Calculus Examples

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

Step 1

Find the first derivative of the function.

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Step 1.1

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

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

Step 1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{4}$ and $g (x) = \frac{x}{2} - 4$ .

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Step 1.2.1

To apply the Chain Rule, set $u$ as $\frac{x}{2} - 4$ .

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

Step 1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 4$ .

$x (4 u^{3} \frac{d}{d x} [\frac{x}{2} - 4]) + {(\frac{x}{2} - 4)}^{4} \frac{d}{d x} [x]$

Step 1.2.3

Replace all occurrences of $u$ with $\frac{x}{2} - 4$ .

$x (4 {(\frac{x}{2} - 4)}^{3} \frac{d}{d x} [\frac{x}{2} - 4]) + {(\frac{x}{2} - 4)}^{4} \frac{d}{d x} [x]$

Step 1.3

Differentiate.

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Step 1.3.1

By the Sum Rule, the derivative of $\frac{x}{2} - 4$ with respect to $x$ is $\frac{d}{d x} [\frac{x}{2}] + \frac{d}{d x} [- 4]$ .

$x (4 {(\frac{x}{2} - 4)}^{3} (\frac{d}{d x} [\frac{x}{2}] + \frac{d}{d x} [- 4])) + {(\frac{x}{2} - 4)}^{4} \frac{d}{d x} [x]$

Step 1.3.2

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

$x (4 {(\frac{x}{2} - 4)}^{3} (\frac{1}{2} \frac{d}{d x} [x] + \frac{d}{d x} [- 4])) + {(\frac{x}{2} - 4)}^{4} \frac{d}{d x} [x]$

Step 1.3.3

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

$x (4 {(\frac{x}{2} - 4)}^{3} (\frac{1}{2} \cdot 1 + \frac{d}{d x} [- 4])) + {(\frac{x}{2} - 4)}^{4} \frac{d}{d x} [x]$

Step 1.3.4

Multiply $\frac{1}{2}$ by $1$ .

$x (4 {(\frac{x}{2} - 4)}^{3} (\frac{1}{2} + \frac{d}{d x} [- 4])) + {(\frac{x}{2} - 4)}^{4} \frac{d}{d x} [x]$

Step 1.3.5

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

$x (4 {(\frac{x}{2} - 4)}^{3} (\frac{1}{2} + 0)) + {(\frac{x}{2} - 4)}^{4} \frac{d}{d x} [x]$

Step 1.3.6

Simplify terms.

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Step 1.3.6.1

Add $\frac{1}{2}$ and $0$ .

$x (4 {(\frac{x}{2} - 4)}^{3} \frac{1}{2}) + {(\frac{x}{2} - 4)}^{4} \frac{d}{d x} [x]$

Step 1.3.6.2

Combine $\frac{1}{2}$ and $4$ .

$x (\frac{4}{2} {(\frac{x}{2} - 4)}^{3}) + {(\frac{x}{2} - 4)}^{4} \frac{d}{d x} [x]$

Step 1.3.6.3

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

$x \frac{4 {(\frac{x}{2} - 4)}^{3}}{2} + {(\frac{x}{2} - 4)}^{4} \frac{d}{d x} [x]$

Step 1.3.6.4

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

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Step 1.3.6.4.1

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

$x \frac{2 (2 {(\frac{x}{2} - 4)}^{3})}{2} + {(\frac{x}{2} - 4)}^{4} \frac{d}{d x} [x]$

Step 1.3.6.4.2

Cancel the common factors.

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Step 1.3.6.4.2.1

Factor $2$ out of $2$ .

$x \frac{2 (2 {(\frac{x}{2} - 4)}^{3})}{2 (1)} + {(\frac{x}{2} - 4)}^{4} \frac{d}{d x} [x]$

Step 1.3.6.4.2.2

Cancel the common factor.

$x \frac{2 (2 {(\frac{x}{2} - 4)}^{3})}{2 \cdot 1} + {(\frac{x}{2} - 4)}^{4} \frac{d}{d x} [x]$

Step 1.3.6.4.2.3

Rewrite the expression.

$x \frac{2 {(\frac{x}{2} - 4)}^{3}}{1} + {(\frac{x}{2} - 4)}^{4} \frac{d}{d x} [x]$

Step 1.3.6.4.2.4

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

$x (2 {(\frac{x}{2} - 4)}^{3}) + {(\frac{x}{2} - 4)}^{4} \frac{d}{d x} [x]$

Step 1.3.7

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

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

Step 1.3.8

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

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

Step 1.4

Simplify.

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Step 1.4.1

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

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Step 1.4.1.1

Reorder $x$ and $2$ .

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

Step 1.4.1.2

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

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

Step 1.4.1.3

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

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

Step 1.4.1.4

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

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

Step 1.4.2

Combine terms.

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Step 1.4.2.1

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

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

Step 1.4.2.2

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

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

Step 1.4.2.3

Combine the numerators over the common denominator.

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

Step 1.4.2.4

Multiply $2$ by $2$ .

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

Step 1.4.2.5

Add $4 x$ and $x$ .

${(\frac{x}{2} - 4)}^{3} (\frac{5 x}{2} - 4)$

Step 2

Find the second derivative of the function.

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Step 2.1

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 $f (x) = {(\frac{x}{2} - 4)}^{3}$ and $g (x) = \frac{5 x}{2} - 4$ .

$f'' (x) = {(\frac{x}{2} - 4)}^{3} \frac{d}{d x} (\frac{5 x}{2} - 4) + (\frac{5 x}{2} - 4) \frac{d}{d x} ({(\frac{x}{2} - 4)}^{3})$

Step 2.2

Differentiate.

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Step 2.2.1

By the Sum Rule, the derivative of $\frac{5 x}{2} - 4$ with respect to $x$ is $\frac{d}{d x} [\frac{5 x}{2}] + \frac{d}{d x} [- 4]$ .

$f'' (x) = {(\frac{x}{2} - 4)}^{3} (\frac{d}{d x} (\frac{5 x}{2}) + \frac{d}{d x} (- 4)) + (\frac{5 x}{2} - 4) \frac{d}{d x} ({(\frac{x}{2} - 4)}^{3})$

Step 2.2.2

Since $\frac{5}{2}$ is constant with respect to $x$ , the derivative of $\frac{5 x}{2}$ with respect to $x$ is $\frac{5}{2} \frac{d}{d x} [x]$ .

$f'' (x) = {(\frac{x}{2} - 4)}^{3} (\frac{5}{2} \cdot \frac{d}{d x} (x) + \frac{d}{d x} (- 4)) + (\frac{5 x}{2} - 4) \frac{d}{d x} ({(\frac{x}{2} - 4)}^{3})$

Step 2.2.3

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

$f'' (x) = {(\frac{x}{2} - 4)}^{3} (\frac{5}{2} \cdot 1 + \frac{d}{d x} (- 4)) + (\frac{5 x}{2} - 4) \frac{d}{d x} ({(\frac{x}{2} - 4)}^{3})$

Step 2.2.4

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

$f'' (x) = {(\frac{x}{2} - 4)}^{3} (\frac{5}{2} + \frac{d}{d x} (- 4)) + (\frac{5 x}{2} - 4) \frac{d}{d x} ({(\frac{x}{2} - 4)}^{3})$

Step 2.2.5

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

$f'' (x) = {(\frac{x}{2} - 4)}^{3} (\frac{5}{2} + 0) + (\frac{5 x}{2} - 4) \frac{d}{d x} ({(\frac{x}{2} - 4)}^{3})$

Step 2.2.6

Combine fractions.

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Step 2.2.6.1

Add $\frac{5}{2}$ and $0$ .

$f'' (x) = {(\frac{x}{2} - 4)}^{3} (\frac{5}{2}) + (\frac{5 x}{2} - 4) \frac{d}{d x} ({(\frac{x}{2} - 4)}^{3})$

Step 2.2.6.2

Combine ${(\frac{x}{2} - 4)}^{3}$ and $\frac{5}{2}$ .

$f'' (x) = \frac{{(\frac{x}{2} - 4)}^{3} \cdot 5}{2} + (\frac{5 x}{2} - 4) \frac{d}{d x} ({(\frac{x}{2} - 4)}^{3})$

Step 2.2.6.3

Move $5$ to the left of ${(\frac{x}{2} - 4)}^{3}$ .

$f'' (x) = \frac{5 \cdot {(\frac{x}{2} - 4)}^{3}}{2} + (\frac{5 x}{2} - 4) \frac{d}{d x} ({(\frac{x}{2} - 4)}^{3})$

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

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Step 2.3.1

To apply the Chain Rule, set $u$ as $\frac{x}{2} - 4$ .

$f'' (x) = \frac{5 {(\frac{x}{2} - 4)}^{3}}{2} + (\frac{5 x}{2} - 4) (\frac{d}{d u} (u^{3}) \frac{d}{d x} (\frac{x}{2} - 4))$

Step 2.3.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 3$ .

$f'' (x) = \frac{5 {(\frac{x}{2} - 4)}^{3}}{2} + (\frac{5 x}{2} - 4) (3 u^{2} \frac{d}{d x} (\frac{x}{2} - 4))$

Step 2.3.3

Replace all occurrences of $u$ with $\frac{x}{2} - 4$ .

$f'' (x) = \frac{5 {(\frac{x}{2} - 4)}^{3}}{2} + (\frac{5 x}{2} - 4) (3 {(\frac{x}{2} - 4)}^{2} \frac{d}{d x} (\frac{x}{2} - 4))$

Step 2.4

Differentiate.

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Step 2.4.1

Move $3$ to the left of $\frac{5 x}{2} - 4$ .

$f'' (x) = \frac{5 {(\frac{x}{2} - 4)}^{3}}{2} + 3 \cdot ((\frac{5 x}{2} - 4) {(\frac{x}{2} - 4)}^{2}) \frac{d}{d x} (\frac{x}{2} - 4)$

Step 2.4.2

By the Sum Rule, the derivative of $\frac{x}{2} - 4$ with respect to $x$ is $\frac{d}{d x} [\frac{x}{2}] + \frac{d}{d x} [- 4]$ .

$f'' (x) = \frac{5 {(\frac{x}{2} - 4)}^{3}}{2} + 3 (\frac{5 x}{2} - 4) {(\frac{x}{2} - 4)}^{2} (\frac{d}{d x} (\frac{x}{2}) + \frac{d}{d x} (- 4))$

Step 2.4.3

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

$f'' (x) = \frac{5 {(\frac{x}{2} - 4)}^{3}}{2} + 3 (\frac{5 x}{2} - 4) {(\frac{x}{2} - 4)}^{2} (\frac{1}{2} \cdot \frac{d}{d x} (x) + \frac{d}{d x} (- 4))$

Step 2.4.4

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

$f'' (x) = \frac{5 {(\frac{x}{2} - 4)}^{3}}{2} + 3 (\frac{5 x}{2} - 4) {(\frac{x}{2} - 4)}^{2} (\frac{1}{2} \cdot 1 + \frac{d}{d x} (- 4))$

Step 2.4.5

Multiply $\frac{1}{2}$ by $1$ .

$f'' (x) = \frac{5 {(\frac{x}{2} - 4)}^{3}}{2} + 3 (\frac{5 x}{2} - 4) {(\frac{x}{2} - 4)}^{2} (\frac{1}{2} + \frac{d}{d x} (- 4))$

Step 2.4.6

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

$f'' (x) = \frac{5 {(\frac{x}{2} - 4)}^{3}}{2} + 3 (\frac{5 x}{2} - 4) {(\frac{x}{2} - 4)}^{2} (\frac{1}{2} + 0)$

Step 2.4.7

Combine fractions.

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Step 2.4.7.1

Add $\frac{1}{2}$ and $0$ .

$f'' (x) = \frac{5 {(\frac{x}{2} - 4)}^{3}}{2} + 3 (\frac{5 x}{2} - 4) {(\frac{x}{2} - 4)}^{2} (\frac{1}{2})$

Step 2.4.7.2

Combine $\frac{1}{2}$ and $3$ .

$f'' (x) = \frac{5 {(\frac{x}{2} - 4)}^{3}}{2} + \frac{3}{2} \cdot ((\frac{5 x}{2} - 4) {(\frac{x}{2} - 4)}^{2})$

Step 2.4.7.3

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

$f'' (x) = \frac{5 {(\frac{x}{2} - 4)}^{3}}{2} + \frac{{(\frac{x}{2} - 4)}^{2} \cdot 3}{2} \cdot (\frac{5 x}{2} - 4)$

Step 2.4.7.4

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

$f'' (x) = \frac{5 {(\frac{x}{2} - 4)}^{3}}{2} + \frac{3 \cdot {(\frac{x}{2} - 4)}^{2}}{2} \cdot (\frac{5 x}{2} - 4)$

Step 2.5

To write $\frac{3 {(\frac{x}{2} - 4)}^{2}}{2} (\frac{5 x}{2} - 4)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f'' (x) = \frac{5 {(\frac{x}{2} - 4)}^{3}}{2} + \frac{3 {(\frac{x}{2} - 4)}^{2}}{2} \cdot (\frac{5 x}{2} - 4) \cdot \frac{2}{2}$

Step 2.6

Combine $\frac{3 {(\frac{x}{2} - 4)}^{2}}{2} (\frac{5 x}{2} - 4)$ and $\frac{2}{2}$ .

$f'' (x) = \frac{5 {(\frac{x}{2} - 4)}^{3}}{2} + \frac{\frac{3 {(\frac{x}{2} - 4)}^{2}}{2} \cdot (\frac{5 x}{2} - 4) \cdot 2}{2}$

Step 2.7

Combine the numerators over the common denominator.

$f'' (x) = \frac{5 {(\frac{x}{2} - 4)}^{3} + \frac{3 {(\frac{x}{2} - 4)}^{2}}{2} \cdot (\frac{5 x}{2} - 4) \cdot 2}{2}$

Step 2.8

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

$f'' (x) = \frac{5 {(\frac{x}{2} - 4)}^{3} + \frac{2 (3 {(\frac{x}{2} - 4)}^{2})}{2} \cdot (\frac{5 x}{2} - 4)}{2}$

Step 2.9

Multiply $3$ by $2$ .

$f'' (x) = \frac{5 {(\frac{x}{2} - 4)}^{3} + \frac{6 {(\frac{x}{2} - 4)}^{2}}{2} \cdot (\frac{5 x}{2} - 4)}{2}$

Step 2.10

Cancel the common factor of $6$ and $2$ .

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Step 2.10.1

Factor $2$ out of $6 {(\frac{x}{2} - 4)}^{2}$ .

$f'' (x) = \frac{5 {(\frac{x}{2} - 4)}^{3} + \frac{2 (3 {(\frac{x}{2} - 4)}^{2})}{2} \cdot (\frac{5 x}{2} - 4)}{2}$

Step 2.10.2

Cancel the common factors.

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Step 2.10.2.1

Factor $2$ out of $2$ .

$f'' (x) = \frac{5 {(\frac{x}{2} - 4)}^{3} + \frac{2 (3 {(\frac{x}{2} - 4)}^{2})}{2 (1)} \cdot (\frac{5 x}{2} - 4)}{2}$

Step 2.10.2.2

Cancel the common factor.

$f'' (x) = \frac{5 {(\frac{x}{2} - 4)}^{3} + \frac{2 (3 {(\frac{x}{2} - 4)}^{2})}{2 \cdot 1} \cdot (\frac{5 x}{2} - 4)}{2}$

Step 2.10.2.3

Rewrite the expression.

$f'' (x) = \frac{5 {(\frac{x}{2} - 4)}^{3} + \frac{3 {(\frac{x}{2} - 4)}^{2}}{1} \cdot (\frac{5 x}{2} - 4)}{2}$

Step 2.10.2.4

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

$f'' (x) = \frac{5 {(\frac{x}{2} - 4)}^{3} + 3 {(\frac{x}{2} - 4)}^{2} (\frac{5 x}{2} - 4)}{2}$

Step 2.11

Simplify.

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Step 2.11.1

Simplify the numerator.

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Step 2.11.1.1

Factor ${(\frac{x}{2} - 4)}^{2}$ out of $5 {(\frac{x}{2} - 4)}^{3} + 3 {(\frac{x}{2} - 4)}^{2} (\frac{5 x}{2} - 4)$ .

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Step 2.11.1.1.1

Factor ${(\frac{x}{2} - 4)}^{2}$ out of $5 {(\frac{x}{2} - 4)}^{3}$ .

$f'' (x) = \frac{{(\frac{x}{2} - 4)}^{2} (5 (\frac{x}{2} - 4)) + 3 {(\frac{x}{2} - 4)}^{2} (\frac{5 x}{2} - 4)}{2}$

Step 2.11.1.1.2

Factor ${(\frac{x}{2} - 4)}^{2}$ out of $3 {(\frac{x}{2} - 4)}^{2} (\frac{5 x}{2} - 4)$ .

$f'' (x) = \frac{{(\frac{x}{2} - 4)}^{2} (5 (\frac{x}{2} - 4)) + {(\frac{x}{2} - 4)}^{2} (3 (\frac{5 x}{2} - 4))}{2}$

Step 2.11.1.1.3

Factor ${(\frac{x}{2} - 4)}^{2}$ out of ${(\frac{x}{2} - 4)}^{2} (5 (\frac{x}{2} - 4)) + {(\frac{x}{2} - 4)}^{2} (3 (\frac{5 x}{2} - 4))$ .

$f'' (x) = \frac{{(\frac{x}{2} - 4)}^{2} (5 (\frac{x}{2} - 4) + 3 (\frac{5 x}{2} - 4))}{2}$

Step 2.11.1.2

Apply the distributive property.

$f'' (x) = \frac{{(\frac{x}{2} - 4)}^{2} (5 (\frac{x}{2}) + 5 \cdot - 4 + 3 (\frac{5 x}{2} - 4))}{2}$

Step 2.11.1.3

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

$f'' (x) = \frac{{(\frac{x}{2} - 4)}^{2} (\frac{5 x}{2} + 5 \cdot - 4 + 3 (\frac{5 x}{2} - 4))}{2}$

Step 2.11.1.4

Multiply $5$ by $- 4$ .

$f'' (x) = \frac{{(\frac{x}{2} - 4)}^{2} (\frac{5 x}{2} - 20 + 3 (\frac{5 x}{2} - 4))}{2}$

Step 2.11.1.5

Apply the distributive property.

$f'' (x) = \frac{{(\frac{x}{2} - 4)}^{2} (\frac{5 x}{2} - 20 + 3 (\frac{5 x}{2}) + 3 \cdot - 4)}{2}$

Step 2.11.1.6

Multiply $3 \frac{5 x}{2}$ .

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Step 2.11.1.6.1

Combine $3$ and $\frac{5 x}{2}$ .

$f'' (x) = \frac{{(\frac{x}{2} - 4)}^{2} (\frac{5 x}{2} - 20 + \frac{3 (5 x)}{2} + 3 \cdot - 4)}{2}$

Step 2.11.1.6.2

Multiply $5$ by $3$ .

$f'' (x) = \frac{{(\frac{x}{2} - 4)}^{2} (\frac{5 x}{2} - 20 + \frac{15 x}{2} + 3 \cdot - 4)}{2}$

Step 2.11.1.7

Multiply $3$ by $- 4$ .

$f'' (x) = \frac{{(\frac{x}{2} - 4)}^{2} (\frac{5 x}{2} - 20 + \frac{15 x}{2} - 12)}{2}$

Step 2.11.1.8

Combine the numerators over the common denominator.

$f'' (x) = \frac{{(\frac{x}{2} - 4)}^{2} (\frac{5 x + 15 x}{2} - 20 - 12)}{2}$

Step 2.11.1.9

Subtract $12$ from $- 20$ .

$f'' (x) = \frac{{(\frac{x}{2} - 4)}^{2} (\frac{5 x + 15 x}{2} - 32)}{2}$

Step 2.11.1.10

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

$f'' (x) = \frac{{(\frac{x}{2} - 4 \cdot \frac{2}{2})}^{2} (\frac{5 x + 15 x}{2} - 32)}{2}$

Step 2.11.1.11

Combine $- 4$ and $\frac{2}{2}$ .

$f'' (x) = \frac{{(\frac{x}{2} + \frac{- 4 \cdot 2}{2})}^{2} (\frac{5 x + 15 x}{2} - 32)}{2}$

Step 2.11.1.12

Combine the numerators over the common denominator.

$f'' (x) = \frac{{(\frac{x - 4 \cdot 2}{2})}^{2} (\frac{5 x + 15 x}{2} - 32)}{2}$

Step 2.11.1.13

Multiply $- 4$ by $2$ .

$f'' (x) = \frac{{(\frac{x - 8}{2})}^{2} (\frac{5 x + 15 x}{2} - 32)}{2}$

Step 2.11.1.14

Add $5 x$ and $15 x$ .

$f'' (x) = \frac{{(\frac{x - 8}{2})}^{2} (\frac{20 x}{2} - 32)}{2}$

Step 2.11.1.15

Cancel the common factor of $20$ and $2$ .

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Step 2.11.1.15.1

Factor $2$ out of $20 x$ .

$f'' (x) = \frac{{(\frac{x - 8}{2})}^{2} (\frac{2 (10 x)}{2} - 32)}{2}$

Step 2.11.1.15.2

Cancel the common factors.

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Step 2.11.1.15.2.1

Factor $2$ out of $2$ .

$f'' (x) = \frac{{(\frac{x - 8}{2})}^{2} (\frac{2 (10 x)}{2 (1)} - 32)}{2}$

Step 2.11.1.15.2.2

Cancel the common factor.

$f'' (x) = \frac{{(\frac{x - 8}{2})}^{2} (\frac{2 (10 x)}{2 \cdot 1} - 32)}{2}$

Step 2.11.1.15.2.3

Rewrite the expression.

$f'' (x) = \frac{{(\frac{x - 8}{2})}^{2} (\frac{10 x}{1} - 32)}{2}$

Step 2.11.1.15.2.4

Divide $10 x$ by $1$ .

$f'' (x) = \frac{{(\frac{x - 8}{2})}^{2} (10 x - 32)}{2}$

Step 2.11.1.16

Apply the product rule to $\frac{x - 8}{2}$ .

$f'' (x) = \frac{\frac{{(x - 8)}^{2}}{2^{2}} \cdot (10 x - 32)}{2}$

Step 2.11.1.17

Factor $2$ out of $10 x - 32$ .

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Step 2.11.1.17.1

Factor $2$ out of $10 x$ .

$f'' (x) = \frac{\frac{{(x - 8)}^{2}}{2^{2}} \cdot (2 (5 x) - 32)}{2}$

Step 2.11.1.17.2

Factor $2$ out of $- 32$ .

$f'' (x) = \frac{\frac{{(x - 8)}^{2}}{2^{2}} \cdot (2 (5 x) + 2 (- 16))}{2}$

Step 2.11.1.17.3

Factor $2$ out of $2 (5 x) + 2 (- 16)$ .

$f'' (x) = \frac{\frac{{(x - 8)}^{2}}{2^{2}} \cdot (2 (5 x - 16))}{2}$

Step 2.11.1.18

Combine $\frac{{(x - 8)}^{2}}{2^{2}}$ and $2$ .

$f'' (x) = \frac{\frac{{(x - 8)}^{2} \cdot 2}{2^{2}} \cdot (5 x - 16)}{2}$

Step 2.11.1.19

Reduce the expression $\frac{{(x - 8)}^{2} \cdot 2}{2^{2}}$ by cancelling the common factors.

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Step 2.11.1.19.1

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

$f'' (x) = \frac{\frac{2 \cdot {(x - 8)}^{2}}{2^{2}} \cdot (5 x - 16)}{2}$

Step 2.11.1.19.2

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

$f'' (x) = \frac{\frac{2 \cdot {(x - 8)}^{2}}{2 \cdot 2} \cdot (5 x - 16)}{2}$

Step 2.11.1.19.3

Cancel the common factor.

$f'' (x) = \frac{\frac{2 \cdot {(x - 8)}^{2}}{2 \cdot 2} \cdot (5 x - 16)}{2}$

Step 2.11.1.19.4

Rewrite the expression.

$f'' (x) = \frac{\frac{{(x - 8)}^{2}}{2} \cdot (5 x - 16)}{2}$

Step 2.11.2

Reorder terms.

$f'' (x) = \frac{(5 x - 16) (\frac{{(x - 8)}^{2}}{2})}{2}$

Step 2.11.3

Factor $\frac{{(x - 8)}^{2}}{2}$ out of $\frac{(5 x - 16) \frac{{(x - 8)}^{2}}{2}}{2}$ .

$f'' (x) = \frac{{(x - 8)}^{2}}{2} \cdot \frac{5 x - 16}{2}$

Step 2.11.4

Multiply $\frac{{(x - 8)}^{2}}{2} \cdot \frac{5 x - 16}{2}$ .

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Step 2.11.4.1

Multiply $\frac{{(x - 8)}^{2}}{2}$ by $\frac{5 x - 16}{2}$ .

$f'' (x) = \frac{{(x - 8)}^{2} (5 x - 16)}{2 \cdot 2}$

Step 2.11.4.2

Multiply $2$ by $2$ .

$f'' (x) = \frac{{(x - 8)}^{2} (5 x - 16)}{4}$

Step 3

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

${(\frac{x}{2} - 4)}^{3} (\frac{5 x}{2} - 4) = 0$

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

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

Step 4.1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{4}$ and $g (x) = \frac{x}{2} - 4$ .

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Step 4.1.2.1

To apply the Chain Rule, set $u$ as $\frac{x}{2} - 4$ .

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

Step 4.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 4$ .

$x (4 u^{3} \frac{d}{d x} [\frac{x}{2} - 4]) + {(\frac{x}{2} - 4)}^{4} \frac{d}{d x} [x]$

Step 4.1.2.3

Replace all occurrences of $u$ with $\frac{x}{2} - 4$ .

$x (4 {(\frac{x}{2} - 4)}^{3} \frac{d}{d x} [\frac{x}{2} - 4]) + {(\frac{x}{2} - 4)}^{4} \frac{d}{d x} [x]$

Step 4.1.3

Differentiate.

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Step 4.1.3.1

By the Sum Rule, the derivative of $\frac{x}{2} - 4$ with respect to $x$ is $\frac{d}{d x} [\frac{x}{2}] + \frac{d}{d x} [- 4]$ .

$x (4 {(\frac{x}{2} - 4)}^{3} (\frac{d}{d x} [\frac{x}{2}] + \frac{d}{d x} [- 4])) + {(\frac{x}{2} - 4)}^{4} \frac{d}{d x} [x]$

Step 4.1.3.2

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

$x (4 {(\frac{x}{2} - 4)}^{3} (\frac{1}{2} \frac{d}{d x} [x] + \frac{d}{d x} [- 4])) + {(\frac{x}{2} - 4)}^{4} \frac{d}{d x} [x]$

Step 4.1.3.3

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

$x (4 {(\frac{x}{2} - 4)}^{3} (\frac{1}{2} \cdot 1 + \frac{d}{d x} [- 4])) + {(\frac{x}{2} - 4)}^{4} \frac{d}{d x} [x]$

Step 4.1.3.4

Multiply $\frac{1}{2}$ by $1$ .

$x (4 {(\frac{x}{2} - 4)}^{3} (\frac{1}{2} + \frac{d}{d x} [- 4])) + {(\frac{x}{2} - 4)}^{4} \frac{d}{d x} [x]$

Step 4.1.3.5

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

$x (4 {(\frac{x}{2} - 4)}^{3} (\frac{1}{2} + 0)) + {(\frac{x}{2} - 4)}^{4} \frac{d}{d x} [x]$

Step 4.1.3.6

Simplify terms.

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Step 4.1.3.6.1

Add $\frac{1}{2}$ and $0$ .

$x (4 {(\frac{x}{2} - 4)}^{3} \frac{1}{2}) + {(\frac{x}{2} - 4)}^{4} \frac{d}{d x} [x]$

Step 4.1.3.6.2

Combine $\frac{1}{2}$ and $4$ .

$x (\frac{4}{2} {(\frac{x}{2} - 4)}^{3}) + {(\frac{x}{2} - 4)}^{4} \frac{d}{d x} [x]$

Step 4.1.3.6.3

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

$x \frac{4 {(\frac{x}{2} - 4)}^{3}}{2} + {(\frac{x}{2} - 4)}^{4} \frac{d}{d x} [x]$

Step 4.1.3.6.4

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

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Step 4.1.3.6.4.1

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

$x \frac{2 (2 {(\frac{x}{2} - 4)}^{3})}{2} + {(\frac{x}{2} - 4)}^{4} \frac{d}{d x} [x]$

Step 4.1.3.6.4.2

Cancel the common factors.

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Step 4.1.3.6.4.2.1

Factor $2$ out of $2$ .

$x \frac{2 (2 {(\frac{x}{2} - 4)}^{3})}{2 (1)} + {(\frac{x}{2} - 4)}^{4} \frac{d}{d x} [x]$

Step 4.1.3.6.4.2.2

Cancel the common factor.

$x \frac{2 (2 {(\frac{x}{2} - 4)}^{3})}{2 \cdot 1} + {(\frac{x}{2} - 4)}^{4} \frac{d}{d x} [x]$

Step 4.1.3.6.4.2.3

Rewrite the expression.

$x \frac{2 {(\frac{x}{2} - 4)}^{3}}{1} + {(\frac{x}{2} - 4)}^{4} \frac{d}{d x} [x]$

Step 4.1.3.6.4.2.4

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

$x (2 {(\frac{x}{2} - 4)}^{3}) + {(\frac{x}{2} - 4)}^{4} \frac{d}{d x} [x]$

Step 4.1.3.7

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

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

Step 4.1.3.8

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

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

Step 4.1.4

Simplify.

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Step 4.1.4.1

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

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Step 4.1.4.1.1

Reorder $x$ and $2$ .

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

Step 4.1.4.1.2

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

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

Step 4.1.4.1.3

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

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

Step 4.1.4.1.4

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

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

Step 4.1.4.2

Combine terms.

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Step 4.1.4.2.1

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

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

Step 4.1.4.2.2

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

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

Step 4.1.4.2.3

Combine the numerators over the common denominator.

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

Step 4.1.4.2.4

Multiply $2$ by $2$ .

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

Step 4.1.4.2.5

Add $4 x$ and $x$ .

$f' (x) = {(\frac{x}{2} - 4)}^{3} (\frac{5 x}{2} - 4)$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is ${(\frac{x}{2} - 4)}^{3} (\frac{5 x}{2} - 4)$ .

${(\frac{x}{2} - 4)}^{3} (\frac{5 x}{2} - 4)$

Step 5

Set the first derivative equal to $0$ then solve the equation ${(\frac{x}{2} - 4)}^{3} (\frac{5 x}{2} - 4) = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

${(\frac{x}{2} - 4)}^{3} (\frac{5 x}{2} - 4) = 0$

Step 5.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

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

$\frac{5 x}{2} - 4 = 0$

Step 5.3

Set ${(\frac{x}{2} - 4)}^{3}$ equal to $0$ and solve for $x$ .

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Step 5.3.1

Set ${(\frac{x}{2} - 4)}^{3}$ equal to $0$ .

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

Step 5.3.2

Solve ${(\frac{x}{2} - 4)}^{3} = 0$ for $x$ .

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Step 5.3.2.1

Set the $\frac{x}{2} - 4$ equal to $0$ .

$\frac{x}{2} - 4 = 0$

Step 5.3.2.2

Solve for $x$ .

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Step 5.3.2.2.1

Add $4$ to both sides of the equation.

$\frac{x}{2} = 4$

Step 5.3.2.2.2

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 \cdot 4$

Step 5.3.2.2.3

Simplify both sides of the equation.

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Step 5.3.2.2.3.1

Simplify the left side.

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Step 5.3.2.2.3.1.1

Cancel the common factor of $2$ .

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Step 5.3.2.2.3.1.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 \cdot 4$

Step 5.3.2.2.3.1.1.2

Rewrite the expression.

$x = 2 \cdot 4$

Step 5.3.2.2.3.2

Simplify the right side.

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Step 5.3.2.2.3.2.1

Multiply $2$ by $4$ .

$x = 8$

Step 5.4

Set $\frac{5 x}{2} - 4$ equal to $0$ and solve for $x$ .

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Step 5.4.1

Set $\frac{5 x}{2} - 4$ equal to $0$ .

$\frac{5 x}{2} - 4 = 0$

Step 5.4.2

Solve $\frac{5 x}{2} - 4 = 0$ for $x$ .

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Step 5.4.2.1

Add $4$ to both sides of the equation.

$\frac{5 x}{2} = 4$

Step 5.4.2.2

Multiply both sides of the equation by $\frac{2}{5}$ .

$\frac{2}{5} \cdot \frac{5 x}{2} = \frac{2}{5} \cdot 4$

Step 5.4.2.3

Simplify both sides of the equation.

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Step 5.4.2.3.1

Simplify the left side.

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Step 5.4.2.3.1.1

Simplify $\frac{2}{5} \cdot \frac{5 x}{2}$ .

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Step 5.4.2.3.1.1.1

Cancel the common factor of $2$ .

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Step 5.4.2.3.1.1.1.1

Cancel the common factor.

$\frac{2}{5} \cdot \frac{5 x}{2} = \frac{2}{5} \cdot 4$

Step 5.4.2.3.1.1.1.2

Rewrite the expression.

$\frac{1}{5} (5 x) = \frac{2}{5} \cdot 4$

Step 5.4.2.3.1.1.2

Cancel the common factor of $5$ .

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Step 5.4.2.3.1.1.2.1

Factor $5$ out of $5 x$ .

$\frac{1}{5} (5 (x)) = \frac{2}{5} \cdot 4$

Step 5.4.2.3.1.1.2.2

Cancel the common factor.

$\frac{1}{5} (5 x) = \frac{2}{5} \cdot 4$

Step 5.4.2.3.1.1.2.3

Rewrite the expression.

$x = \frac{2}{5} \cdot 4$

Step 5.4.2.3.2

Simplify the right side.

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Step 5.4.2.3.2.1

Multiply $\frac{2}{5} \cdot 4$ .

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Step 5.4.2.3.2.1.1

Combine $\frac{2}{5}$ and $4$ .

$x = \frac{2 \cdot 4}{5}$

Step 5.4.2.3.2.1.2

Multiply $2$ by $4$ .

$x = \frac{8}{5}$

Step 5.5

The final solution is all the values that make ${(\frac{x}{2} - 4)}^{3} (\frac{5 x}{2} - 4) = 0$ true.

$x = 8, \frac{8}{5}$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 7

Critical points to evaluate.

$x = 8, \frac{8}{5}$

Step 8

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

$\frac{{((8) - 8)}^{2} (5 (8) - 16)}{4}$

Step 9

Evaluate the second derivative.

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Step 9.1

Reduce the expression by cancelling the common factors.

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Step 9.1.1

Cancel the common factor of $5 (8) - 16$ and $4$ .

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Step 9.1.1.1

Factor $4$ out of ${((8) - 8)}^{2} (5 (8) - 16)$ .

$\frac{4 ({((8) - 8)}^{2} (5 \cdot (2) - 4))}{4}$

Step 9.1.1.2

Cancel the common factors.

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Step 9.1.1.2.1

Factor $4$ out of $4$ .

$\frac{4 ({((8) - 8)}^{2} (5 \cdot (2) - 4))}{4 (1)}$

Step 9.1.1.2.2

Cancel the common factor.

$\frac{4 ({((8) - 8)}^{2} (5 \cdot (2) - 4))}{4 \cdot 1}$

Step 9.1.1.2.3

Rewrite the expression.

$\frac{{((8) - 8)}^{2} (5 \cdot (2) - 4)}{1}$

Step 9.1.1.2.4

Divide ${((8) - 8)}^{2} (5 \cdot (2) - 4)$ by $1$ .

${((8) - 8)}^{2} (5 \cdot (2) - 4)$

Step 9.1.2

Simplify the expression.

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Step 9.1.2.1

Subtract $8$ from $8$ .

$0^{2} (5 \cdot (2) - 4)$

Step 9.1.2.2

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

$0 (5 \cdot (2) - 4)$

Step 9.2

Multiply $5$ by $2$ .

$0 (10 - 4)$

Step 9.3

Simplify the expression.

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Step 9.3.1

Subtract $4$ from $10$ .

$0 \cdot 6$

Step 9.3.2

Multiply $0$ by $6$ .

$0$

Step 10

Since there is at least one point with $0$ or undefined second derivative, apply the first derivative test.

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Step 10.1

Split $(- \infty, \infty)$ into separate intervals around the $x$ values that make the first derivative $0$ or undefined.

$(- \infty, \frac{8}{5}) \cup (\frac{8}{5}, 8) \cup (8, \infty)$

Step 10.2

Substitute any number, such as $0$ , from the interval $(- \infty, \frac{8}{5})$ in the first derivative ${(\frac{x}{2} - 4)}^{3} (\frac{5 x}{2} - 4)$ to check if the result is negative or positive.

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Step 10.2.1

Replace the variable $x$ with $0$ in the expression.

$f' (0) = {(\frac{0}{2} - 4)}^{3} (\frac{5 (0)}{2} - 4)$

Step 10.2.2

Simplify the result.

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Step 10.2.2.1

Divide $0$ by $2$ .

$f' (0) = {(0 - 4)}^{3} (\frac{5 (0)}{2} - 4)$

Step 10.2.2.2

Subtract $4$ from $0$ .

$f' (0) = {(- 4)}^{3} (\frac{5 (0)}{2} - 4)$

Step 10.2.2.3

Raise $- 4$ to the power of $3$ .

$f' (0) = - 64 (\frac{5 (0)}{2} - 4)$

Step 10.2.2.4

Multiply $5$ by $0$ .

$f' (0) = - 64 (\frac{0}{2} - 4)$

Step 10.2.2.5

Divide $0$ by $2$ .

$f' (0) = - 64 (0 - 4)$

Step 10.2.2.6

Subtract $4$ from $0$ .

$f' (0) = - 64 \cdot - 4$

Step 10.2.2.7

Multiply $- 64$ by $- 4$ .

$f' (0) = 256$

Step 10.2.2.8

The final answer is $256$ .

$256$

Step 10.3

Substitute any number, such as $5$ , from the interval $(\frac{8}{5}, 8)$ in the first derivative ${(\frac{x}{2} - 4)}^{3} (\frac{5 x}{2} - 4)$ to check if the result is negative or positive.

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Step 10.3.1

Replace the variable $x$ with $5$ in the expression.

$f' (5) = {(\frac{5}{2} - 4)}^{3} (\frac{5 (5)}{2} - 4)$

Step 10.3.2

Simplify the result.

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Step 10.3.2.1

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

$f' (5) = {(\frac{5}{2} - 4 \cdot \frac{2}{2})}^{3} (\frac{5 (5)}{2} - 4)$

Step 10.3.2.2

Combine $- 4$ and $\frac{2}{2}$ .

$f' (5) = {(\frac{5}{2} + \frac{- 4 \cdot 2}{2})}^{3} (\frac{5 (5)}{2} - 4)$

Step 10.3.2.3

Combine the numerators over the common denominator.

$f' (5) = {(\frac{5 - 4 \cdot 2}{2})}^{3} (\frac{5 (5)}{2} - 4)$

Step 10.3.2.4

Simplify the numerator.

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Step 10.3.2.4.1

Multiply $- 4$ by $2$ .

$f' (5) = {(\frac{5 - 8}{2})}^{3} (\frac{5 (5)}{2} - 4)$

Step 10.3.2.4.2

Subtract $8$ from $5$ .

$f' (5) = {(\frac{- 3}{2})}^{3} (\frac{5 (5)}{2} - 4)$

Step 10.3.2.5

Move the negative in front of the fraction.

$f' (5) = {(- \frac{3}{2})}^{3} (\frac{5 (5)}{2} - 4)$

Step 10.3.2.6

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

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Step 10.3.2.6.1

Apply the product rule to $- \frac{3}{2}$ .

$f' (5) = {(- 1)}^{3} {(\frac{3}{2})}^{3} (\frac{5 (5)}{2} - 4)$

Step 10.3.2.6.2

Apply the product rule to $\frac{3}{2}$ .

$f' (5) = {(- 1)}^{3} (\frac{3^{3}}{2^{3}}) \cdot (\frac{5 (5)}{2} - 4)$

Step 10.3.2.7

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

$f' (5) = - \frac{3^{3}}{2^{3}} \cdot (\frac{5 (5)}{2} - 4)$

Step 10.3.2.8

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

$f' (5) = - \frac{27}{2^{3}} \cdot (\frac{5 (5)}{2} - 4)$

Step 10.3.2.9

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

$f' (5) = - \frac{27}{8} \cdot (\frac{5 (5)}{2} - 4)$

Step 10.3.2.10

Multiply $5$ by $5$ .

$f' (5) = - \frac{27}{8} \cdot (\frac{25}{2} - 4)$

Step 10.3.2.11

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

$f' (5) = - \frac{27}{8} \cdot (\frac{25}{2} - 4 \cdot \frac{2}{2})$

Step 10.3.2.12

Combine $- 4$ and $\frac{2}{2}$ .

$f' (5) = - \frac{27}{8} \cdot (\frac{25}{2} + \frac{- 4 \cdot 2}{2})$

Step 10.3.2.13

Combine the numerators over the common denominator.

$f' (5) = - \frac{27}{8} \cdot \frac{25 - 4 \cdot 2}{2}$

Step 10.3.2.14

Simplify the numerator.

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Step 10.3.2.14.1

Multiply $- 4$ by $2$ .

$f' (5) = - \frac{27}{8} \cdot \frac{25 - 8}{2}$

Step 10.3.2.14.2

Subtract $8$ from $25$ .

$f' (5) = - \frac{27}{8} \cdot \frac{17}{2}$

Step 10.3.2.15

Multiply $- \frac{27}{8} \cdot \frac{17}{2}$ .

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Step 10.3.2.15.1

Multiply $\frac{17}{2}$ by $\frac{27}{8}$ .

$f' (5) = - \frac{17 \cdot 27}{2 \cdot 8}$

Step 10.3.2.15.2

Multiply $17$ by $27$ .

$f' (5) = - \frac{459}{2 \cdot 8}$

Step 10.3.2.15.3

Multiply $2$ by $8$ .

$f' (5) = - \frac{459}{16}$

Step 10.3.2.16

The final answer is $- \frac{459}{16}$ .

$- \frac{459}{16}$

Step 10.4

Substitute any number, such as $10$ , from the interval $(8, \infty)$ in the first derivative ${(\frac{x}{2} - 4)}^{3} (\frac{5 x}{2} - 4)$ to check if the result is negative or positive.

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Step 10.4.1

Replace the variable $x$ with $10$ in the expression.

$f' (10) = {(\frac{10}{2} - 4)}^{3} (\frac{5 (10)}{2} - 4)$

Step 10.4.2

Simplify the result.

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Step 10.4.2.1

Divide $10$ by $2$ .

$f' (10) = {(5 - 4)}^{3} (\frac{5 (10)}{2} - 4)$

Step 10.4.2.2

Subtract $4$ from $5$ .

$f' (10) = 1^{3} (\frac{5 (10)}{2} - 4)$

Step 10.4.2.3

One to any power is one.

$f' (10) = 1 (\frac{5 (10)}{2} - 4)$

Step 10.4.2.4

Multiply $\frac{5 (10)}{2} - 4$ by $1$ .

$f' (10) = \frac{5 (10)}{2} - 4$

Step 10.4.2.5

Multiply $5$ by $10$ .

$f' (10) = \frac{50}{2} - 4$

Step 10.4.2.6

Divide $50$ by $2$ .

$f' (10) = 25 - 4$

Step 10.4.2.7

Subtract $4$ from $25$ .

$f' (10) = 21$

Step 10.4.2.8

The final answer is $21$ .

$21$

Step 10.5

Since the first derivative changed signs from positive to negative around $x = \frac{8}{5}$ , then $x = \frac{8}{5}$ is a local maximum.

$x = \frac{8}{5}$ is a local maximum

Step 10.6

Since the first derivative changed signs from negative to positive around $x = 8$ , then $x = 8$ is a local minimum.

$x = 8$ is a local minimum

Step 10.7

These are the local extrema for $f (x) = x {(\frac{x}{2} - 4)}^{4}$ .

$x = \frac{8}{5}$ is a local maximum

$x = 8$ is a local minimum

$x = \frac{8}{5}$ is a local maximum

$x = 8$ is a local minimum

Step 11