Calculus Examples

$2 x \sqrt{4 - x}$

Step 1

Write $2 x \sqrt{4 - x}$ as a function.

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

Step 2

Find the first derivative of the function.

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Step 2.1

Differentiate using the Constant Multiple Rule.

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Step 2.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 - x}$ as ${(4 - x)}^{\frac{1}{2}}$ .

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

Step 2.1.2

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

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

Step 2.2

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 $g (x) = {(4 - x)}^{\frac{1}{2}}$ .

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

Step 2.3

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 $g (x) = 4 - x$ .

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Step 2.3.1

To apply the Chain Rule, set $u$ as $4 - x$ .

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

Step 2.3.2

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

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

Step 2.3.3

Replace all occurrences of $u$ with $4 - x$ .

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

Combine the numerators over the common denominator.

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

Step 2.7

Simplify the numerator.

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Step 2.7.1

Multiply $- 1$ by $2$ .

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

Step 2.7.2

Subtract $2$ from $1$ .

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

Step 2.8

Combine fractions.

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Step 2.8.1

Move the negative in front of the fraction.

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

Step 2.8.2

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

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

Step 2.8.3

Move ${(4 - x)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2.8.4

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

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

Step 2.9

By the Sum Rule, the derivative of $4 - x$ with respect to $x$ is $\frac{d}{d x} [4] + \frac{d}{d x} [- x]$ .

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

Step 2.10

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

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

Step 2.11

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

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

Step 2.12

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

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

Step 2.13

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

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

Step 2.14

Combine fractions.

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Step 2.14.1

Multiply $- 1$ by $1$ .

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

Step 2.14.2

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

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

Step 2.14.3

Simplify the expression.

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Step 2.14.3.1

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

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

Step 2.14.3.2

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

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

Step 2.14.3.3

Move the negative in front of the fraction.

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

Step 2.15

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

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

Step 2.16

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

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

Step 2.17

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

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

Step 2.18

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

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

Step 2.19

Combine the numerators over the common denominator.

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

Step 2.20

Multiply ${(4 - x)}^{\frac{1}{2}}$ by ${(4 - x)}^{\frac{1}{2}}$ by adding the exponents.

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Step 2.20.1

Move ${(4 - x)}^{\frac{1}{2}}$ .

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

Step 2.20.2

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

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

Step 2.20.3

Combine the numerators over the common denominator.

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

Step 2.20.4

Add $1$ and $1$ .

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

Step 2.20.5

Divide $2$ by $2$ .

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

Step 2.21

Simplify the expression.

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Step 2.21.1

Simplify ${(4 - x)}^{1} \cdot 2$ .

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

Step 2.21.2

Move $2$ to the left of $4 - x$ .

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

Step 2.22

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

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

Step 2.23

Cancel the common factor.

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

Step 2.24

Rewrite the expression.

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

Step 2.25

Simplify.

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Step 2.25.1

Apply the distributive property.

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

Step 2.25.2

Simplify the numerator.

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Step 2.25.2.1

Simplify each term.

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Step 2.25.2.1.1

Multiply $2$ by $4$ .

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

Step 2.25.2.1.2

Multiply $- 1$ by $2$ .

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

Step 2.25.2.2

Subtract $2 x$ from $- x$ .

$\frac{- 3 x + 8}{{(4 - x)}^{\frac{1}{2}}}$

Step 2.25.3

Reorder terms.

$\frac{- 3 x + 8}{{(- x + 4)}^{\frac{1}{2}}}$

Step 2.25.4

Factor $- 1$ out of $- 3 x$ .

$\frac{- (3 x) + 8}{{(- x + 4)}^{\frac{1}{2}}}$

Step 2.25.5

Rewrite $8$ as $- 1 (- 8)$ .

$\frac{- (3 x) - 1 (- 8)}{{(- x + 4)}^{\frac{1}{2}}}$

Step 2.25.6

Factor $- 1$ out of $- (3 x) - 1 (- 8)$ .

$\frac{- (3 x - 8)}{{(- x + 4)}^{\frac{1}{2}}}$

Step 2.25.7

Rewrite $- (3 x - 8)$ as $- 1 (3 x - 8)$ .

$\frac{- 1 (3 x - 8)}{{(- x + 4)}^{\frac{1}{2}}}$

Step 2.25.8

Move the negative in front of the fraction.

$- \frac{3 x - 8}{{(- x + 4)}^{\frac{1}{2}}}$

Step 3

Find the second derivative of the function.

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Step 3.1

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 $f (x) = - 1$ and $g (x) = \frac{3 x - 8}{{(- x + 4)}^{\frac{1}{2}}}$ .

$f'' (x) = - \frac{d}{d x} \cdot \frac{3 x - 8}{{(- x + 4)}^{\frac{1}{2}}} + \frac{3 x - 8}{{(- x + 4)}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.2

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 $f (x) = 3 x - 8$ and $g (x) = {(- x + 4)}^{\frac{1}{2}}$ .

$f'' (x) = - \frac{{(- x + 4)}^{\frac{1}{2}} \frac{d}{d x} (3 x - 8) - (3 x - 8) \frac{d}{d x} {(- x + 4)}^{\frac{1}{2}}}{{({(- x + 4)}^{\frac{1}{2}})}^{2}} + \frac{3 x - 8}{{(- x + 4)}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.3

Multiply the exponents in ${({(- x + 4)}^{\frac{1}{2}})}^{2}$ .

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Step 3.3.1

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

$f'' (x) = - \frac{{(- x + 4)}^{\frac{1}{2}} \frac{d}{d x} (3 x - 8) - (3 x - 8) \frac{d}{d x} {(- x + 4)}^{\frac{1}{2}}}{{(- x + 4)}^{\frac{1}{2} \cdot 2}} + \frac{3 x - 8}{{(- x + 4)}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.3.2

Cancel the common factor of $2$ .

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Step 3.3.2.1

Cancel the common factor.

Step 3.3.2.2

Rewrite the expression.

$f'' (x) = - \frac{{(- x + 4)}^{\frac{1}{2}} \frac{d}{d x} (3 x - 8) - (3 x - 8) \frac{d}{d x} {(- x + 4)}^{\frac{1}{2}}}{- x + 4} + \frac{3 x - 8}{{(- x + 4)}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.4

Simplify.

Step 3.5

Differentiate.

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Step 3.5.1

By the Sum Rule, the derivative of $3 x - 8$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [- 8]$ .

$f'' (x) = - \frac{{(- x + 4)}^{\frac{1}{2}} (\frac{d}{d x} (3 x) + \frac{d}{d x} (- 8)) - (3 x - 8) \frac{d}{d x} {(- x + 4)}^{\frac{1}{2}}}{- x + 4} + \frac{3 x - 8}{{(- x + 4)}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.5.2

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$f'' (x) = - \frac{{(- x + 4)}^{\frac{1}{2}} (3 \frac{d}{d x} (x) + \frac{d}{d x} (- 8)) - (3 x - 8) \frac{d}{d x} {(- x + 4)}^{\frac{1}{2}}}{- x + 4} + \frac{3 x - 8}{{(- x + 4)}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.5.3

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

$f'' (x) = - \frac{{(- x + 4)}^{\frac{1}{2}} (3 \cdot 1 + \frac{d}{d x} (- 8)) - (3 x - 8) \frac{d}{d x} {(- x + 4)}^{\frac{1}{2}}}{- x + 4} + \frac{3 x - 8}{{(- x + 4)}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.5.4

Multiply $3$ by $1$ .

$f'' (x) = - \frac{{(- x + 4)}^{\frac{1}{2}} (3 + \frac{d}{d x} (- 8)) - (3 x - 8) \frac{d}{d x} {(- x + 4)}^{\frac{1}{2}}}{- x + 4} + \frac{3 x - 8}{{(- x + 4)}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.5.5

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8$ with respect to $x$ is $0$ .

$f'' (x) = - \frac{{(- x + 4)}^{\frac{1}{2}} (3 + 0) - (3 x - 8) \frac{d}{d x} {(- x + 4)}^{\frac{1}{2}}}{- x + 4} + \frac{3 x - 8}{{(- x + 4)}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.5.6

Simplify the expression.

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Step 3.5.6.1

Add $3$ and $0$ .

$f'' (x) = - \frac{{(- x + 4)}^{\frac{1}{2}} \cdot 3 - (3 x - 8) \frac{d}{d x} {(- x + 4)}^{\frac{1}{2}}}{- x + 4} + \frac{3 x - 8}{{(- x + 4)}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.5.6.2

Move $3$ to the left of ${(- x + 4)}^{\frac{1}{2}}$ .

$f'' (x) = - \frac{3 \cdot {(- x + 4)}^{\frac{1}{2}} - (3 x - 8) \frac{d}{d x} {(- x + 4)}^{\frac{1}{2}}}{- x + 4} + \frac{3 x - 8}{{(- x + 4)}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.6

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 $g (x) = - x + 4$ .

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Step 3.6.1

To apply the Chain Rule, set $u_{1}$ as $- x + 4$ .

$f'' (x) = - \frac{3 {(- x + 4)}^{\frac{1}{2}} - (3 x - 8) (\frac{d}{d u (1)} ({u_{1}}^{\frac{1}{2}}) \frac{d}{d x} (- x + 4))}{- x + 4} + \frac{3 x - 8}{{(- x + 4)}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.6.2

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

$f'' (x) = - \frac{3 {(- x + 4)}^{\frac{1}{2}} - (3 x - 8) (\frac{1}{2} {u_{1}}^{\frac{1}{2} - 1} \frac{d}{d x} (- x + 4))}{- x + 4} + \frac{3 x - 8}{{(- x + 4)}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.6.3

Replace all occurrences of $u_{1}$ with $- x + 4$ .

$f'' (x) = - \frac{3 {(- x + 4)}^{\frac{1}{2}} - (3 x - 8) (\frac{1}{2} \cdot {(- x + 4)}^{\frac{1}{2} - 1} \frac{d}{d x} (- x + 4))}{- x + 4} + \frac{3 x - 8}{{(- x + 4)}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.7

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

$f'' (x) = - \frac{3 {(- x + 4)}^{\frac{1}{2}} - (3 x - 8) (\frac{1}{2} \cdot {(- x + 4)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} (- x + 4))}{- x + 4} + \frac{3 x - 8}{{(- x + 4)}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.8

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

$f'' (x) = - \frac{3 {(- x + 4)}^{\frac{1}{2}} - (3 x - 8) (\frac{1}{2} \cdot {(- x + 4)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} (- x + 4))}{- x + 4} + \frac{3 x - 8}{{(- x + 4)}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.9

Combine the numerators over the common denominator.

$f'' (x) = - \frac{3 {(- x + 4)}^{\frac{1}{2}} - (3 x - 8) (\frac{1}{2} \cdot {(- x + 4)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} (- x + 4))}{- x + 4} + \frac{3 x - 8}{{(- x + 4)}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.10

Simplify the numerator.

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Step 3.10.1

Multiply $- 1$ by $2$ .

$f'' (x) = - \frac{3 {(- x + 4)}^{\frac{1}{2}} - (3 x - 8) (\frac{1}{2} \cdot {(- x + 4)}^{\frac{1 - 2}{2}} \frac{d}{d x} (- x + 4))}{- x + 4} + \frac{3 x - 8}{{(- x + 4)}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.10.2

Subtract $2$ from $1$ .

$f'' (x) = - \frac{3 {(- x + 4)}^{\frac{1}{2}} - (3 x - 8) (\frac{1}{2} \cdot {(- x + 4)}^{\frac{- 1}{2}} \frac{d}{d x} (- x + 4))}{- x + 4} + \frac{3 x - 8}{{(- x + 4)}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.11

Combine fractions.

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Step 3.11.1

Move the negative in front of the fraction.

$f'' (x) = - \frac{3 {(- x + 4)}^{\frac{1}{2}} - (3 x - 8) (\frac{1}{2} \cdot {(- x + 4)}^{- \frac{1}{2}} \frac{d}{d x} (- x + 4))}{- x + 4} + \frac{3 x - 8}{{(- x + 4)}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.11.2

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

$f'' (x) = - \frac{3 {(- x + 4)}^{\frac{1}{2}} - (3 x - 8) (\frac{{(- x + 4)}^{- \frac{1}{2}}}{2} \cdot \frac{d}{d x} (- x + 4))}{- x + 4} + \frac{3 x - 8}{{(- x + 4)}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.11.3

Move ${(- x + 4)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (x) = - \frac{3 {(- x + 4)}^{\frac{1}{2}} - (3 x - 8) (\frac{1}{2 {(- x + 4)}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- x + 4))}{- x + 4} + \frac{3 x - 8}{{(- x + 4)}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.12

By the Sum Rule, the derivative of $- x + 4$ with respect to $x$ is $\frac{d}{d x} [- x] + \frac{d}{d x} [4]$ .

$f'' (x) = - \frac{3 {(- x + 4)}^{\frac{1}{2}} - (3 x - 8) (\frac{1}{2 {(- x + 4)}^{\frac{1}{2}}} \cdot (\frac{d}{d x} (- x) + \frac{d}{d x} (4)))}{- x + 4} + \frac{3 x - 8}{{(- x + 4)}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.13

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

$f'' (x) = - \frac{3 {(- x + 4)}^{\frac{1}{2}} - (3 x - 8) (\frac{1}{2 {(- x + 4)}^{\frac{1}{2}}} \cdot (- \frac{d}{d x} x + \frac{d}{d x} (4)))}{- x + 4} + \frac{3 x - 8}{{(- x + 4)}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.14

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

$f'' (x) = - \frac{3 {(- x + 4)}^{\frac{1}{2}} - (3 x - 8) (\frac{1}{2 {(- x + 4)}^{\frac{1}{2}}} \cdot (- 1 \cdot 1 + \frac{d}{d x} (4)))}{- x + 4} + \frac{3 x - 8}{{(- x + 4)}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.15

Multiply $- 1$ by $1$ .

$f'' (x) = - \frac{3 {(- x + 4)}^{\frac{1}{2}} - (3 x - 8) (\frac{1}{2 {(- x + 4)}^{\frac{1}{2}}} \cdot (- 1 + \frac{d}{d x} (4)))}{- x + 4} + \frac{3 x - 8}{{(- x + 4)}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.16

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$f'' (x) = - \frac{3 {(- x + 4)}^{\frac{1}{2}} - (3 x - 8) (\frac{1}{2 {(- x + 4)}^{\frac{1}{2}}} \cdot (- 1 + 0))}{- x + 4} + \frac{3 x - 8}{{(- x + 4)}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.17

Combine fractions.

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Step 3.17.1

Add $- 1$ and $0$ .

$f'' (x) = - \frac{3 {(- x + 4)}^{\frac{1}{2}} - (3 x - 8) (\frac{1}{2 {(- x + 4)}^{\frac{1}{2}}} \cdot - 1)}{- x + 4} + \frac{3 x - 8}{{(- x + 4)}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.17.2

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

$f'' (x) = - \frac{3 {(- x + 4)}^{\frac{1}{2}} - (3 x - 8) \frac{- 1}{2 {(- x + 4)}^{\frac{1}{2}}}}{- x + 4} + \frac{3 x - 8}{{(- x + 4)}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.17.3

Simplify the expression.

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Step 3.17.3.1

Move the negative in front of the fraction.

$f'' (x) = - \frac{3 {(- x + 4)}^{\frac{1}{2}} - (3 x - 8) (- \frac{1}{2 {(- x + 4)}^{\frac{1}{2}}})}{- x + 4} + \frac{3 x - 8}{{(- x + 4)}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.17.3.2

Multiply $- 1$ by $- 1$ .

$f'' (x) = - \frac{3 {(- x + 4)}^{\frac{1}{2}} + 1 (3 x - 8) (\frac{1}{2 {(- x + 4)}^{\frac{1}{2}}})}{- x + 4} + \frac{3 x - 8}{{(- x + 4)}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.17.3.3

Multiply $3 x - 8$ by $1$ .

$f'' (x) = - \frac{3 {(- x + 4)}^{\frac{1}{2}} + (3 x - 8) (\frac{1}{2 {(- x + 4)}^{\frac{1}{2}}})}{- x + 4} + \frac{3 x - 8}{{(- x + 4)}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.18

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$f'' (x) = - \frac{3 {(- x + 4)}^{\frac{1}{2}} + (3 x - 8) (\frac{1}{2 {(- x + 4)}^{\frac{1}{2}}})}{- x + 4} + \frac{3 x - 8}{{(- x + 4)}^{\frac{1}{2}}} \cdot 0$

Step 3.19

Simplify the expression.

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Step 3.19.1

Multiply $\frac{3 x - 8}{{(- x + 4)}^{\frac{1}{2}}}$ by $0$ .

$f'' (x) = - \frac{3 {(- x + 4)}^{\frac{1}{2}} + (3 x - 8) (\frac{1}{2 {(- x + 4)}^{\frac{1}{2}}})}{- x + 4} + 0$

Step 3.19.2

Add $- \frac{3 {(- x + 4)}^{\frac{1}{2}} + (3 x - 8) \frac{1}{2 {(- x + 4)}^{\frac{1}{2}}}}{- x + 4}$ and $0$ .

$f'' (x) = - \frac{3 {(- x + 4)}^{\frac{1}{2}} + (3 x - 8) (\frac{1}{2 {(- x + 4)}^{\frac{1}{2}}})}{- x + 4}$

Step 3.20

Simplify.

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Step 3.20.1

Simplify the numerator.

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Step 3.20.1.1

Let $u_{2} = {(- x + 4)}^{\frac{1}{2}}$ . Substitute $u_{2}$ for all occurrences of ${(- x + 4)}^{\frac{1}{2}}$ .

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Step 3.20.1.1.1

Rewrite using the commutative property of multiplication.

$f'' (x) = - \frac{\frac{3 \cdot (2 u_{2} u_{2}) + 3 x - 8}{2 u_{2}}}{- x + 4}$

Step 3.20.1.1.2

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

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Step 3.20.1.1.2.1

Move $u_{2}$ .

$f'' (x) = - \frac{\frac{3 \cdot (2 (u_{2} u_{2})) + 3 x - 8}{2 u_{2}}}{- x + 4}$

Step 3.20.1.1.2.2

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

$f'' (x) = - \frac{\frac{3 \cdot (2 {u_{2}}^{2}) + 3 x - 8}{2 u_{2}}}{- x + 4}$

Step 3.20.1.1.3

Multiply $3$ by $2$ .

$f'' (x) = - \frac{\frac{6 {u_{2}}^{2} + 3 x - 8}{2 u_{2}}}{- x + 4}$

Step 3.20.1.2

Replace all occurrences of $u_{2}$ with ${(- x + 4)}^{\frac{1}{2}}$ .

$f'' (x) = - \frac{\frac{6 {({(- x + 4)}^{\frac{1}{2}})}^{2} + 3 x - 8}{2 {(- x + 4)}^{\frac{1}{2}}}}{- x + 4}$

Step 3.20.1.3

Simplify.

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Step 3.20.1.3.1

Simplify each term.

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Step 3.20.1.3.1.1

Multiply the exponents in ${({(- x + 4)}^{\frac{1}{2}})}^{2}$ .

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Step 3.20.1.3.1.1.1

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

$f'' (x) = - \frac{\frac{6 {(- x + 4)}^{\frac{1}{2} \cdot 2} + 3 x - 8}{2 {(- x + 4)}^{\frac{1}{2}}}}{- x + 4}$

Step 3.20.1.3.1.1.2

Cancel the common factor of $2$ .

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Step 3.20.1.3.1.1.2.1

Cancel the common factor.

$f'' (x) = - \frac{\frac{6 {(- x + 4)}^{\frac{1}{2} \cdot 2} + 3 x - 8}{2 {(- x + 4)}^{\frac{1}{2}}}}{- x + 4}$

Step 3.20.1.3.1.1.2.2

Rewrite the expression.

$f'' (x) = - \frac{\frac{6 (- x + 4) + 3 x - 8}{2 {(- x + 4)}^{\frac{1}{2}}}}{- x + 4}$

Step 3.20.1.3.1.2

Simplify.

$f'' (x) = - \frac{\frac{6 (- x + 4) + 3 x - 8}{2 {(- x + 4)}^{\frac{1}{2}}}}{- x + 4}$

Step 3.20.1.3.1.3

Apply the distributive property.

$f'' (x) = - \frac{\frac{6 (- x) + 6 \cdot 4 + 3 x - 8}{2 {(- x + 4)}^{\frac{1}{2}}}}{- x + 4}$

Step 3.20.1.3.1.4

Multiply $- 1$ by $6$ .

$f'' (x) = - \frac{\frac{- 6 x + 6 \cdot 4 + 3 x - 8}{2 {(- x + 4)}^{\frac{1}{2}}}}{- x + 4}$

Step 3.20.1.3.1.5

Multiply $6$ by $4$ .

$f'' (x) = - \frac{\frac{- 6 x + 24 + 3 x - 8}{2 {(- x + 4)}^{\frac{1}{2}}}}{- x + 4}$

Step 3.20.1.3.2

Add $- 6 x$ and $3 x$ .

$f'' (x) = - \frac{\frac{- 3 x + 24 - 8}{2 {(- x + 4)}^{\frac{1}{2}}}}{- x + 4}$

Step 3.20.1.3.3

Subtract $8$ from $24$ .

$f'' (x) = - \frac{\frac{- 3 x + 16}{2 {(- x + 4)}^{\frac{1}{2}}}}{- x + 4}$

Step 3.20.2

Combine terms.

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Step 3.20.2.1

Rewrite $\frac{\frac{- 3 x + 16}{2 {(- x + 4)}^{\frac{1}{2}}}}{- x + 4}$ as a product.

$f'' (x) = - (\frac{- 3 x + 16}{2 {(- x + 4)}^{\frac{1}{2}}} \cdot \frac{1}{- x + 4})$

Step 3.20.2.2

Multiply $\frac{- 3 x + 16}{2 {(- x + 4)}^{\frac{1}{2}}}$ by $\frac{1}{- x + 4}$ .

$f'' (x) = - \frac{- 3 x + 16}{2 {(- x + 4)}^{\frac{1}{2}} (- x + 4)}$

Step 3.20.2.3

Raise $- x + 4$ to the power of $1$ .

$f'' (x) = - \frac{- 3 x + 16}{2 ((- x + 4) {(- x + 4)}^{\frac{1}{2}})}$

Step 3.20.2.4

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

$f'' (x) = - \frac{- 3 x + 16}{2 {(- x + 4)}^{1 + \frac{1}{2}}}$

Step 3.20.2.5

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

$f'' (x) = - \frac{- 3 x + 16}{2 {(- x + 4)}^{\frac{2}{2} + \frac{1}{2}}}$

Step 3.20.2.6

Combine the numerators over the common denominator.

$f'' (x) = - \frac{- 3 x + 16}{2 {(- x + 4)}^{\frac{2 + 1}{2}}}$

Step 3.20.2.7

Add $2$ and $1$ .

$f'' (x) = - \frac{- 3 x + 16}{2 {(- x + 4)}^{\frac{3}{2}}}$

Step 3.20.3

Factor $- 1$ out of $- 3 x$ .

$f'' (x) = - \frac{- (3 x) + 16}{2 {(- x + 4)}^{\frac{3}{2}}}$

Step 3.20.4

Rewrite $16$ as $- 1 (- 16)$ .

$f'' (x) = - \frac{- (3 x) - 1 \cdot - 16}{2 {(- x + 4)}^{\frac{3}{2}}}$

Step 3.20.5

Factor $- 1$ out of $- (3 x) - 1 (- 16)$ .

$f'' (x) = - \frac{- (3 x - 16)}{2 {(- x + 4)}^{\frac{3}{2}}}$

Step 3.20.6

Rewrite $- (3 x - 16)$ as $- 1 (3 x - 16)$ .

$f'' (x) = - \frac{- 1 (3 x - 16)}{2 {(- x + 4)}^{\frac{3}{2}}}$

Step 3.20.7

Move the negative in front of the fraction.

$f'' (x) = \frac{3 x - 16}{2 {(- x + 4)}^{\frac{3}{2}}}$

Step 3.20.8

Multiply $- 1$ by $- 1$ .

$f'' (x) = 1 (\frac{3 x - 16}{2 {(- x + 4)}^{\frac{3}{2}}})$

Step 3.20.9

Multiply $\frac{3 x - 16}{2 {(- x + 4)}^{\frac{3}{2}}}$ by $1$ .

$f'' (x) = \frac{3 x - 16}{2 {(- x + 4)}^{\frac{3}{2}}}$

Step 4

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

$- \frac{3 x - 8}{{(- x + 4)}^{\frac{1}{2}}} = 0$

Step 5

Find the first derivative.

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Step 5.1

Find the first derivative.

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Step 5.1.1

Differentiate using the Constant Multiple Rule.

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Step 5.1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 - x}$ as ${(4 - x)}^{\frac{1}{2}}$ .

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

Step 5.1.1.2

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

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

Step 5.1.2

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

Step 5.1.3

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 $g (x) = 4 - x$ .

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Step 5.1.3.1

To apply the Chain Rule, set $u$ as $4 - x$ .

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

Step 5.1.3.2

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

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

Step 5.1.3.3

Replace all occurrences of $u$ with $4 - x$ .

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

Step 5.1.4

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

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

Step 5.1.5

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

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

Step 5.1.6

Combine the numerators over the common denominator.

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

Step 5.1.7

Simplify the numerator.

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Step 5.1.7.1

Multiply $- 1$ by $2$ .

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

Step 5.1.7.2

Subtract $2$ from $1$ .

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

Step 5.1.8

Combine fractions.

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Step 5.1.8.1

Move the negative in front of the fraction.

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

Step 5.1.8.2

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

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

Step 5.1.8.3

Move ${(4 - x)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 5.1.8.4

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

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

Step 5.1.9

By the Sum Rule, the derivative of $4 - x$ with respect to $x$ is $\frac{d}{d x} [4] + \frac{d}{d x} [- x]$ .

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

Step 5.1.10

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

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

Step 5.1.11

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

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

Step 5.1.12

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

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

Step 5.1.13

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

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

Step 5.1.14

Combine fractions.

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Step 5.1.14.1

Multiply $- 1$ by $1$ .

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

Step 5.1.14.2

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

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

Step 5.1.14.3

Simplify the expression.

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Step 5.1.14.3.1

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

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

Step 5.1.14.3.2

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

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

Step 5.1.14.3.3

Move the negative in front of the fraction.

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

Step 5.1.15

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

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

Step 5.1.16

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

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

Step 5.1.17

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

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

Step 5.1.18

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

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

Step 5.1.19

Combine the numerators over the common denominator.

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

Step 5.1.20

Multiply ${(4 - x)}^{\frac{1}{2}}$ by ${(4 - x)}^{\frac{1}{2}}$ by adding the exponents.

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Step 5.1.20.1

Move ${(4 - x)}^{\frac{1}{2}}$ .

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

Step 5.1.20.2

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

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

Step 5.1.20.3

Combine the numerators over the common denominator.

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

Step 5.1.20.4

Add $1$ and $1$ .

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

Step 5.1.20.5

Divide $2$ by $2$ .

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

Step 5.1.21

Simplify the expression.

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Step 5.1.21.1

Simplify ${(4 - x)}^{1} \cdot 2$ .

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

Step 5.1.21.2

Move $2$ to the left of $4 - x$ .

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

Step 5.1.22

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

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

Step 5.1.23

Cancel the common factor.

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

Step 5.1.24

Rewrite the expression.

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

Step 5.1.25

Simplify.

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Step 5.1.25.1

Apply the distributive property.

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

Step 5.1.25.2

Simplify the numerator.

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Step 5.1.25.2.1

Simplify each term.

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Step 5.1.25.2.1.1

Multiply $2$ by $4$ .

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

Step 5.1.25.2.1.2

Multiply $- 1$ by $2$ .

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

Step 5.1.25.2.2

Subtract $2 x$ from $- x$ .

$\frac{- 3 x + 8}{{(4 - x)}^{\frac{1}{2}}}$

Step 5.1.25.3

Reorder terms.

$\frac{- 3 x + 8}{{(- x + 4)}^{\frac{1}{2}}}$

Step 5.1.25.4

Factor $- 1$ out of $- 3 x$ .

$\frac{- (3 x) + 8}{{(- x + 4)}^{\frac{1}{2}}}$

Step 5.1.25.5

Rewrite $8$ as $- 1 (- 8)$ .

$\frac{- (3 x) - 1 (- 8)}{{(- x + 4)}^{\frac{1}{2}}}$

Step 5.1.25.6

Factor $- 1$ out of $- (3 x) - 1 (- 8)$ .

$\frac{- (3 x - 8)}{{(- x + 4)}^{\frac{1}{2}}}$

Step 5.1.25.7

Rewrite $- (3 x - 8)$ as $- 1 (3 x - 8)$ .

$\frac{- 1 (3 x - 8)}{{(- x + 4)}^{\frac{1}{2}}}$

Step 5.1.25.8

Move the negative in front of the fraction.

$f' (x) = - \frac{3 x - 8}{{(- x + 4)}^{\frac{1}{2}}}$

Step 5.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{3 x - 8}{{(- x + 4)}^{\frac{1}{2}}}$ .

$- \frac{3 x - 8}{{(- x + 4)}^{\frac{1}{2}}}$

Step 6

Set the first derivative equal to $0$ then solve the equation $- \frac{3 x - 8}{{(- x + 4)}^{\frac{1}{2}}} = 0$ .

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Step 6.1

Set the first derivative equal to $0$ .

$- \frac{3 x - 8}{{(- x + 4)}^{\frac{1}{2}}} = 0$

Step 6.2

Set the numerator equal to zero.

$3 x - 8 = 0$

Step 6.3

Solve the equation for $x$ .

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Step 6.3.1

Add $8$ to both sides of the equation.

$3 x = 8$

Step 6.3.2

Divide each term in $3 x = 8$ by $3$ and simplify.

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Step 6.3.2.1

Divide each term in $3 x = 8$ by $3$ .

$\frac{3 x}{3} = \frac{8}{3}$

Step 6.3.2.2

Simplify the left side.

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Step 6.3.2.2.1

Cancel the common factor of $3$ .

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Step 6.3.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{8}{3}$

Step 6.3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{8}{3}$

Step 7

Find the values where the derivative is undefined.

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Step 7.1

Convert expressions with fractional exponents to radicals.

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Step 7.1.1

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

$- \frac{3 x - 8}{\sqrt{{(- x + 4)}^{1}}}$

Step 7.1.2

Anything raised to $1$ is the base itself.

$- \frac{3 x - 8}{\sqrt{- x + 4}}$

Step 7.2

Set the denominator in $\frac{3 x - 8}{\sqrt{- x + 4}}$ equal to $0$ to find where the expression is undefined.

$\sqrt{- x + 4} = 0$

Step 7.3

Solve for $x$ .

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Step 7.3.1

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

${\sqrt{- x + 4}}^{2} = 0^{2}$

Step 7.3.2

Simplify each side of the equation.

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Step 7.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{- x + 4}$ as ${(- x + 4)}^{\frac{1}{2}}$ .

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

Step 7.3.2.2

Simplify the left side.

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Step 7.3.2.2.1

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

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Step 7.3.2.2.1.1

Multiply the exponents in ${({(- x + 4)}^{\frac{1}{2}})}^{2}$ .

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Step 7.3.2.2.1.1.1

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

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

Step 7.3.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 7.3.2.2.1.1.2.1

Cancel the common factor.

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

Step 7.3.2.2.1.1.2.2

Rewrite the expression.

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

Step 7.3.2.2.1.2

Simplify.

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

Step 7.3.2.3

Simplify the right side.

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Step 7.3.2.3.1

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

$- x + 4 = 0$

Step 7.3.3

Solve for $x$ .

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Step 7.3.3.1

Subtract $4$ from both sides of the equation.

$- x = - 4$

Step 7.3.3.2

Divide each term in $- x = - 4$ by $- 1$ and simplify.

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Step 7.3.3.2.1

Divide each term in $- x = - 4$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 4}{- 1}$

Step 7.3.3.2.2

Simplify the left side.

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Step 7.3.3.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 4}{- 1}$

Step 7.3.3.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 4}{- 1}$

Step 7.3.3.2.3

Simplify the right side.

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Step 7.3.3.2.3.1

Divide $- 4$ by $- 1$ .

$x = 4$

Step 7.4

Set the radicand in $\sqrt{- x + 4}$ less than $0$ to find where the expression is undefined.

$- x + 4 < 0$

Step 7.5

Solve for $x$ .

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Step 7.5.1

Subtract $4$ from both sides of the inequality.

$- x < - 4$

Step 7.5.2

Divide each term in $- x < - 4$ by $- 1$ and simplify.

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Step 7.5.2.1

Divide each term in $- x < - 4$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} > \frac{- 4}{- 1}$

Step 7.5.2.2

Simplify the left side.

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Step 7.5.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} > \frac{- 4}{- 1}$

Step 7.5.2.2.2

Divide $x$ by $1$ .

$x > \frac{- 4}{- 1}$

Step 7.5.2.3

Simplify the right side.

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Step 7.5.2.3.1

Divide $- 4$ by $- 1$ .

$x > 4$

Step 7.6

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 $0$ .

$x \geq 4$

$[4, \infty)$

$x \geq 4$

$[4, \infty)$

Step 8

Critical points to evaluate.

$x = \frac{8}{3}, 4$

Step 9

Evaluate the second derivative at $x = \frac{8}{3}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{3 (\frac{8}{3}) - 16}{2 {(- (\frac{8}{3}) + 4)}^{\frac{3}{2}}}$

Step 10

Evaluate the second derivative.

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Step 10.1

Simplify the numerator.

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Step 10.1.1

Cancel the common factor of $3$ .

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Step 10.1.1.1

Cancel the common factor.

$\frac{3 (\frac{8}{3}) - 16}{2 {(- \frac{8}{3} + 4)}^{\frac{3}{2}}}$

Step 10.1.1.2

Rewrite the expression.

$\frac{8 - 16}{2 {(- \frac{8}{3} + 4)}^{\frac{3}{2}}}$

Step 10.1.2

Subtract $16$ from $8$ .

$\frac{- 8}{2 {(- \frac{8}{3} + 4)}^{\frac{3}{2}}}$

Step 10.2

Simplify the denominator.

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Step 10.2.1

To write $4$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{- 8}{2 {(- \frac{8}{3} + 4 \cdot \frac{3}{3})}^{\frac{3}{2}}}$

Step 10.2.2

Combine $4$ and $\frac{3}{3}$ .

$\frac{- 8}{2 {(- \frac{8}{3} + \frac{4 \cdot 3}{3})}^{\frac{3}{2}}}$

Step 10.2.3

Combine the numerators over the common denominator.

$\frac{- 8}{2 {(\frac{- 8 + 4 \cdot 3}{3})}^{\frac{3}{2}}}$

Step 10.2.4

Simplify the numerator.

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Step 10.2.4.1

Multiply $4$ by $3$ .

$\frac{- 8}{2 {(\frac{- 8 + 12}{3})}^{\frac{3}{2}}}$

Step 10.2.4.2

Add $- 8$ and $12$ .

$\frac{- 8}{2 {(\frac{4}{3})}^{\frac{3}{2}}}$

Step 10.2.5

Apply the product rule to $\frac{4}{3}$ .

$\frac{- 8}{2 \frac{4^{\frac{3}{2}}}{3^{\frac{3}{2}}}}$

Step 10.2.6

Simplify the numerator.

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Step 10.2.6.1

Rewrite $4$ as $2^{2}$ .

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

Step 10.2.6.2

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

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

Step 10.2.6.3

Cancel the common factor of $2$ .

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Step 10.2.6.3.1

Cancel the common factor.

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

Step 10.2.6.3.2

Rewrite the expression.

$\frac{- 8}{2 \frac{2^{3}}{3^{\frac{3}{2}}}}$

Step 10.2.6.4

Raise $2$ to the power of $3$ .

$\frac{- 8}{2 \frac{8}{3^{\frac{3}{2}}}}$

Step 10.3

Combine fractions.

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Step 10.3.1

Combine $2$ and $\frac{8}{3^{\frac{3}{2}}}$ .

$\frac{- 8}{\frac{2 \cdot 8}{3^{\frac{3}{2}}}}$

Step 10.3.2

Multiply $2$ by $8$ .

$\frac{- 8}{\frac{16}{3^{\frac{3}{2}}}}$

Step 10.4

Multiply the numerator by the reciprocal of the denominator.

$- 8 \frac{3^{\frac{3}{2}}}{16}$

Step 10.5

Cancel the common factor of $8$ .

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Step 10.5.1

Factor $8$ out of $- 8$ .

$8 (- 1) \frac{3^{\frac{3}{2}}}{16}$

Step 10.5.2

Factor $8$ out of $16$ .

$8 \cdot - 1 \frac{3^{\frac{3}{2}}}{8 \cdot 2}$

Step 10.5.3

Cancel the common factor.

$8 \cdot - 1 \frac{3^{\frac{3}{2}}}{8 \cdot 2}$

Step 10.5.4

Rewrite the expression.

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

Step 11

$x = \frac{8}{3}$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = \frac{8}{3}$ is a local maximum

Step 12

Find the y-value when $x = \frac{8}{3}$ .

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Step 12.1

Replace the variable $x$ with $\frac{8}{3}$ in the expression.

$f (\frac{8}{3}) = 2 (\frac{8}{3}) \sqrt{4 - (\frac{8}{3})}$

Step 12.2

Simplify the result.

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Step 12.2.1

Multiply $2 (\frac{8}{3})$ .

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Step 12.2.1.1

Combine $2$ and $\frac{8}{3}$ .

$f (\frac{8}{3}) = \frac{2 \cdot 8}{3} \cdot \sqrt{4 - (\frac{8}{3})}$

Step 12.2.1.2

Multiply $2$ by $8$ .

$f (\frac{8}{3}) = \frac{16}{3} \cdot \sqrt{4 - (\frac{8}{3})}$

Step 12.2.2

To write $4$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (\frac{8}{3}) = \frac{16}{3} \cdot \sqrt{4 \cdot \frac{3}{3} - \frac{8}{3}}$

Step 12.2.3

Combine $4$ and $\frac{3}{3}$ .

$f (\frac{8}{3}) = \frac{16}{3} \cdot \sqrt{\frac{4 \cdot 3}{3} - \frac{8}{3}}$

Step 12.2.4

Combine the numerators over the common denominator.

$f (\frac{8}{3}) = \frac{16}{3} \cdot \sqrt{\frac{4 \cdot 3 - 8}{3}}$

Step 12.2.5

Simplify the numerator.

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Step 12.2.5.1

Multiply $4$ by $3$ .

$f (\frac{8}{3}) = \frac{16}{3} \cdot \sqrt{\frac{12 - 8}{3}}$

Step 12.2.5.2

Subtract $8$ from $12$ .

$f (\frac{8}{3}) = \frac{16}{3} \cdot \sqrt{\frac{4}{3}}$

Step 12.2.6

Rewrite $\sqrt{\frac{4}{3}}$ as $\frac{\sqrt{4}}{\sqrt{3}}$ .

$f (\frac{8}{3}) = \frac{16}{3} \cdot \frac{\sqrt{4}}{\sqrt{3}}$

Step 12.2.7

Simplify the numerator.

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Step 12.2.7.1

Rewrite $4$ as $2^{2}$ .

$f (\frac{8}{3}) = \frac{16}{3} \cdot \frac{\sqrt{2^{2}}}{\sqrt{3}}$

Step 12.2.7.2

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

$f (\frac{8}{3}) = \frac{16}{3} \cdot \frac{2}{\sqrt{3}}$

Step 12.2.8

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$f (\frac{8}{3}) = \frac{16}{3} \cdot (\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})$

Step 12.2.9

Combine and simplify the denominator.

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Step 12.2.9.1

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$f (\frac{8}{3}) = \frac{16}{3} \cdot \frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 12.2.9.2

Raise $\sqrt{3}$ to the power of $1$ .

$f (\frac{8}{3}) = \frac{16}{3} \cdot \frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 12.2.9.3

Raise $\sqrt{3}$ to the power of $1$ .

$f (\frac{8}{3}) = \frac{16}{3} \cdot \frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 12.2.9.4

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

$f (\frac{8}{3}) = \frac{16}{3} \cdot \frac{2 \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 12.2.9.5

Add $1$ and $1$ .

$f (\frac{8}{3}) = \frac{16}{3} \cdot \frac{2 \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 12.2.9.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 12.2.9.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$f (\frac{8}{3}) = \frac{16}{3} \cdot \frac{2 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 12.2.9.6.2

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

$f (\frac{8}{3}) = \frac{16}{3} \cdot \frac{2 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 12.2.9.6.3

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

$f (\frac{8}{3}) = \frac{16}{3} \cdot \frac{2 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 12.2.9.6.4

Cancel the common factor of $2$ .

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Step 12.2.9.6.4.1

Cancel the common factor.

$f (\frac{8}{3}) = \frac{16}{3} \cdot \frac{2 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 12.2.9.6.4.2

Rewrite the expression.

$f (\frac{8}{3}) = \frac{16}{3} \cdot \frac{2 \sqrt{3}}{3}$

Step 12.2.9.6.5

Evaluate the exponent.

$f (\frac{8}{3}) = \frac{16}{3} \cdot \frac{2 \sqrt{3}}{3}$

Step 12.2.10

Multiply $\frac{16}{3} \cdot \frac{2 \sqrt{3}}{3}$ .

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Step 12.2.10.1

Multiply $\frac{16}{3}$ by $\frac{2 \sqrt{3}}{3}$ .

$f (\frac{8}{3}) = \frac{16 (2 \sqrt{3})}{3 \cdot 3}$

Step 12.2.10.2

Multiply $2$ by $16$ .

$f (\frac{8}{3}) = \frac{32 \sqrt{3}}{3 \cdot 3}$

Step 12.2.10.3

Multiply $3$ by $3$ .

$f (\frac{8}{3}) = \frac{32 \sqrt{3}}{9}$

Step 12.2.11

The final answer is $\frac{32 \sqrt{3}}{9}$ .

$y = \frac{32 \sqrt{3}}{9}$

Step 13

Evaluate the second derivative at $x = 4$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{3 (4) - 16}{2 {(- (4) + 4)}^{\frac{3}{2}}}$

Step 14

Evaluate the second derivative.

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Step 14.1

Simplify the expression.

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Step 14.1.1

Multiply $- 1$ by $4$ .

$\frac{3 (4) - 16}{2 {(- 4 + 4)}^{\frac{3}{2}}}$

Step 14.1.2

Add $- 4$ and $4$ .

$\frac{3 (4) - 16}{2 \cdot 0^{\frac{3}{2}}}$

Step 14.1.3

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

$\frac{3 (4) - 16}{2 \cdot {(0^{2})}^{\frac{3}{2}}}$

Step 14.1.4

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

$\frac{3 (4) - 16}{2 \cdot 0^{2 (\frac{3}{2})}}$

Step 14.2

Cancel the common factor of $2$ .

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Step 14.2.1

Cancel the common factor.

$\frac{3 (4) - 16}{2 \cdot 0^{2 (\frac{3}{2})}}$

Step 14.2.2

Rewrite the expression.

$\frac{3 (4) - 16}{2 \cdot 0^{3}}$

Step 14.3

Simplify the expression.

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Step 14.3.1

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

$\frac{3 (4) - 16}{2 \cdot 0}$

Step 14.3.2

Multiply $2$ by $0$ .

$\frac{3 (4) - 16}{0}$

Step 14.3.3

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

Undefined

$\frac{3 (4) - 16}{0}$

Step 14.4

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

Undefined

Step 15

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

No Local Extrema

Step 16