Calculus Examples

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

Step 1

Write $y = x {(\frac{x}{2} - 5)}^{4}$ as a function.

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

Step 2

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

$x \frac{d}{d x} [{(\frac{x}{2} - 5)}^{4}] + {(\frac{x}{2} - 5)}^{4} \frac{d}{d x} [x]$

Step 2.2

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 $g (x) = \frac{x}{2} - 5$ .

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Step 2.2.1

To apply the Chain Rule, set $u$ as $\frac{x}{2} - 5$ .

$x (\frac{d}{d u} [u^{4}] \frac{d}{d x} [\frac{x}{2} - 5]) + {(\frac{x}{2} - 5)}^{4} \frac{d}{d x} [x]$

Step 2.2.2

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

$x (4 u^{3} \frac{d}{d x} [\frac{x}{2} - 5]) + {(\frac{x}{2} - 5)}^{4} \frac{d}{d x} [x]$

Step 2.2.3

Replace all occurrences of $u$ with $\frac{x}{2} - 5$ .

$x (4 {(\frac{x}{2} - 5)}^{3} \frac{d}{d x} [\frac{x}{2} - 5]) + {(\frac{x}{2} - 5)}^{4} \frac{d}{d x} [x]$

Step 2.3

Differentiate.

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Step 2.3.1

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

$x (4 {(\frac{x}{2} - 5)}^{3} (\frac{d}{d x} [\frac{x}{2}] + \frac{d}{d x} [- 5])) + {(\frac{x}{2} - 5)}^{4} \frac{d}{d x} [x]$

Step 2.3.2

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

$x (4 {(\frac{x}{2} - 5)}^{3} (\frac{1}{2} \frac{d}{d x} [x] + \frac{d}{d x} [- 5])) + {(\frac{x}{2} - 5)}^{4} \frac{d}{d x} [x]$

Step 2.3.3

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

$x (4 {(\frac{x}{2} - 5)}^{3} (\frac{1}{2} \cdot 1 + \frac{d}{d x} [- 5])) + {(\frac{x}{2} - 5)}^{4} \frac{d}{d x} [x]$

Step 2.3.4

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

$x (4 {(\frac{x}{2} - 5)}^{3} (\frac{1}{2} + \frac{d}{d x} [- 5])) + {(\frac{x}{2} - 5)}^{4} \frac{d}{d x} [x]$

Step 2.3.5

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5$ with respect to $x$ is $0$ .

$x (4 {(\frac{x}{2} - 5)}^{3} (\frac{1}{2} + 0)) + {(\frac{x}{2} - 5)}^{4} \frac{d}{d x} [x]$

Step 2.3.6

Simplify terms.

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Step 2.3.6.1

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

$x (4 {(\frac{x}{2} - 5)}^{3} \frac{1}{2}) + {(\frac{x}{2} - 5)}^{4} \frac{d}{d x} [x]$

Step 2.3.6.2

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

$x (\frac{4}{2} {(\frac{x}{2} - 5)}^{3}) + {(\frac{x}{2} - 5)}^{4} \frac{d}{d x} [x]$

Step 2.3.6.3

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

$x \frac{4 {(\frac{x}{2} - 5)}^{3}}{2} + {(\frac{x}{2} - 5)}^{4} \frac{d}{d x} [x]$

Step 2.3.6.4

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

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Step 2.3.6.4.1

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

$x \frac{2 (2 {(\frac{x}{2} - 5)}^{3})}{2} + {(\frac{x}{2} - 5)}^{4} \frac{d}{d x} [x]$

Step 2.3.6.4.2

Cancel the common factors.

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Step 2.3.6.4.2.1

Factor $2$ out of $2$ .

$x \frac{2 (2 {(\frac{x}{2} - 5)}^{3})}{2 (1)} + {(\frac{x}{2} - 5)}^{4} \frac{d}{d x} [x]$

Step 2.3.6.4.2.2

Cancel the common factor.

$x \frac{2 (2 {(\frac{x}{2} - 5)}^{3})}{2 \cdot 1} + {(\frac{x}{2} - 5)}^{4} \frac{d}{d x} [x]$

Step 2.3.6.4.2.3

Rewrite the expression.

$x \frac{2 {(\frac{x}{2} - 5)}^{3}}{1} + {(\frac{x}{2} - 5)}^{4} \frac{d}{d x} [x]$

Step 2.3.6.4.2.4

Divide $2 {(\frac{x}{2} - 5)}^{3}$ by $1$ .

$x (2 {(\frac{x}{2} - 5)}^{3}) + {(\frac{x}{2} - 5)}^{4} \frac{d}{d x} [x]$

Step 2.3.7

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

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

Step 2.3.8

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

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

Step 2.4

Simplify.

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Step 2.4.1

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

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Step 2.4.1.1

Reorder $x$ and $2$ .

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

Step 2.4.1.2

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

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

Step 2.4.1.3

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

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

Step 2.4.1.4

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

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

Step 2.4.2

Combine terms.

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Step 2.4.2.1

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

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

Step 2.4.2.2

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

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

Step 2.4.2.3

Combine the numerators over the common denominator.

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

Step 2.4.2.4

Multiply $2$ by $2$ .

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

Step 2.4.2.5

Add $4 x$ and $x$ .

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

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

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

Step 3.2

Differentiate.

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.2.4

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

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

Step 3.2.5

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5$ with respect to $x$ is $0$ .

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

Step 3.2.6

Combine fractions.

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Step 3.2.6.1

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

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

Step 3.2.6.2

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

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

Step 3.2.6.3

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

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

Step 3.3

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 $g (x) = \frac{x}{2} - 5$ .

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Step 3.3.1

To apply the Chain Rule, set $u$ as $\frac{x}{2} - 5$ .

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

Step 3.3.2

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

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

Step 3.3.3

Replace all occurrences of $u$ with $\frac{x}{2} - 5$ .

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

Step 3.4

Differentiate.

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Step 3.4.1

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

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

Step 3.4.2

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

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

Step 3.4.3

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

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

Step 3.4.4

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

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

Step 3.4.5

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

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

Step 3.4.6

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5$ with respect to $x$ is $0$ .

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

Step 3.4.7

Combine fractions.

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Step 3.4.7.1

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

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

Step 3.4.7.2

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

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

Step 3.4.7.3

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

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

Step 3.4.7.4

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

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

Combine the numerators over the common denominator.

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

Step 3.8

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

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

Step 3.9

Multiply $3$ by $2$ .

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

Step 3.10

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

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Step 3.10.1

Factor $2$ out of $6 {(\frac{x}{2} - 5)}^{2}$ .

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

Step 3.10.2

Cancel the common factors.

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Step 3.10.2.1

Factor $2$ out of $2$ .

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

Step 3.10.2.2

Cancel the common factor.

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

Step 3.10.2.3

Rewrite the expression.

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

Step 3.10.2.4

Divide $3 {(\frac{x}{2} - 5)}^{2}$ by $1$ .

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

Step 3.11

Simplify.

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Step 3.11.1

Simplify the numerator.

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Step 3.11.1.1

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

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Step 3.11.1.1.1

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

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

Step 3.11.1.1.2

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

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

Step 3.11.1.1.3

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

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

Step 3.11.1.2

Apply the distributive property.

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

Step 3.11.1.3

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

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

Step 3.11.1.4

Multiply $5$ by $- 5$ .

$f'' (x) = \frac{{(\frac{x}{2} - 5)}^{2} (\frac{5 x}{2} - 25 + 3 (\frac{5 x}{2} - 5))}{2}$

Step 3.11.1.5

Apply the distributive property.

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

Step 3.11.1.6

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

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Step 3.11.1.6.1

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

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

Step 3.11.1.6.2

Multiply $5$ by $3$ .

$f'' (x) = \frac{{(\frac{x}{2} - 5)}^{2} (\frac{5 x}{2} - 25 + \frac{15 x}{2} + 3 \cdot - 5)}{2}$

Step 3.11.1.7

Multiply $3$ by $- 5$ .

$f'' (x) = \frac{{(\frac{x}{2} - 5)}^{2} (\frac{5 x}{2} - 25 + \frac{15 x}{2} - 15)}{2}$

Step 3.11.1.8

Combine the numerators over the common denominator.

$f'' (x) = \frac{{(\frac{x}{2} - 5)}^{2} (\frac{5 x + 15 x}{2} - 25 - 15)}{2}$

Step 3.11.1.9

Subtract $15$ from $- 25$ .

$f'' (x) = \frac{{(\frac{x}{2} - 5)}^{2} (\frac{5 x + 15 x}{2} - 40)}{2}$

Step 3.11.1.10

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f'' (x) = \frac{{(\frac{x}{2} - 5 \cdot \frac{2}{2})}^{2} (\frac{5 x + 15 x}{2} - 40)}{2}$

Step 3.11.1.11

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

$f'' (x) = \frac{{(\frac{x}{2} + \frac{- 5 \cdot 2}{2})}^{2} (\frac{5 x + 15 x}{2} - 40)}{2}$

Step 3.11.1.12

Combine the numerators over the common denominator.

$f'' (x) = \frac{{(\frac{x - 5 \cdot 2}{2})}^{2} (\frac{5 x + 15 x}{2} - 40)}{2}$

Step 3.11.1.13

Multiply $- 5$ by $2$ .

$f'' (x) = \frac{{(\frac{x - 10}{2})}^{2} (\frac{5 x + 15 x}{2} - 40)}{2}$

Step 3.11.1.14

Add $5 x$ and $15 x$ .

$f'' (x) = \frac{{(\frac{x - 10}{2})}^{2} (\frac{20 x}{2} - 40)}{2}$

Step 3.11.1.15

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

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Step 3.11.1.15.1

Factor $2$ out of $20 x$ .

$f'' (x) = \frac{{(\frac{x - 10}{2})}^{2} (\frac{2 (10 x)}{2} - 40)}{2}$

Step 3.11.1.15.2

Cancel the common factors.

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Step 3.11.1.15.2.1

Factor $2$ out of $2$ .

$f'' (x) = \frac{{(\frac{x - 10}{2})}^{2} (\frac{2 (10 x)}{2 (1)} - 40)}{2}$

Step 3.11.1.15.2.2

Cancel the common factor.

$f'' (x) = \frac{{(\frac{x - 10}{2})}^{2} (\frac{2 (10 x)}{2 \cdot 1} - 40)}{2}$

Step 3.11.1.15.2.3

Rewrite the expression.

$f'' (x) = \frac{{(\frac{x - 10}{2})}^{2} (\frac{10 x}{1} - 40)}{2}$

Step 3.11.1.15.2.4

Divide $10 x$ by $1$ .

$f'' (x) = \frac{{(\frac{x - 10}{2})}^{2} (10 x - 40)}{2}$

Step 3.11.1.16

Apply the product rule to $\frac{x - 10}{2}$ .

$f'' (x) = \frac{\frac{{(x - 10)}^{2}}{2^{2}} \cdot (10 x - 40)}{2}$

Step 3.11.1.17

Factor $10$ out of $10 x - 40$ .

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Step 3.11.1.17.1

Factor $10$ out of $10 x$ .

$f'' (x) = \frac{\frac{{(x - 10)}^{2}}{2^{2}} \cdot (10 (x) - 40)}{2}$

Step 3.11.1.17.2

Factor $10$ out of $- 40$ .

$f'' (x) = \frac{\frac{{(x - 10)}^{2}}{2^{2}} \cdot (10 x + 10 \cdot - 4)}{2}$

Step 3.11.1.17.3

Factor $10$ out of $10 x + 10 \cdot - 4$ .

$f'' (x) = \frac{\frac{{(x - 10)}^{2}}{2^{2}} \cdot (10 (x - 4))}{2}$

Step 3.11.1.18

Combine $\frac{{(x - 10)}^{2}}{2^{2}}$ and $10$ .

$f'' (x) = \frac{\frac{{(x - 10)}^{2} \cdot 10}{2^{2}} \cdot (x - 4)}{2}$

Step 3.11.1.19

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

$f'' (x) = \frac{\frac{{(x - 10)}^{2} \cdot 10}{4} \cdot (x - 4)}{2}$

Step 3.11.1.20

Cancel the common factor of $10$ and $4$ .

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Step 3.11.1.20.1

Factor $2$ out of ${(x - 10)}^{2} \cdot 10$ .

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

Step 3.11.1.20.2

Cancel the common factors.

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Step 3.11.1.20.2.1

Factor $2$ out of $4$ .

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

Step 3.11.1.20.2.2

Cancel the common factor.

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

Step 3.11.1.20.2.3

Rewrite the expression.

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

Step 3.11.1.21

Move $5$ to the left of ${(x - 10)}^{2}$ .

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

Step 3.11.2

Reorder terms.

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

Step 3.11.3

Factor $\frac{5 {(x - 10)}^{2}}{2}$ out of $\frac{(x - 4) \frac{5 {(x - 10)}^{2}}{2}}{2}$ .

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

Step 3.11.4

Multiply $\frac{5 {(x - 10)}^{2}}{2} \cdot \frac{x - 4}{2}$ .

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Step 3.11.4.1

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

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

Step 3.11.4.2

Multiply $2$ by $2$ .

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

Step 4

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

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

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

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

Step 5.1.2

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 $g (x) = \frac{x}{2} - 5$ .

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Step 5.1.2.1

To apply the Chain Rule, set $u$ as $\frac{x}{2} - 5$ .

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

Step 5.1.2.2

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

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

Step 5.1.2.3

Replace all occurrences of $u$ with $\frac{x}{2} - 5$ .

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

Step 5.1.3

Differentiate.

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Step 5.1.3.1

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

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

Step 5.1.3.2

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

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

Step 5.1.3.3

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

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

Step 5.1.3.4

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

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

Step 5.1.3.5

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5$ with respect to $x$ is $0$ .

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

Step 5.1.3.6

Simplify terms.

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Step 5.1.3.6.1

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

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

Step 5.1.3.6.2

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

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

Step 5.1.3.6.3

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

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

Step 5.1.3.6.4

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

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Step 5.1.3.6.4.1

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

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

Step 5.1.3.6.4.2

Cancel the common factors.

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Step 5.1.3.6.4.2.1

Factor $2$ out of $2$ .

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

Step 5.1.3.6.4.2.2

Cancel the common factor.

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

Step 5.1.3.6.4.2.3

Rewrite the expression.

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

Step 5.1.3.6.4.2.4

Divide $2 {(\frac{x}{2} - 5)}^{3}$ by $1$ .

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

Step 5.1.3.7

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

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

Step 5.1.3.8

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

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

Step 5.1.4

Simplify.

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Step 5.1.4.1

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

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Step 5.1.4.1.1

Reorder $x$ and $2$ .

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

Step 5.1.4.1.2

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

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

Step 5.1.4.1.3

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

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

Step 5.1.4.1.4

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

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

Step 5.1.4.2

Combine terms.

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Step 5.1.4.2.1

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

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

Step 5.1.4.2.2

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

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

Step 5.1.4.2.3

Combine the numerators over the common denominator.

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

Step 5.1.4.2.4

Multiply $2$ by $2$ .

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

Step 5.1.4.2.5

Add $4 x$ and $x$ .

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

Step 5.2

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

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

Step 6

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

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Step 6.1

Set the first derivative equal to $0$ .

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

Step 6.2

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

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

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

Step 6.3

Set ${(\frac{x}{2} - 5)}^{3}$ equal to $0$ and solve for $x$ .

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Step 6.3.1

Set ${(\frac{x}{2} - 5)}^{3}$ equal to $0$ .

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

Step 6.3.2

Solve ${(\frac{x}{2} - 5)}^{3} = 0$ for $x$ .

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Step 6.3.2.1

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

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

Step 6.3.2.2

Solve for $x$ .

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Step 6.3.2.2.1

Add $5$ to both sides of the equation.

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

Step 6.3.2.2.2

Multiply both sides of the equation by $2$ .

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

Step 6.3.2.2.3

Simplify both sides of the equation.

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Step 6.3.2.2.3.1

Simplify the left side.

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Step 6.3.2.2.3.1.1

Cancel the common factor of $2$ .

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Step 6.3.2.2.3.1.1.1

Cancel the common factor.

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

Step 6.3.2.2.3.1.1.2

Rewrite the expression.

$x = 2 \cdot 5$

Step 6.3.2.2.3.2

Simplify the right side.

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Step 6.3.2.2.3.2.1

Multiply $2$ by $5$ .

$x = 10$

Step 6.4

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

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Step 6.4.1

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

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

Step 6.4.2

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

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Step 6.4.2.1

Add $5$ to both sides of the equation.

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

Step 6.4.2.2

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

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

Step 6.4.2.3

Simplify both sides of the equation.

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Step 6.4.2.3.1

Simplify the left side.

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Step 6.4.2.3.1.1

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

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Step 6.4.2.3.1.1.1

Cancel the common factor of $2$ .

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Step 6.4.2.3.1.1.1.1

Cancel the common factor.

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

Step 6.4.2.3.1.1.1.2

Rewrite the expression.

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

Step 6.4.2.3.1.1.2

Cancel the common factor of $5$ .

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Step 6.4.2.3.1.1.2.1

Factor $5$ out of $5 x$ .

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

Step 6.4.2.3.1.1.2.2

Cancel the common factor.

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

Step 6.4.2.3.1.1.2.3

Rewrite the expression.

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

Step 6.4.2.3.2

Simplify the right side.

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Step 6.4.2.3.2.1

Cancel the common factor of $5$ .

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Step 6.4.2.3.2.1.1

Cancel the common factor.

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

Step 6.4.2.3.2.1.2

Rewrite the expression.

$x = 2$

Step 6.5

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

$x = 10, 2$

Step 7

Find the values where the derivative is undefined.

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Step 7.1

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 8

Critical points to evaluate.

$x = 10, 2$

Step 9

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

$\frac{5 {((10) - 10)}^{2} ((10) - 4)}{4}$

Step 10

Evaluate the second derivative.

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Step 10.1

Cancel the common factor of $(10) - 4$ and $4$ .

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Step 10.1.1

Factor $2$ out of $5 {((10) - 10)}^{2} ((10) - 4)$ .

$\frac{2 (5 {((10) - 10)}^{2} (5 - 2))}{4}$

Step 10.1.2

Cancel the common factors.

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Step 10.1.2.1

Factor $2$ out of $4$ .

$\frac{2 (5 {((10) - 10)}^{2} (5 - 2))}{2 (2)}$

Step 10.1.2.2

Cancel the common factor.

$\frac{2 (5 {((10) - 10)}^{2} (5 - 2))}{2 \cdot 2}$

Step 10.1.2.3

Rewrite the expression.

$\frac{5 {((10) - 10)}^{2} (5 - 2)}{2}$

Step 10.2

Simplify the numerator.

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Step 10.2.1

Subtract $10$ from $10$ .

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

Step 10.2.2

Subtract $2$ from $5$ .

$\frac{5 \cdot 0^{2} \cdot 3}{2}$

Step 10.2.3

Multiply $3$ by $5$ .

$\frac{15 \cdot 0^{2}}{2}$

Step 10.2.4

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

$\frac{15 \cdot 0}{2}$

Step 10.3

Simplify the expression.

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Step 10.3.1

Multiply $15$ by $0$ .

$\frac{0}{2}$

Step 10.3.2

Divide $0$ by $2$ .

$0$

Step 11

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

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Step 11.1

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

$(- \infty, 2) \cup (2, 10) \cup (10, \infty)$

Step 11.2

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

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Step 11.2.1

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

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

Step 11.2.2

Simplify the result.

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Step 11.2.2.1

Divide $0$ by $2$ .

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

Step 11.2.2.2

Subtract $5$ from $0$ .

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

Step 11.2.2.3

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

$f' (0) = - 125 (\frac{5 (0)}{2} - 5)$

Step 11.2.2.4

Multiply $5$ by $0$ .

$f' (0) = - 125 (\frac{0}{2} - 5)$

Step 11.2.2.5

Divide $0$ by $2$ .

$f' (0) = - 125 (0 - 5)$

Step 11.2.2.6

Subtract $5$ from $0$ .

$f' (0) = - 125 \cdot - 5$

Step 11.2.2.7

Multiply $- 125$ by $- 5$ .

$f' (0) = 625$

Step 11.2.2.8

The final answer is $625$ .

$625$

Step 11.3

Substitute any number, such as $6$ , from the interval $(2, 10)$ in the first derivative ${(\frac{x}{2} - 5)}^{3} (\frac{5 x}{2} - 5)$ to check if the result is negative or positive.

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Step 11.3.1

Replace the variable $x$ with $6$ in the expression.

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

Step 11.3.2

Simplify the result.

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Step 11.3.2.1

Divide $6$ by $2$ .

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

Step 11.3.2.2

Subtract $5$ from $3$ .

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

Step 11.3.2.3

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

$f' (6) = - 8 (\frac{5 (6)}{2} - 5)$

Step 11.3.2.4

Multiply $5$ by $6$ .

$f' (6) = - 8 (\frac{30}{2} - 5)$

Step 11.3.2.5

Divide $30$ by $2$ .

$f' (6) = - 8 (15 - 5)$

Step 11.3.2.6

Subtract $5$ from $15$ .

$f' (6) = - 8 \cdot 10$

Step 11.3.2.7

Multiply $- 8$ by $10$ .

$f' (6) = - 80$

Step 11.3.2.8

The final answer is $- 80$ .

$- 80$

Step 11.4

Substitute any number, such as $12$ , from the interval $(10, \infty)$ in the first derivative ${(\frac{x}{2} - 5)}^{3} (\frac{5 x}{2} - 5)$ to check if the result is negative or positive.

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Step 11.4.1

Replace the variable $x$ with $12$ in the expression.

$f' (12) = {(\frac{12}{2} - 5)}^{3} (\frac{5 (12)}{2} - 5)$

Step 11.4.2

Simplify the result.

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Step 11.4.2.1

Divide $12$ by $2$ .

$f' (12) = {(6 - 5)}^{3} (\frac{5 (12)}{2} - 5)$

Step 11.4.2.2

Subtract $5$ from $6$ .

$f' (12) = 1^{3} (\frac{5 (12)}{2} - 5)$

Step 11.4.2.3

One to any power is one.

$f' (12) = 1 (\frac{5 (12)}{2} - 5)$

Step 11.4.2.4

Multiply $\frac{5 (12)}{2} - 5$ by $1$ .

$f' (12) = \frac{5 (12)}{2} - 5$

Step 11.4.2.5

Multiply $5$ by $12$ .

$f' (12) = \frac{60}{2} - 5$

Step 11.4.2.6

Divide $60$ by $2$ .

$f' (12) = 30 - 5$

Step 11.4.2.7

Subtract $5$ from $30$ .

$f' (12) = 25$

Step 11.4.2.8

The final answer is $25$ .

$25$

Step 11.5

Since the first derivative changed signs from positive to negative around $x = 2$ , then $x = 2$ is a local maximum.

$x = 2$ is a local maximum

Step 11.6

Since the first derivative changed signs from negative to positive around $x = 10$ , then $x = 10$ is a local minimum.

$x = 10$ is a local minimum

Step 11.7

These are the local extrema for $f (x) = x {(\frac{x}{2} - 5)}^{4}$ .

$x = 2$ is a local maximum

$x = 10$ is a local minimum

$x = 2$ is a local maximum

$x = 10$ is a local minimum

Step 12