac{- 1 \cdot x}{2 {(3 - x)}^{\frac{1}{2}}} + {(3 - x)}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 1.14.3.2

Rewrite $- 1 x$ as $- x$ .

$7 (\frac{- x}{2 {(3 - x)}^{\frac{1}{2}}} + {(3 - x)}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 1.14.3.3

Move the negative in front of the fraction.

$7 (- \frac{x}{2 {(3 - x)}^{\frac{1}{2}}} + {(3 - x)}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 1.15

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

$7 (- \frac{x}{2 {(3 - x)}^{\frac{1}{2}}} + {(3 - x)}^{\frac{1}{2}} \cdot 1)$

Step 1.16

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

$7 (- \frac{x}{2 {(3 - x)}^{\frac{1}{2}}} + {(3 - x)}^{\frac{1}{2}})$

Step 1.17

To write ${(3 - x)}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{2 {(3 - x)}^{\frac{1}{2}}}{2 {(3 - x)}^{\frac{1}{2}}}$ .

$7 (- \frac{x}{2 {(3 - x)}^{\frac{1}{2}}} + {(3 - x)}^{\frac{1}{2}} \cdot \frac{2 {(3 - x)}^{\frac{1}{2}}}{2 {(3 - x)}^{\frac{1}{2}}})$

Step 1.18

Combine ${(3 - x)}^{\frac{1}{2}}$ and $\frac{2 {(3 - x)}^{\frac{1}{2}}}{2 {(3 - x)}^{\frac{1}{2}}}$ .

$7 (- \frac{x}{2 {(3 - x)}^{\frac{1}{2}}} + \frac{{(3 - x)}^{\frac{1}{2}} (2 {(3 - x)}^{\frac{1}{2}})}{2 {(3 - x)}^{\frac{1}{2}}})$

Step 1.19

Combine the numerators over the common denominator.

$7 \frac{- x + {(3 - x)}^{\frac{1}{2}} (2 {(3 - x)}^{\frac{1}{2}})}{2 {(3 - x)}^{\frac{1}{2}}}$

Step 1.20

Multiply ${(3 - x)}^{\frac{1}{2}}$ by ${(3 - x)}^{\frac{1}{2}}$ by adding the exponents.

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Step 1.20.1

Move ${(3 - x)}^{\frac{1}{2}}$ .

$7 \frac{- x + {(3 - x)}^{\frac{1}{2}} {(3 - x)}^{\frac{1}{2}} \cdot 2}{2 {(3 - x)}^{\frac{1}{2}}}$

Step 1.20.2

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

$7 \frac{- x + {(3 - x)}^{\frac{1}{2} + \frac{1}{2}} \cdot 2}{2 {(3 - x)}^{\frac{1}{2}}}$

Step 1.20.3

Combine the numerators over the common denominator.

$7 \frac{- x + {(3 - x)}^{\frac{1 + 1}{2}} \cdot 2}{2 {(3 - x)}^{\frac{1}{2}}}$

Step 1.20.4

Add $1$ and $1$ .

$7 \frac{- x + {(3 - x)}^{\frac{2}{2}} \cdot 2}{2 {(3 - x)}^{\frac{1}{2}}}$

Step 1.20.5

Divide $2$ by $2$ .

$7 \frac{- x + {(3 - x)}^{1} \cdot 2}{2 {(3 - x)}^{\frac{1}{2}}}$

Step 1.21

Simplify the expression.

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Step 1.21.1

Simplify ${(3 - x)}^{1} \cdot 2$ .

$7 \frac{- x + (3 - x) \cdot 2}{2 {(3 - x)}^{\frac{1}{2}}}$

Step 1.21.2

Move $2$ to the left of $3 - x$ .

$7 \frac{- x + 2 \cdot (3 - x)}{2 {(3 - x)}^{\frac{1}{2}}}$

Step 1.22

Combine $7$ and $\frac{- x + 2 (3 - x)}{2 {(3 - x)}^{\frac{1}{2}}}$ .

$\frac{7 (- x + 2 (3 - x))}{2 {(3 - x)}^{\frac{1}{2}}}$

Step 1.23

Simplify.

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Step 1.23.1

Apply the distributive property.

$\frac{7 (- x + 2 \cdot 3 + 2 (- x))}{2 {(3 - x)}^{\frac{1}{2}}}$

Step 1.23.2

Apply the distributive property.

$\frac{7 (- x) + 7 (2 \cdot 3) + 7 (2 (- x))}{2 {(3 - x)}^{\frac{1}{2}}}$

Step 1.23.3

Simplify the numerator.

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Step 1.23.3.1

Simplify each term.

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Step 1.23.3.1.1

Multiply $- 1$ by $7$ .

$\frac{- 7 x + 7 (2 \cdot 3) + 7 (2 (- x))}{2 {(3 - x)}^{\frac{1}{2}}}$

Step 1.23.3.1.2

Multiply $7 (2 \cdot 3)$ .

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Step 1.23.3.1.2.1

Multiply $2$ by $3$ .

$\frac{- 7 x + 7 \cdot 6 + 7 (2 (- x))}{2 {(3 - x)}^{\frac{1}{2}}}$

Step 1.23.3.1.2.2

Multiply $7$ by $6$ .

$\frac{- 7 x + 42 + 7 (2 (- x))}{2 {(3 - x)}^{\frac{1}{2}}}$

Step 1.23.3.1.3

Multiply $- 1$ by $2$ .

$\frac{- 7 x + 42 + 7 (- 2 x)}{2 {(3 - x)}^{\frac{1}{2}}}$

Step 1.23.3.1.4

Multiply $- 2$ by $7$ .

$\frac{- 7 x + 42 - 14 x}{2 {(3 - x)}^{\frac{1}{2}}}$

Step 1.23.3.2

Subtract $14 x$ from $- 7 x$ .

$\frac{- 21 x + 42}{2 {(3 - x)}^{\frac{1}{2}}}$

Step 1.23.4

Factor $21$ out of $- 21 x + 42$ .

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Step 1.23.4.1

Factor $21$ out of $- 21 x$ .

$\frac{21 (- x) + 42}{2 {(3 - x)}^{\frac{1}{2}}}$

Step 1.23.4.2

Factor $21$ out of $42$ .

$\frac{21 (- x) + 21 (2)}{2 {(3 - x)}^{\frac{1}{2}}}$

Step 1.23.4.3

Factor $21$ out of $21 (- x) + 21 (2)$ .

$\frac{21 (- x + 2)}{2 {(3 - x)}^{\frac{1}{2}}}$

Step 1.23.5

Factor $- 1$ out of $- x$ .

$\frac{21 (- (x) + 2)}{2 {(3 - x)}^{\frac{1}{2}}}$

Step 1.23.6

Rewrite $2$ as $- 1 (- 2)$ .

$\frac{21 (- (x) - 1 (- 2))}{2 {(3 - x)}^{\frac{1}{2}}}$

Step 1.23.7

Factor $- 1$ out of $- (x) - 1 (- 2)$ .

$\frac{21 (- (x - 2))}{2 {(3 - x)}^{\frac{1}{2}}}$

Step 1.23.8

Rewrite $- (x - 2)$ as $- 1 (x - 2)$ .

$\frac{21 (- 1 (x - 2))}{2 {(3 - x)}^{\frac{1}{2}}}$

Step 1.23.9

Move the negative in front of the fraction.

$- \frac{21 (x - 2)}{2 {(3 - x)}^{\frac{1}{2}}}$

Step 2

Find the second derivative of the function.

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Step 2.1

Since $- \frac{21}{2}$ is constant with respect to $x$ , the derivative of $- \frac{21 (x - 2)}{2 {(3 - x)}^{\frac{1}{2}}}$ with respect to $x$ is $- \frac{21}{2} \frac{d}{d x} [\frac{x - 2}{{(3 - x)}^{\frac{1}{2}}}]$ .

$f'' (x) = - \frac{21}{2} \cdot \frac{d}{d x} \frac{x - 2}{{(3 - x)}^{\frac{1}{2}}}$

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 $f (x) = x - 2$ and $g (x) = {(3 - x)}^{\frac{1}{2}}$ .

$f'' (x) = - \frac{21}{2} \cdot \frac{{(3 - x)}^{\frac{1}{2}} \frac{d}{d x} (x - 2) - (x - 2) \frac{d}{d x} {(3 - x)}^{\frac{1}{2}}}{{({(3 - x)}^{\frac{1}{2}})}^{2}}$

Step 2.3

Multiply the exponents in ${({(3 - x)}^{\frac{1}{2}})}^{2}$ .

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Step 2.3.1

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

$f'' (x) = - \frac{21}{2} \cdot \frac{{(3 - x)}^{\frac{1}{2}} \frac{d}{d x} (x - 2) - (x - 2) \frac{d}{d x} {(3 - x)}^{\frac{1}{2}}}{{(3 - x)}^{\frac{1}{2} \cdot 2}}$

Step 2.3.2

Cancel the common factor of $2$ .

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Step 2.3.2.1

Cancel the common factor.

$f'' (x) = - \frac{21}{2} \cdot \frac{{(3 - x)}^{\frac{1}{2}} \frac{d}{d x} (x - 2) - (x - 2) \frac{d}{d x} {(3 - x)}^{\frac{1}{2}}}{{(3 - x)}^{\frac{1}{2} \cdot 2}}$

Step 2.3.2.2

Rewrite the expression.

$f'' (x) = - \frac{21}{2} \cdot \frac{{(3 - x)}^{\frac{1}{2}} \frac{d}{d x} (x - 2) - (x - 2) \frac{d}{d x} {(3 - x)}^{\frac{1}{2}}}{3 - x}$

Step 2.4

Simplify.

$f'' (x) = - \frac{21}{2} \cdot \frac{{(3 - x)}^{\frac{1}{2}} \frac{d}{d x} (x - 2) - (x - 2) \frac{d}{d x} {(3 - x)}^{\frac{1}{2}}}{3 - x}$

Step 2.5

Differentiate.

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Step 2.5.1

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

$f'' (x) = - \frac{21}{2} \cdot \frac{{(3 - x)}^{\frac{1}{2}} (\frac{d}{d x} (x) + \frac{d}{d x} (- 2)) - (x - 2) \frac{d}{d x} {(3 - x)}^{\frac{1}{2}}}{3 - x}$

Step 2.5.2

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

$f'' (x) = - \frac{21}{2} \cdot \frac{{(3 - x)}^{\frac{1}{2}} (1 + \frac{d}{d x} (- 2)) - (x - 2) \frac{d}{d x} {(3 - x)}^{\frac{1}{2}}}{3 - x}$

Step 2.5.3

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$f'' (x) = - \frac{21}{2} \cdot \frac{{(3 - x)}^{\frac{1}{2}} (1 + 0) - (x - 2) \frac{d}{d x} {(3 - x)}^{\frac{1}{2}}}{3 - x}$

Step 2.5.4

Simplify the expression.

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Step 2.5.4.1

Add $1$ and $0$ .

$f'' (x) = - \frac{21}{2} \cdot \frac{{(3 - x)}^{\frac{1}{2}} \cdot 1 - (x - 2) \frac{d}{d x} {(3 - x)}^{\frac{1}{2}}}{3 - x}$

Step 2.5.4.2

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

$f'' (x) = - \frac{21}{2} \cdot \frac{{(3 - x)}^{\frac{1}{2}} - (x - 2) \frac{d}{d x} {(3 - x)}^{\frac{1}{2}}}{3 - x}$

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 $g (x) = 3 - x$ .

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Step 2.6.1

To apply the Chain Rule, set $u_{1}$ as $3 - x$ .

$f'' (x) = - \frac{21}{2} \cdot \frac{{(3 - x)}^{\frac{1}{2}} - (x - 2) (\frac{d}{d u (1)} ({u_{1}}^{\frac{1}{2}}) \frac{d}{d x} (3 - x))}{3 - x}$

Step 2.6.2

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

$f'' (x) = - \frac{21}{2} \cdot \frac{{(3 - x)}^{\frac{1}{2}} - (x - 2) (\frac{1}{2} {u_{1}}^{\frac{1}{2} - 1} \frac{d}{d x} (3 - x))}{3 - x}$

Step 2.6.3

Replace all occurrences of $u_{1}$ with $3 - x$ .

$f'' (x) = - \frac{21}{2} \cdot \frac{{(3 - x)}^{\frac{1}{2}} - (x - 2) (\frac{1}{2} \cdot {(3 - x)}^{\frac{1}{2} - 1} \frac{d}{d x} (3 - x))}{3 - x}$

Step 2.7

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

$f'' (x) = - \frac{21}{2} \cdot \frac{{(3 - x)}^{\frac{1}{2}} - (x - 2) (\frac{1}{2} \cdot {(3 - x)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} (3 - x))}{3 - x}$

Step 2.8

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

$f'' (x) = - \frac{21}{2} \cdot \frac{{(3 - x)}^{\frac{1}{2}} - (x - 2) (\frac{1}{2} \cdot {(3 - x)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} (3 - x))}{3 - x}$

Step 2.9

Combine the numerators over the common denominator.

$f'' (x) = - \frac{21}{2} \cdot \frac{{(3 - x)}^{\frac{1}{2}} - (x - 2) (\frac{1}{2} \cdot {(3 - x)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} (3 - x))}{3 - x}$

Step 2.10

Simplify the numerator.

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Step 2.10.1

Multiply $- 1$ by $2$ .

$f'' (x) = - \frac{21}{2} \cdot \frac{{(3 - x)}^{\frac{1}{2}} - (x - 2) (\frac{1}{2} \cdot {(3 - x)}^{\frac{1 - 2}{2}} \frac{d}{d x} (3 - x))}{3 - x}$

Step 2.10.2

Subtract $2$ from $1$ .

$f'' (x) = - \frac{21}{2} \cdot \frac{{(3 - x)}^{\frac{1}{2}} - (x - 2) (\frac{1}{2} \cdot {(3 - x)}^{\frac{- 1}{2}} \frac{d}{d x} (3 - x))}{3 - x}$

Step 2.11

Combine fractions.

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Step 2.11.1

Move the negative in front of the fraction.

$f'' (x) = - \frac{21}{2} \cdot \frac{{(3 - x)}^{\frac{1}{2}} - (x - 2) (\frac{1}{2} \cdot {(3 - x)}^{- \frac{1}{2}} \frac{d}{d x} (3 - x))}{3 - x}$

Step 2.11.2

Combine $\frac{1}{2}$ and ${(3 - x)}^{- \frac{1}{2}}$ .

$f'' (x) = - \frac{21}{2} \cdot \frac{{(3 - x)}^{\frac{1}{2}} - (x - 2) (\frac{{(3 - x)}^{- \frac{1}{2}}}{2} \cdot \frac{d}{d x} (3 - x))}{3 - x}$

Step 2.11.3

Move ${(3 - x)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (x) = - \frac{21}{2} \cdot \frac{{(3 - x)}^{\frac{1}{2}} - (x - 2) (\frac{1}{2 {(3 - x)}^{\frac{1}{2}}} \cdot \frac{d}{d x} (3 - x))}{3 - x}$

Step 2.12

By the Sum Rule, the derivative of $3 - x$ with respect to $x$ is $\frac{d}{d x} [3] + \frac{d}{d x} [- x]$ .

$f'' (x) = - \frac{21}{2} \cdot \frac{{(3 - x)}^{\frac{1}{2}} - (x - 2) (\frac{1}{2 {(3 - x)}^{\frac{1}{2}}} \cdot (\frac{d}{d x} (3) + \frac{d}{d x} (- x)))}{3 - x}$

Step 2.13

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$f'' (x) = - \frac{21}{2} \cdot \frac{{(3 - x)}^{\frac{1}{2}} - (x - 2) (\frac{1}{2 {(3 - x)}^{\frac{1}{2}}} \cdot (0 + \frac{d}{d x} (- x)))}{3 - x}$

Step 2.14

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

$f'' (x) = - \frac{21}{2} \cdot \frac{{(3 - x)}^{\frac{1}{2}} - (x - 2) (\frac{1}{2 {(3 - x)}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- x))}{3 - x}$

Step 2.15

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

$f'' (x) = - \frac{21}{2} \cdot \frac{{(3 - x)}^{\frac{1}{2}} - (x - 2) (\frac{1}{2 {(3 - x)}^{\frac{1}{2}}} \cdot (- \frac{d}{d x} x))}{3 - x}$

Step 2.16

Multiply.

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Step 2.16.1

Multiply $- 1$ by $- 1$ .

$f'' (x) = - \frac{21}{2} \cdot \frac{{(3 - x)}^{\frac{1}{2}} + 1 (x - 2) (\frac{1}{2 {(3 - x)}^{\frac{1}{2}}} \cdot (\frac{d}{d x} (x)))}{3 - x}$

Step 2.16.2

Multiply $x - 2$ by $1$ .

$f'' (x) = - \frac{21}{2} \cdot \frac{{(3 - x)}^{\frac{1}{2}} + (x - 2) (\frac{1}{2 {(3 - x)}^{\frac{1}{2}}} \cdot (\frac{d}{d x} (x)))}{3 - x}$

Step 2.17

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

$f'' (x) = - \frac{21}{2} \cdot \frac{{(3 - x)}^{\frac{1}{2}} + (x - 2) (\frac{1}{2 {(3 - x)}^{\frac{1}{2}}}) \cdot 1}{3 - x}$

Step 2.18

Combine fractions.

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Step 2.18.1

Multiply $x - 2$ by $1$ .

$f'' (x) = - \frac{21}{2} \cdot \frac{{(3 - x)}^{\frac{1}{2}} + (x - 2) (\frac{1}{2 {(3 - x)}^{\frac{1}{2}}})}{3 - x}$

Step 2.18.2

Multiply $\frac{{(3 - x)}^{\frac{1}{2}} + (x - 2) \frac{1}{2 {(3 - x)}^{\frac{1}{2}}}}{3 - x}$ by $\frac{21}{2}$ .

$f'' (x) = - \frac{({(3 - x)}^{\frac{1}{2}} + (x - 2) (\frac{1}{2 {(3 - x)}^{\frac{1}{2}}})) \cdot 21}{(3 - x) \cdot 2}$

Step 2.18.3

Reorder.

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Step 2.18.3.1

Move $21$ to the left of ${(3 - x)}^{\frac{1}{2}} + (x - 2) \frac{1}{2 {(3 - x)}^{\frac{1}{2}}}$ .

$f'' (x) = - \frac{21 \cdot ({(3 - x)}^{\frac{1}{2}} + (x - 2) (\frac{1}{2 {(3 - x)}^{\frac{1}{2}}}))}{(3 - x) \cdot 2}$

Step 2.18.3.2

Move $2$ to the left of $3 - x$ .

$f'' (x) = - \frac{21 ({(3 - x)}^{\frac{1}{2}} + (x - 2) (\frac{1}{2 {(3 - x)}^{\frac{1}{2}}}))}{2 \cdot (3 - x)}$

Step 2.19

Simplify.

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Step 2.19.1

Apply the distributive property.

$f'' (x) = - \frac{21 {(3 - x)}^{\frac{1}{2}} + 21 ((x - 2) (\frac{1}{2 {(3 - x)}^{\frac{1}{2}}}))}{2 (3 - x)}$

Step 2.19.2

Apply the distributive property.

$f'' (x) = - \frac{21 {(3 - x)}^{\frac{1}{2}} + 21 ((x - 2) (\frac{1}{2 {(3 - x)}^{\frac{1}{2}}}))}{2 \cdot 3 + 2 (- x)}$

Step 2.19.3

Simplify the numerator.

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Step 2.19.3.1

Factor $21$ out of $21 {(3 - x)}^{\frac{1}{2}} + 21 (x - 2) \frac{1}{2 {(3 - x)}^{\frac{1}{2}}}$ .

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Step 2.19.3.1.1

Factor $21$ out of $21 {(3 - x)}^{\frac{1}{2}}$ .

$f'' (x) = - \frac{21 ({(3 - x)}^{\frac{1}{2}}) + 21 (x - 2) (\frac{1}{2 {(3 - x)}^{\frac{1}{2}}})}{2 \cdot 3 + 2 (- x)}$

Step 2.19.3.1.2

Factor $21$ out of $21 (x - 2) \frac{1}{2 {(3 - x)}^{\frac{1}{2}}}$ .

$f'' (x) = - \frac{21 ({(3 - x)}^{\frac{1}{2}}) + 21 ((x - 2) (\frac{1}{2 {(3 - x)}^{\frac{1}{2}}}))}{2 \cdot 3 + 2 (- x)}$

Step 2.19.3.1.3

Factor $21$ out of $21 ({(3 - x)}^{\frac{1}{2}}) + 21 ((x - 2) \frac{1}{2 {(3 - x)}^{\frac{1}{2}}})$ .

$f'' (x) = - \frac{21 ({(3 - x)}^{\frac{1}{2}} + (x - 2) (\frac{1}{2 {(3 - x)}^{\frac{1}{2}}}))}{2 \cdot 3 + 2 (- x)}$

Step 2.19.3.2

Let $u_{2} = {(3 - x)}^{\frac{1}{2}}$ . Substitute $u_{2}$ for all occurrences of ${(3 - x)}^{\frac{1}{2}}$ .

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Step 2.19.3.2.1

Rewrite using the commutative property of multiplication.

$f'' (x) = - \frac{21 (\frac{2 u_{2} u_{2} + x - 2}{2 u_{2}})}{2 \cdot 3 + 2 (- x)}$

Step 2.19.3.2.2

Multiply $u_{2}$ by $u_{2}$ by adding the exponents.

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Step 2.19.3.2.2.1

Move $u_{2}$ .

$f'' (x) = - \frac{21 (\frac{2 (u_{2} u_{2}) + x - 2}{2 u_{2}})}{2 \cdot 3 + 2 (- x)}$

Step 2.19.3.2.2.2

Multiply $u_{2}$ by $u_{2}$ .

$f'' (x) = - \frac{21 (\frac{2 {u_{2}}^{2} + x - 2}{2 u_{2}})}{2 \cdot 3 + 2 (- x)}$

Step 2.19.3.3

Replace all occurrences of $u_{2}$ with ${(3 - x)}^{\frac{1}{2}}$ .

$f'' (x) = - \frac{21 (\frac{2 {({(3 - x)}^{\frac{1}{2}})}^{2} + x - 2}{2 {(3 - x)}^{\frac{1}{2}}})}{2 \cdot 3 + 2 (- x)}$

Step 2.19.3.4

Simplify.

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Step 2.19.3.4.1

Simplify each term.

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Step 2.19.3.4.1.1

Multiply the exponents in ${({(3 - x)}^{\frac{1}{2}})}^{2}$ .

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Step 2.19.3.4.1.1.1

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

$f'' (x) = - \frac{21 (\frac{2 {(3 - x)}^{\frac{1}{2} \cdot 2} + x - 2}{2 {(3 - x)}^{\frac{1}{2}}})}{2 \cdot 3 + 2 (- x)}$

Step 2.19.3.4.1.1.2

Cancel the common factor of $2$ .

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Step 2.19.3.4.1.1.2.1

Cancel the common factor.

$f'' (x) = - \frac{21 (\frac{2 {(3 - x)}^{\frac{1}{2} \cdot 2} + x - 2}{2 {(3 - x)}^{\frac{1}{2}}})}{2 \cdot 3 + 2 (- x)}$

Step 2.19.3.4.1.1.2.2

Rewrite the expression.

$f'' (x) = - \frac{21 (\frac{2 (3 - x) + x - 2}{2 {(3 - x)}^{\frac{1}{2}}})}{2 \cdot 3 + 2 (- x)}$

Step 2.19.3.4.1.2

Simplify.

$f'' (x) = - \frac{21 (\frac{2 (3 - x) + x - 2}{2 {(3 - x)}^{\frac{1}{2}}})}{2 \cdot 3 + 2 (- x)}$

Step 2.19.3.4.1.3

Apply the distributive property.

$f'' (x) = - \frac{21 (\frac{2 \cdot 3 + 2 (- x) + x - 2}{2 {(3 - x)}^{\frac{1}{2}}})}{2 \cdot 3 + 2 (- x)}$

Step 2.19.3.4.1.4

Multiply $2$ by $3$ .

$f'' (x) = - \frac{21 (\frac{6 + 2 (- x) + x - 2}{2 {(3 - x)}^{\frac{1}{2}}})}{2 \cdot 3 + 2 (- x)}$

Step 2.19.3.4.1.5

Multiply $- 1$ by $2$ .

$f'' (x) = - \frac{21 (\frac{6 - 2 x + x - 2}{2 {(3 - x)}^{\frac{1}{2}}})}{2 \cdot 3 + 2 (- x)}$

Step 2.19.3.4.2

Subtract $2$ from $6$ .

$f'' (x) = - \frac{21 (\frac{- 2 x + x + 4}{2 {(3 - x)}^{\frac{1}{2}}})}{2 \cdot 3 + 2 (- x)}$

Step 2.19.3.4.3

Add $- 2 x$ and $x$ .

$f'' (x) = - \frac{21 (\frac{- x + 4}{2 {(3 - x)}^{\frac{1}{2}}})}{2 \cdot 3 + 2 (- x)}$

Step 2.19.4

Combine terms.

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Step 2.19.4.1

Combine $21$ and $\frac{- x + 4}{2 {(3 - x)}^{\frac{1}{2}}}$ .

$f'' (x) = - \frac{\frac{21 (- x + 4)}{2 {(3 - x)}^{\frac{1}{2}}}}{2 \cdot 3 + 2 (- x)}$

Step 2.19.4.2

Multiply $2$ by $3$ .

$f'' (x) = - \frac{\frac{21 (- x + 4)}{2 {(3 - x)}^{\frac{1}{2}}}}{6 + 2 (- x)}$

Step 2.19.4.3

Multiply $- 1$ by $2$ .

$f'' (x) = - \frac{\frac{21 (- x + 4)}{2 {(3 - x)}^{\frac{1}{2}}}}{6 - 2 x}$

Step 2.19.4.4

Rewrite $\frac{\frac{21 (- x + 4)}{2 {(3 - x)}^{\frac{1}{2}}}}{6 - 2 x}$ as a product.

$f'' (x) = - (\frac{21 (- x + 4)}{2 {(3 - x)}^{\frac{1}{2}}} \cdot \frac{1}{6 - 2 x})$

Step 2.19.4.5

Multiply $\frac{21 (- x + 4)}{2 {(3 - x)}^{\frac{1}{2}}}$ by $\frac{1}{6 - 2 x}$ .

$f'' (x) = - \frac{21 (- x + 4)}{2 {(3 - x)}^{\frac{1}{2}} (6 - 2 x)}$

Step 2.19.5

Simplify the denominator.

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Step 2.19.5.1

Factor $2$ out of $6 - 2 x$ .

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Step 2.19.5.1.1

Factor $2$ out of $6$ .

$f'' (x) = - \frac{21 (- x + 4)}{2 {(3 - x)}^{\frac{1}{2}} (2 (3) - 2 x)}$

Step 2.19.5.1.2

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

$f'' (x) = - \frac{21 (- x + 4)}{2 {(3 - x)}^{\frac{1}{2}} (2 (3) + 2 (- x))}$

Step 2.19.5.1.3

Factor $2$ out of $2 (3) + 2 (- x)$ .

$f'' (x) = - \frac{21 (- x + 4)}{2 {(3 - x)}^{\frac{1}{2}} (2 (3 - x))}$

$f'' (x) = - \frac{21 (- x + 4)}{2 {(3 - x)}^{\frac{1}{2}} \cdot (2 (3 - x))}$

Step 2.19.5.2

Combine exponents.

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Step 2.19.5.2.1

Multiply $2$ by $2$ .

$f'' (x) = - \frac{21 (- x + 4)}{4 {(3 - x)}^{\frac{1}{2}} (3 - x)}$

Step 2.19.5.2.2

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

$f'' (x) = - \frac{21 (- x + 4)}{4 ((3 - x) {(3 - x)}^{\frac{1}{2}})}$

Step 2.19.5.2.3

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

$f'' (x) = - \frac{21 (- x + 4)}{4 {(3 - x)}^{1 + \frac{1}{2}}}$

Step 2.19.5.2.4

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

$f'' (x) = - \frac{21 (- x + 4)}{4 {(3 - x)}^{\frac{2}{2} + \frac{1}{2}}}$

Step 2.19.5.2.5

Combine the numerators over the common denominator.

$f'' (x) = - \frac{21 (- x + 4)}{4 {(3 - x)}^{\frac{2 + 1}{2}}}$

Step 2.19.5.2.6

Add $2$ and $1$ .

$f'' (x) = - \frac{21 (- x + 4)}{4 {(3 - x)}^{\frac{3}{2}}}$

Step 2.19.6

Factor $- 1$ out of $- x$ .

$f'' (x) = - \frac{21 (- (x) + 4)}{4 {(3 - x)}^{\frac{3}{2}}}$

Step 2.19.7

Rewrite $4$ as $- 1 (- 4)$ .

$f'' (x) = - \frac{21 (- (x) - 1 \cdot - 4)}{4 {(3 - x)}^{\frac{3}{2}}}$

Step 2.19.8

Factor $- 1$ out of $- (x) - 1 (- 4)$ .

$f'' (x) = - \frac{21 (- (x - 4))}{4 {(3 - x)}^{\frac{3}{2}}}$

Step 2.19.9

Rewrite $- (x - 4)$ as $- 1 (x - 4)$ .

$f'' (x) = - \frac{21 (- 1 (x - 4))}{4 {(3 - x)}^{\frac{3}{2}}}$

Step 2.19.10

Move the negative in front of the fraction.

$f'' (x) = \frac{21 (x - 4)}{4 {(3 - x)}^{\frac{3}{2}}}$

Step 2.19.11

Multiply $- 1$ by $- 1$ .

$f'' (x) = 1 (\frac{21 (x - 4)}{4 {(3 - x)}^{\frac{3}{2}}})$

Step 2.19.12

Multiply $\frac{21 (x - 4)}{4 {(3 - x)}^{\frac{3}{2}}}$ by $1$ .

$f'' (x) = \frac{21 (x - 4)}{4 {(3 - x)}^{\frac{3}{2}}}$

Step 3

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

$- \frac{21 (x - 2)}{2 {(3 - x)}^{\frac{1}{2}}} = 0$

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 $\sqrt{3 - x}$ as ${(3 - x)}^{\frac{1}{2}}$ .

$f' (x) = \frac{d}{d x} (7 x {(3 - x)}^{\frac{1}{2}})$

Step 4.1.1.2

Since $7$ is constant with respect to $x$ , the derivative of $7 x {(3 - x)}^{\frac{1}{2}}$ with respect to $x$ is $7 \frac{d}{d x} [x {(3 - x)}^{\frac{1}{2}}]$ .

$f' (x) = 7 \frac{d}{d x} (x {(3 - x)}^{\frac{1}{2}})$

Step 4.1.2

$f' (x) = 7 (x \frac{d}{d x} ({(3 - x)}^{\frac{1}{2}}) + {(3 - x)}^{\frac{1}{2}} \frac{d}{d x} (x))$

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 $g (x) = 3 - x$ .

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Step 4.1.3.1

To apply the Chain Rule, set $u$ as $3 - x$ .

$f' (x) = 7 (x (\frac{d}{d u} (u^{\frac{1}{2}}) \frac{d}{d x} (3 - x)) + {(3 - x)}^{\frac{1}{2}} \frac{d}{d x} (x))$

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

$f' (x) = 7 (x (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} (3 - x)) + {(3 - x)}^{\frac{1}{2}} \frac{d}{d x} (x))$

Step 4.1.3.3

Replace all occurrences of $u$ with $3 - x$ .

$f' (x) = 7 (x (\frac{1}{2} \cdot {(3 - x)}^{\frac{1}{2} - 1} \frac{d}{d x} (3 - x)) + {(3 - x)}^{\frac{1}{2}} \frac{d}{d x} (x))$

Step 4.1.4

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

$f' (x) = 7 (x (\frac{1}{2} \cdot {(3 - x)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} (3 - x)) + {(3 - x)}^{\frac{1}{2}} \frac{d}{d x} (x))$

Step 4.1.5

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

$f' (x) = 7 (x (\frac{1}{2} \cdot {(3 - x)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} (3 - x)) + {(3 - x)}^{\frac{1}{2}} \frac{d}{d x} (x))$

Step 4.1.6

Combine the numerators over the common denominator.

$f' (x) = 7 (x (\frac{1}{2} \cdot {(3 - x)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} (3 - x)) + {(3 - x)}^{\frac{1}{2}} \frac{d}{d x} (x))$

Step 4.1.7

Simplify the numerator.

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Step 4.1.7.1

Multiply $- 1$ by $2$ .

$f' (x) = 7 (x (\frac{1}{2} \cdot {(3 - x)}^{\frac{1 - 2}{2}} \frac{d}{d x} (3 - x)) + {(3 - x)}^{\frac{1}{2}} \frac{d}{d x} (x))$

Step 4.1.7.2

Subtract $2$ from $1$ .

$f' (x) = 7 (x (\frac{1}{2} \cdot {(3 - x)}^{\frac{- 1}{2}} \frac{d}{d x} (3 - x)) + {(3 - x)}^{\frac{1}{2}} \frac{d}{d x} (x))$

Step 4.1.8

Combine fractions.

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Step 4.1.8.1

Move the negative in front of the fraction.

$f' (x) = 7 (x (\frac{1}{2} \cdot {(3 - x)}^{- \frac{1}{2}} \frac{d}{d x} (3 - x)) + {(3 - x)}^{\frac{1}{2}} \frac{d}{d x} (x))$

Step 4.1.8.2

Combine $\frac{1}{2}$ and ${(3 - x)}^{- \frac{1}{2}}$ .

$f' (x) = 7 (x (\frac{{(3 - x)}^{- \frac{1}{2}}}{2} \cdot \frac{d}{d x} (3 - x)) + {(3 - x)}^{\frac{1}{2}} \frac{d}{d x} (x))$

Step 4.1.8.3

Move ${(3 - x)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f' (x) = 7 (x (\frac{1}{2 {(3 - x)}^{\frac{1}{2}}} \cdot \frac{d}{d x} (3 - x)) + {(3 - x)}^{\frac{1}{2}} \frac{d}{d x} (x))$

Step 4.1.8.4

Combine $\frac{1}{2 {(3 - x)}^{\frac{1}{2}}}$ and $x$ .

$f' (x) = 7 (\frac{x}{2 {(3 - x)}^{\frac{1}{2}}} \cdot \frac{d}{d x} (3 - x) + {(3 - x)}^{\frac{1}{2}} \frac{d}{d x} (x))$

Step 4.1.9

By the Sum Rule, the derivative of $3 - x$ with respect to $x$ is $\frac{d}{d x} [3] + \frac{d}{d x} [- x]$ .

$f' (x) = 7 (\frac{x}{2 {(3 - x)}^{\frac{1}{2}}} \cdot (\frac{d}{d x} (3) + \frac{d}{d x} (- x)) + {(3 - x)}^{\frac{1}{2}} \frac{d}{d x} (x))$

Step 4.1.10

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$f' (x) = 7 (\frac{x}{2 {(3 - x)}^{\frac{1}{2}}} \cdot (0 + \frac{d}{d x} (- x)) + {(3 - x)}^{\frac{1}{2}} \frac{d}{d x} (x))$

Step 4.1.11

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

$f' (x) = 7 (\frac{x}{2 {(3 - x)}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- x) + {(3 - x)}^{\frac{1}{2}} \frac{d}{d x} (x))$

Step 4.1.12

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

$f' (x) = 7 (\frac{x}{2 {(3 - x)}^{\frac{1}{2}}} \cdot (- \frac{d}{d x} x) + {(3 - x)}^{\frac{1}{2}} \frac{d}{d x} (x))$

Step 4.1.13

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

$f' (x) = 7 (\frac{x}{2 {(3 - x)}^{\frac{1}{2}}} \cdot (- 1 \cdot 1) + {(3 - x)}^{\frac{1}{2}} \frac{d}{d x} (x))$

Step 4.1.14

Combine fractions.

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Step 4.1.14.1

Multiply $- 1$ by $1$ .

$f' (x) = 7 (\frac{x}{2 {(3 - x)}^{\frac{1}{2}}} \cdot - 1 + {(3 - x)}^{\frac{1}{2}} \frac{d}{d x} (x))$

Step 4.1.14.2

Combine $\frac{x}{2 {(3 - x)}^{\frac{1}{2}}}$ and $- 1$ .

$f' (x) = 7 (\frac{x \cdot - 1}{2 {(3 - x)}^{\frac{1}{2}}} + {(3 - x)}^{\frac{1}{2}} \frac{d}{d x} (x))$

Step 4.1.14.3

Simplify the expression.

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Step 4.1.14.3.1

Move $- 1$ to the left of $x$ .

$f' (x) = 7 (\frac{- 1 \cdot x}{2 {(3 - x)}^{\frac{1}{2}}} + {(3 - x)}^{\frac{1}{2}} \frac{d}{d x} (x))$

Step 4.1.14.3.2

Rewrite $- 1 x$ as $- x$ .

$f' (x) = 7 (\frac{- x}{2 {(3 - x)}^{\frac{1}{2}}} + {(3 - x)}^{\frac{1}{2}} \frac{d}{d x} (x))$

Step 4.1.14.3.3

Move the negative in front of the fraction.

$f' (x) = 7 (- \frac{x}{2 {(3 - x)}^{\frac{1}{2}}} + {(3 - x)}^{\frac{1}{2}} \frac{d}{d x} (x))$

Step 4.1.15

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

$f' (x) = 7 (- \frac{x}{2 {(3 - x)}^{\frac{1}{2}}} + {(3 - x)}^{\frac{1}{2}} \cdot 1)$

Step 4.1.16

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

$f' (x) = 7 (- \frac{x}{2 {(3 - x)}^{\frac{1}{2}}} + {(3 - x)}^{\frac{1}{2}})$

Step 4.1.17

To write ${(3 - x)}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{2 {(3 - x)}^{\frac{1}{2}}}{2 {(3 - x)}^{\frac{1}{2}}}$ .

$f' (x) = 7 (- \frac{x}{2 {(3 - x)}^{\frac{1}{2}}} + {(3 - x)}^{\frac{1}{2}} \cdot \frac{2 {(3 - x)}^{\frac{1}{2}}}{2 {(3 - x)}^{\frac{1}{2}}})$

Step 4.1.18

Combine ${(3 - x)}^{\frac{1}{2}}$ and $\frac{2 {(3 - x)}^{\frac{1}{2}}}{2 {(3 - x)}^{\frac{1}{2}}}$ .

$f' (x) = 7 (- \frac{x}{2 {(3 - x)}^{\frac{1}{2}}} + \frac{{(3 - x)}^{\frac{1}{2}} (2 {(3 - x)}^{\frac{1}{2}})}{2 {(3 - x)}^{\frac{1}{2}}})$

Step 4.1.19

Combine the numerators over the common denominator.

$f' (x) = 7 (\frac{- x + {(3 - x)}^{\frac{1}{2}} (2 {(3 - x)}^{\frac{1}{2}})}{2 {(3 - x)}^{\frac{1}{2}}})$

Step 4.1.20

Multiply ${(3 - x)}^{\frac{1}{2}}$ by ${(3 - x)}^{\frac{1}{2}}$ by adding the exponents.

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Step 4.1.20.1

Move ${(3 - x)}^{\frac{1}{2}}$ .

$f' (x) = 7 (\frac{- x + {(3 - x)}^{\frac{1}{2}} {(3 - x)}^{\frac{1}{2}} \cdot 2}{2 {(3 - x)}^{\frac{1}{2}}})$

Step 4.1.20.2

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

$f' (x) = 7 (\frac{- x + {(3 - x)}^{\frac{1}{2} + \frac{1}{2}} \cdot 2}{2 {(3 - x)}^{\frac{1}{2}}})$

Step 4.1.20.3

Combine the numerators over the common denominator.

$f' (x) = 7 (\frac{- x + {(3 - x)}^{\frac{1 + 1}{2}} \cdot 2}{2 {(3 - x)}^{\frac{1}{2}}})$

Step 4.1.20.4

Add $1$ and $1$ .

$f' (x) = 7 (\frac{- x + {(3 - x)}^{\frac{2}{2}} \cdot 2}{2 {(3 - x)}^{\frac{1}{2}}})$

Step 4.1.20.5

Divide $2$ by $2$ .

$f' (x) = 7 (\frac{- x + (3 - x) \cdot 2}{2 {(3 - x)}^{\frac{1}{2}}})$

Step 4.1.21

Simplify the expression.

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Step 4.1.21.1

Simplify ${(3 - x)}^{1} \cdot 2$ .

$f' (x) = 7 (\frac{- x + (3 - x) \cdot 2}{2 {(3 - x)}^{\frac{1}{2}}})$

Step 4.1.21.2

Move $2$ to the left of $3 - x$ .

$f' (x) = 7 (\frac{- x + 2 \cdot (3 - x)}{2 {(3 - x)}^{\frac{1}{2}}})$

Step 4.1.22

Combine $7$ and $\frac{- x + 2 (3 - x)}{2 {(3 - x)}^{\frac{1}{2}}}$ .

$f' (x) = \frac{7 (- x + 2 (3 - x))}{2 {(3 - x)}^{\frac{1}{2}}}$

Step 4.1.23

Simplify.

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Step 4.1.23.1

Apply the distributive property.

$f' (x) = \frac{7 (- x + 2 \cdot 3 + 2 (- x))}{2 {(3 - x)}^{\frac{1}{2}}}$

Step 4.1.23.2

Apply the distributive property.

$f' (x) = \frac{7 (- x) + 7 (2 \cdot 3) + 7 (2 (- x))}{2 {(3 - x)}^{\frac{1}{2}}}$

Step 4.1.23.3

Simplify the numerator.

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Step 4.1.23.3.1

Simplify each term.

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Step 4.1.23.3.1.1

Multiply $- 1$ by $7$ .

$f' (x) = \frac{- 7 x + 7 (2 \cdot 3) + 7 (2 (- x))}{2 {(3 - x)}^{\frac{1}{2}}}$

Step 4.1.23.3.1.2

Multiply $7 (2 \cdot 3)$ .

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Step 4.1.23.3.1.2.1

Multiply $2$ by $3$ .

$f' (x) = \frac{- 7 x + 7 \cdot 6 + 7 (2 (- x))}{2 {(3 - x)}^{\frac{1}{2}}}$

Step 4.1.23.3.1.2.2

Multiply $7$ by $6$ .

$f' (x) = \frac{- 7 x + 42 + 7 (2 (- x))}{2 {(3 - x)}^{\frac{1}{2}}}$

Step 4.1.23.3.1.3

Multiply $- 1$ by $2$ .

$f' (x) = \frac{- 7 x + 42 + 7 (- 2 x)}{2 {(3 - x)}^{\frac{1}{2}}}$

Step 4.1.23.3.1.4

Multiply $- 2$ by $7$ .

$f' (x) = \frac{- 7 x + 42 - 14 x}{2 {(3 - x)}^{\frac{1}{2}}}$

Step 4.1.23.3.2

Subtract $14 x$ from $- 7 x$ .

$f' (x) = \frac{- 21 x + 42}{2 {(3 - x)}^{\frac{1}{2}}}$

Step 4.1.23.4

Factor $21$ out of $- 21 x + 42$ .

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Step 4.1.23.4.1

Factor $21$ out of $- 21 x$ .

$f' (x) = \frac{21 (- x) + 42}{2 {(3 - x)}^{\frac{1}{2}}}$

Step 4.1.23.4.2

Factor $21$ out of $42$ .

$f' (x) = \frac{21 (- x) + 21 (2)}{2 {(3 - x)}^{\frac{1}{2}}}$

Step 4.1.23.4.3

Factor $21$ out of $21 (- x) + 21 (2)$ .

$f' (x) = \frac{21 (- x + 2)}{2 {(3 - x)}^{\frac{1}{2}}}$

Step 4.1.23.5

Factor $- 1$ out of $- x$ .

$f' (x) = \frac{21 (- (x) + 2)}{2 {(3 - x)}^{\frac{1}{2}}}$

Step 4.1.23.6

Rewrite $2$ as $- 1 (- 2)$ .

$f' (x) = \frac{21 (- (x) - 1 \cdot - 2)}{2 {(3 - x)}^{\frac{1}{2}}}$

Step 4.1.23.7

Factor $- 1$ out of $- (x) - 1 (- 2)$ .

$f' (x) = \frac{21 (- (x - 2))}{2 {(3 - x)}^{\frac{1}{2}}}$

Step 4.1.23.8

Rewrite $- (x - 2)$ as $- 1 (x - 2)$ .

$f' (x) = \frac{21 (- 1 (x - 2))}{2 {(3 - x)}^{\frac{1}{2}}}$

Step 4.1.23.9

Move the negative in front of the fraction.

$f' (x) = - \frac{21 (x - 2)}{2 {(3 - x)}^{\frac{1}{2}}}$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{21 (x - 2)}{2 {(3 - x)}^{\frac{1}{2}}}$ .

$- \frac{21 (x - 2)}{2 {(3 - x)}^{\frac{1}{2}}}$

Step 5

Set the first derivative equal to $0$ then solve the equation $- \frac{21 (x - 2)}{2 {(3 - x)}^{\frac{1}{2}}} = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$- \frac{21 (x - 2)}{2 {(3 - x)}^{\frac{1}{2}}} = 0$

Step 5.2

Set the numerator equal to zero.

$21 (x - 2) = 0$

Step 5.3

Solve the equation for $x$ .

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Step 5.3.1

Divide each term in $21 (x - 2) = 0$ by $21$ and simplify.

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Step 5.3.1.1

Divide each term in $21 (x - 2) = 0$ by $21$ .

$\frac{21 (x - 2)}{21} = \frac{0}{21}$

Step 5.3.1.2

Simplify the left side.

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Step 5.3.1.2.1

Cancel the common factor of $21$ .

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Step 5.3.1.2.1.1

Cancel the common factor.

$\frac{21 (x - 2)}{21} = \frac{0}{21}$

Step 5.3.1.2.1.2

Divide $x - 2$ by $1$ .

$x - 2 = \frac{0}{21}$

Step 5.3.1.3

Simplify the right side.

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Step 5.3.1.3.1

Divide $0$ by $21$ .

$x - 2 = 0$

Step 5.3.2

Add $2$ to both sides of the equation.

$x = 2$

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.

$- \frac{21 (x - 2)}{2 \sqrt{{(3 - x)}^{1}}}$

Step 6.1.2

Anything raised to $1$ is the base itself.

$- \frac{21 (x - 2)}{2 \sqrt{3 - x}}$

Step 6.2

Set the denominator in $\frac{21 (x - 2)}{2 \sqrt{3 - x}}$ equal to $0$ to find where the expression is undefined.

$2 \sqrt{3 - x} = 0$

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.

${(2 \sqrt{3 - x})}^{2} = 0^{2}$

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 $\sqrt{3 - x}$ as ${(3 - x)}^{\frac{1}{2}}$ .

${(2 {(3 - x)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 6.3.2.2

Simplify the left side.

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Step 6.3.2.2.1

Simplify ${(2 {(3 - x)}^{\frac{1}{2}})}^{2}$ .

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Step 6.3.2.2.1.1

Apply the product rule to $2 {(3 - x)}^{\frac{1}{2}}$ .

$2^{2} {({(3 - x)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 6.3.2.2.1.2

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

$4 {({(3 - x)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 6.3.2.2.1.3

Multiply the exponents in ${({(3 - x)}^{\frac{1}{2}})}^{2}$ .

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Step 6.3.2.2.1.3.1

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

$4 {(3 - x)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 6.3.2.2.1.3.2

Cancel the common factor of $2$ .

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Step 6.3.2.2.1.3.2.1

Cancel the common factor.

$4 {(3 - x)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 6.3.2.2.1.3.2.2

Rewrite the expression.

$4 {(3 - x)}^{1} = 0^{2}$

Step 6.3.2.2.1.4

Simplify.

$4 (3 - x) = 0^{2}$

Step 6.3.2.2.1.5

Apply the distributive property.

$4 \cdot 3 + 4 (- x) = 0^{2}$

Step 6.3.2.2.1.6

Multiply.

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Step 6.3.2.2.1.6.1

Multiply $4$ by $3$ .

$12 + 4 (- x) = 0^{2}$

Step 6.3.2.2.1.6.2

Multiply $- 1$ by $4$ .

$12 - 4 x = 0^{2}$

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 $0$ .

$12 - 4 x = 0$

Step 6.3.3

Solve for $x$ .

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Step 6.3.3.1

Subtract $12$ from both sides of the equation.

$- 4 x = - 12$

Step 6.3.3.2

Divide each term in $- 4 x = - 12$ by $- 4$ and simplify.

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Step 6.3.3.2.1

Divide each term in $- 4 x = - 12$ by $- 4$ .

$\frac{- 4 x}{- 4} = \frac{- 12}{- 4}$

Step 6.3.3.2.2

Simplify the left side.

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Step 6.3.3.2.2.1

Cancel the common factor of $- 4$ .

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Step 6.3.3.2.2.1.1

Cancel the common factor.

$\frac{- 4 x}{- 4} = \frac{- 12}{- 4}$

Step 6.3.3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 12}{- 4}$

Step 6.3.3.2.3

Simplify the right side.

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Step 6.3.3.2.3.1

Divide $- 12$ by $- 4$ .

$x = 3$

Step 6.4

Set the radicand in $\sqrt{3 - x}$ less than $0$ to find where the expression is undefined.

$3 - x < 0$

Step 6.5

Solve for $x$ .

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Step 6.5.1

Subtract $3$ from both sides of the inequality.

$- x < - 3$

Step 6.5.2

Divide each term in $- x < - 3$ by $- 1$ and simplify.

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Step 6.5.2.1

Divide each term in $- x < - 3$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} > \frac{- 3}{- 1}$

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.

$\frac{x}{1} > \frac{- 3}{- 1}$

Step 6.5.2.2.2

Divide $x$ by $1$ .

$x > \frac{- 3}{- 1}$

Step 6.5.2.3

Simplify the right side.

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Step 6.5.2.3.1

Divide $- 3$ by $- 1$ .

$x > 3$

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 $0$ .

$x \geq 3$

$[3, \infty)$

$x \geq 3$

$[3, \infty)$

Step 7

Critical points to evaluate.

$x = 2, 3$

Step 8

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

$\frac{21 ((2) - 4)}{4 {(3 - (2))}^{\frac{3}{2}}}$

Step 9

Evaluate the second derivative.

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Step 9.1

Factor $2$ out of $21 ((2) - 4)$ .

$\frac{2 (21 (1 - 2))}{4 {(3 - (2))}^{\frac{3}{2}}}$

Step 9.2

Cancel the common factors.

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Step 9.2.1

Factor $2$ out of $4 {(3 - (2))}^{\frac{3}{2}}$ .

$\frac{2 (21 (1 - 2))}{2 (2 {(3 - (2))}^{\frac{3}{2}})}$

Step 9.2.2

Cancel the common factor.

$\frac{2 (21 (1 - 2))}{2 (2 {(3 - (2))}^{\frac{3}{2}})}$

Step 9.2.3

Rewrite the expression.

$\frac{21 (1 - 2)}{2 {(3 - (2))}^{\frac{3}{2}}}$

Step 9.3

Subtract $2$ from $1$ .

$\frac{21 \cdot - 1}{2 {(3 - (2))}^{\frac{3}{2}}}$

Step 9.4

Simplify the denominator.

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Step 9.4.1

Multiply $- 1$ by $2$ .

$\frac{21 \cdot - 1}{2 {(3 - 2)}^{\frac{3}{2}}}$

Step 9.4.2

Subtract $2$ from $3$ .

$\frac{21 \cdot - 1}{2 \cdot 1^{\frac{3}{2}}}$

Step 9.4.3

One to any power is one.

$\frac{21 \cdot - 1}{2 \cdot 1}$

Step 9.5

Simplify the expression.

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Step 9.5.1

Multiply $21$ by $- 1$ .

$\frac{- 21}{2 \cdot 1}$

Step 9.5.2

Multiply $2$ by $1$ .

$\frac{- 21}{2}$

Step 9.5.3

Move the negative in front of the fraction.

$- \frac{21}{2}$

Step 10

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

$x = 2$ is a local maximum

Step 11

Find the y-value when $x = 2$ .

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Step 11.1

Replace the variable $x$ with $2$ in the expression.

$f (2) = 7 (2) \sqrt{3 - (2)}$

Step 11.2

Simplify the result.

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Step 11.2.1

Multiply $7$ by $2$ .

$f (2) = 14 \sqrt{3 - (2)}$

Step 11.2.2

Multiply $- 1$ by $2$ .

$f (2) = 14 \sqrt{3 - 2}$

Step 11.2.3

Subtract $2$ from $3$ .

$f (2) = 14 \sqrt{1}$

Step 11.2.4

Any root of $1$ is $1$ .

$f (2) = 14 \cdot 1$

Step 11.2.5

Multiply $14$ by $1$ .

$f (2) = 14$

Step 11.2.6

The final answer is $14$ .

$y = 14$

Step 12

Evaluate the second derivative at $x = 3$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{21 ((3) - 4)}{4 {(3 - (3))}^{\frac{3}{2}}}$

Step 13

Evaluate the second derivative.

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Step 13.1

Simplify the expression.

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Step 13.1.1

Multiply $- 1$ by $3$ .

$\frac{21 ((3) - 4)}{4 {(3 - 3)}^{\frac{3}{2}}}$

Step 13.1.2

Subtract $3$ from $3$ .

$\frac{21 ((3) - 4)}{4 \cdot 0^{\frac{3}{2}}}$

Step 13.1.3

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

$\frac{21 ((3) - 4)}{4 \cdot {(0^{2})}^{\frac{3}{2}}}$

Step 13.1.4

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

$\frac{21 ((3) - 4)}{4 \cdot 0^{2 (\frac{3}{2})}}$

Step 13.2

Cancel the common factor of $2$ .

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Step 13.2.1

Cancel the common factor.

$\frac{21 ((3) - 4)}{4 \cdot 0^{2 (\frac{3}{2})}}$

Step 13.2.2

Rewrite the expression.

$\frac{21 ((3) - 4)}{4 \cdot 0^{3}}$

Step 13.3

Simplify the expression.

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Step 13.3.1

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

$\frac{21 ((3) - 4)}{4 \cdot 0}$

Step 13.3.2

Multiply $4$ by $0$ .

$\frac{21 ((3) - 4)}{0}$

Step 13.3.3

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

Undefined

$\frac{21 ((3) - 4)}{0}$

Step 13.4

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

Undefined

Step 14

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

No Local Extrema

Step 15