Calculus Examples

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

Step 1

Write $x {(8 - x)}^{\frac{1}{3}}$ as a function.

$f (x) = x {(8 - x)}^{\frac{1}{3}}$

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 $g (x) = {(8 - x)}^{\frac{1}{3}}$ .

$x \frac{d}{d x} [{(8 - x)}^{\frac{1}{3}}] + {(8 - x)}^{\frac{1}{3}} \frac{d}{d x} [x]$

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

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Step 2.2.1

To apply the Chain Rule, set $u$ as $8 - x$ .

$x (\frac{d}{d u} [u^{\frac{1}{3}}] \frac{d}{d x} [8 - x]) + {(8 - x)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 2.2.2

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

$x (\frac{1}{3} u^{\frac{1}{3} - 1} \frac{d}{d x} [8 - x]) + {(8 - x)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 2.2.3

Replace all occurrences of $u$ with $8 - x$ .

$x (\frac{1}{3} {(8 - x)}^{\frac{1}{3} - 1} \frac{d}{d x} [8 - x]) + {(8 - x)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 2.3

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

$x (\frac{1}{3} {(8 - x)}^{\frac{1}{3} - 1 \cdot \frac{3}{3}} \frac{d}{d x} [8 - x]) + {(8 - x)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 2.4

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

$x (\frac{1}{3} {(8 - x)}^{\frac{1}{3} + \frac{- 1 \cdot 3}{3}} \frac{d}{d x} [8 - x]) + {(8 - x)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 2.5

Combine the numerators over the common denominator.

$x (\frac{1}{3} {(8 - x)}^{\frac{1 - 1 \cdot 3}{3}} \frac{d}{d x} [8 - x]) + {(8 - x)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Multiply $- 1$ by $3$ .

$x (\frac{1}{3} {(8 - x)}^{\frac{1 - 3}{3}} \frac{d}{d x} [8 - x]) + {(8 - x)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 2.6.2

Subtract $3$ from $1$ .

$x (\frac{1}{3} {(8 - x)}^{\frac{- 2}{3}} \frac{d}{d x} [8 - x]) + {(8 - x)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 2.7

Combine fractions.

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Step 2.7.1

Move the negative in front of the fraction.

$x (\frac{1}{3} {(8 - x)}^{- \frac{2}{3}} \frac{d}{d x} [8 - x]) + {(8 - x)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 2.7.2

Combine $\frac{1}{3}$ and ${(8 - x)}^{- \frac{2}{3}}$ .

$x (\frac{{(8 - x)}^{- \frac{2}{3}}}{3} \frac{d}{d x} [8 - x]) + {(8 - x)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 2.7.3

Move ${(8 - x)}^{- \frac{2}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$x (\frac{1}{3 {(8 - x)}^{\frac{2}{3}}} \frac{d}{d x} [8 - x]) + {(8 - x)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 2.7.4

Combine $\frac{1}{3 {(8 - x)}^{\frac{2}{3}}}$ and $x$ .

$\frac{x}{3 {(8 - x)}^{\frac{2}{3}}} \frac{d}{d x} [8 - x] + {(8 - x)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 2.8

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

$\frac{x}{3 {(8 - x)}^{\frac{2}{3}}} (\frac{d}{d x} [8] + \frac{d}{d x} [- x]) + {(8 - x)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 2.9

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

$\frac{x}{3 {(8 - x)}^{\frac{2}{3}}} (0 + \frac{d}{d x} [- x]) + {(8 - x)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 2.10

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

$\frac{x}{3 {(8 - x)}^{\frac{2}{3}}} \frac{d}{d x} [- x] + {(8 - x)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 2.11

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

$\frac{x}{3 {(8 - x)}^{\frac{2}{3}}} (- \frac{d}{d x} [x]) + {(8 - x)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 2.12

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

$\frac{x}{3 {(8 - x)}^{\frac{2}{3}}} (- 1 \cdot 1) + {(8 - x)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 2.13

Combine fractions.

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Step 2.13.1

Multiply $- 1$ by $1$ .

$\frac{x}{3 {(8 - x)}^{\frac{2}{3}}} \cdot - 1 + {(8 - x)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 2.13.2

Combine $\frac{x}{3 {(8 - x)}^{\frac{2}{3}}}$ and $- 1$ .

$\frac{x \cdot - 1}{3 {(8 - x)}^{\frac{2}{3}}} + {(8 - x)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 2.13.3

Simplify the expression.

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Step 2.13.3.1

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

$\frac{- 1 \cdot x}{3 {(8 - x)}^{\frac{2}{3}}} + {(8 - x)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 2.13.3.2

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

$\frac{- x}{3 {(8 - x)}^{\frac{2}{3}}} + {(8 - x)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 2.13.3.3

Move the negative in front of the fraction.

$- \frac{x}{3 {(8 - x)}^{\frac{2}{3}}} + {(8 - x)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 2.14

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

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

Step 2.15

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

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

Step 2.16

To write ${(8 - x)}^{\frac{1}{3}}$ as a fraction with a common denominator, multiply by $\frac{3 {(8 - x)}^{\frac{2}{3}}}{3 {(8 - x)}^{\frac{2}{3}}}$ .

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

Step 2.17

Combine ${(8 - x)}^{\frac{1}{3}}$ and $\frac{3 {(8 - x)}^{\frac{2}{3}}}{3 {(8 - x)}^{\frac{2}{3}}}$ .

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

Step 2.18

Combine the numerators over the common denominator.

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

Step 2.19

Multiply ${(8 - x)}^{\frac{1}{3}}$ by ${(8 - x)}^{\frac{2}{3}}$ by adding the exponents.

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Step 2.19.1

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

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

Step 2.19.2

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

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

Step 2.19.3

Combine the numerators over the common denominator.

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

Step 2.19.4

Add $2$ and $1$ .

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

Step 2.19.5

Divide $3$ by $3$ .

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

Step 2.20

Simplify ${(8 - x)}^{1} \cdot 3$ .

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

Step 2.21

Move $3$ to the left of $8 - x$ .

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

Step 2.22

Simplify.

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Step 2.22.1

Apply the distributive property.

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

Step 2.22.2

Simplify the numerator.

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Step 2.22.2.1

Simplify each term.

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Step 2.22.2.1.1

Multiply $3$ by $8$ .

$\frac{- x + 24 + 3 (- x)}{3 {(8 - x)}^{\frac{2}{3}}}$

Step 2.22.2.1.2

Multiply $- 1$ by $3$ .

$\frac{- x + 24 - 3 x}{3 {(8 - x)}^{\frac{2}{3}}}$

Step 2.22.2.2

Subtract $3 x$ from $- x$ .

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

Step 2.22.3

Factor $4$ out of $- 4 x + 24$ .

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Step 2.22.3.1

Factor $4$ out of $- 4 x$ .

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

Step 2.22.3.2

Factor $4$ out of $24$ .

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

Step 2.22.3.3

Factor $4$ out of $4 (- x) + 4 (6)$ .

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

Step 2.22.4

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

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

Step 2.22.5

Rewrite $6$ as $- 1 (- 6)$ .

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

Step 2.22.6

Factor $- 1$ out of $- (x) - 1 (- 6)$ .

$\frac{4 (- (x - 6))}{3 {(8 - x)}^{\frac{2}{3}}}$

Step 2.22.7

Rewrite $- (x - 6)$ as $- 1 (x - 6)$ .

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

Step 2.22.8

Move the negative in front of the fraction.

$- \frac{4 (x - 6)}{3 {(8 - x)}^{\frac{2}{3}}}$

Step 3

Find the second derivative of the function.

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Step 3.1

Since $- \frac{4}{3}$ is constant with respect to $x$ , the derivative of $- \frac{4 (x - 6)}{3 {(8 - x)}^{\frac{2}{3}}}$ with respect to $x$ is $- \frac{4}{3} \frac{d}{d x} [\frac{x - 6}{{(8 - x)}^{\frac{2}{3}}}]$ .

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

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 $f (x) = x - 6$ and $g (x) = {(8 - x)}^{\frac{2}{3}}$ .

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

Step 3.3

Differentiate.

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Step 3.3.1

Multiply the exponents in ${({(8 - x)}^{\frac{2}{3}})}^{2}$ .

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Step 3.3.1.1

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

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

Step 3.3.1.2

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

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Step 3.3.1.2.1

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

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

Step 3.3.1.2.2

Multiply $2$ by $2$ .

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

Step 3.3.2

By the Sum Rule, the derivative of $x - 6$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 6]$ .

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

Step 3.3.3

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

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

Step 3.3.4

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6$ with respect to $x$ is $0$ .

$f'' (x) = - \frac{4}{3} \cdot \frac{{(8 - x)}^{\frac{2}{3}} (1 + 0) - (x - 6) \frac{d}{d x} {(8 - x)}^{\frac{2}{3}}}{{(8 - x)}^{\frac{4}{3}}}$

Step 3.3.5

Simplify the expression.

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Step 3.3.5.1

Add $1$ and $0$ .

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

Step 3.3.5.2

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

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

Step 3.4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{2}{3}}$ and $g (x) = 8 - x$ .

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Step 3.4.1

To apply the Chain Rule, set $u$ as $8 - x$ .

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

Step 3.4.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{2}{3}$ .

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

Step 3.4.3

Replace all occurrences of $u$ with $8 - x$ .

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

Combine the numerators over the common denominator.

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

Step 3.8

Simplify the numerator.

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Step 3.8.1

Multiply $- 1$ by $3$ .

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

Step 3.8.2

Subtract $3$ from $2$ .

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

Step 3.9

Combine fractions.

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Step 3.9.1

Move the negative in front of the fraction.

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

Step 3.9.2

Combine $\frac{2}{3}$ and ${(8 - x)}^{- \frac{1}{3}}$ .

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

Step 3.9.3

Move ${(8 - x)}^{- \frac{1}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 3.10

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

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

Step 3.11

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

$f'' (x) = - \frac{4}{3} \cdot \frac{{(8 - x)}^{\frac{2}{3}} - (x - 6) (\frac{2}{3 {(8 - x)}^{\frac{1}{3}}} \cdot (0 + \frac{d}{d x} (- x)))}{{(8 - x)}^{\frac{4}{3}}}$

Step 3.12

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

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

Step 3.13

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

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

Step 3.14

Multiply.

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Step 3.14.1

Multiply $- 1$ by $- 1$ .

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

Step 3.14.2

Multiply $x - 6$ by $1$ .

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

Step 3.15

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

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

Step 3.16

Combine fractions.

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Step 3.16.1

Multiply $x - 6$ by $1$ .

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

Step 3.16.2

Multiply $\frac{{(8 - x)}^{\frac{2}{3}} + (x - 6) \frac{2}{3 {(8 - x)}^{\frac{1}{3}}}}{{(8 - x)}^{\frac{4}{3}}}$ by $\frac{4}{3}$ .

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

Step 3.16.3

Reorder.

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Step 3.16.3.1

Move $4$ to the left of ${(8 - x)}^{\frac{2}{3}} + (x - 6) \frac{2}{3 {(8 - x)}^{\frac{1}{3}}}$ .

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

Step 3.16.3.2

Move $3$ to the left of ${(8 - x)}^{\frac{4}{3}}$ .

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

Step 3.17

Simplify.

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Step 3.17.1

Apply the distributive property.

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

Step 3.17.2

Simplify the numerator.

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Step 3.17.2.1

Factor $4$ out of $4 {(8 - x)}^{\frac{2}{3}} + 4 (x - 6) \frac{2}{3 {(8 - x)}^{\frac{1}{3}}}$ .

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Step 3.17.2.1.1

Factor $4$ out of $4 {(8 - x)}^{\frac{2}{3}}$ .

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

Step 3.17.2.1.2

Factor $4$ out of $4 (x - 6) \frac{2}{3 {(8 - x)}^{\frac{1}{3}}}$ .

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

Step 3.17.2.1.3

Factor $4$ out of $4 ({(8 - x)}^{\frac{2}{3}}) + 4 ((x - 6) \frac{2}{3 {(8 - x)}^{\frac{1}{3}}})$ .

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

Step 3.17.2.2

Multiply $x - 6$ by $\frac{2}{3 {(8 - x)}^{\frac{1}{3}}}$ .

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

Step 3.17.2.3

Move $2$ to the left of $x - 6$ .

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

Step 3.17.2.4

To write ${(8 - x)}^{\frac{2}{3}}$ as a fraction with a common denominator, multiply by $\frac{3 {(8 - x)}^{\frac{1}{3}}}{3 {(8 - x)}^{\frac{1}{3}}}$ .

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

Step 3.17.2.5

Combine ${(8 - x)}^{\frac{2}{3}}$ and $\frac{3 {(8 - x)}^{\frac{1}{3}}}{3 {(8 - x)}^{\frac{1}{3}}}$ .

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

Step 3.17.2.6

Combine the numerators over the common denominator.

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

Step 3.17.2.7

Rewrite $\frac{{(8 - x)}^{\frac{2}{3}} (3 {(8 - x)}^{\frac{1}{3}}) + 2 (x - 6)}{3 {(8 - x)}^{\frac{1}{3}}}$ in a factored form.

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Step 3.17.2.7.1

Rewrite using the commutative property of multiplication.

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

Step 3.17.2.7.2

Multiply ${(8 - x)}^{\frac{2}{3}}$ by ${(8 - x)}^{\frac{1}{3}}$ by adding the exponents.

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Step 3.17.2.7.2.1

Move ${(8 - x)}^{\frac{1}{3}}$ .

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

Step 3.17.2.7.2.2

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

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

Step 3.17.2.7.2.3

Combine the numerators over the common denominator.

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

Step 3.17.2.7.2.4

Add $1$ and $2$ .

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

Step 3.17.2.7.2.5

Divide $3$ by $3$ .

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

Step 3.17.2.7.3

Simplify $3 {(8 - x)}^{1}$ .

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

Step 3.17.2.7.4

Apply the distributive property.

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

Step 3.17.2.7.5

Multiply $3$ by $8$ .

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

Step 3.17.2.7.6

Multiply $- 1$ by $3$ .

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

Step 3.17.2.7.7

Apply the distributive property.

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

Step 3.17.2.7.8

Multiply $2$ by $- 6$ .

$f'' (x) = - \frac{4 (\frac{24 - 3 x + 2 x - 12}{3 {(8 - x)}^{\frac{1}{3}}})}{3 {(8 - x)}^{\frac{4}{3}}}$

Step 3.17.2.7.9

Subtract $12$ from $24$ .

$f'' (x) = - \frac{4 (\frac{- 3 x + 2 x + 12}{3 {(8 - x)}^{\frac{1}{3}}})}{3 {(8 - x)}^{\frac{4}{3}}}$

Step 3.17.2.7.10

Add $- 3 x$ and $2 x$ .

$f'' (x) = - \frac{4 (\frac{- x + 12}{3 {(8 - x)}^{\frac{1}{3}}})}{3 {(8 - x)}^{\frac{4}{3}}}$

Step 3.17.3

Combine terms.

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Step 3.17.3.1

Combine $4$ and $\frac{- x + 12}{3 {(8 - x)}^{\frac{1}{3}}}$ .

$f'' (x) = - \frac{\frac{4 (- x + 12)}{3 {(8 - x)}^{\frac{1}{3}}}}{3 {(8 - x)}^{\frac{4}{3}}}$

Step 3.17.3.2

Rewrite $\frac{\frac{4 (- x + 12)}{3 {(8 - x)}^{\frac{1}{3}}}}{3 {(8 - x)}^{\frac{4}{3}}}$ as a product.

$f'' (x) = - (\frac{4 (- x + 12)}{3 {(8 - x)}^{\frac{1}{3}}} \cdot \frac{1}{3 {(8 - x)}^{\frac{4}{3}}})$

Step 3.17.3.3

Multiply $\frac{4 (- x + 12)}{3 {(8 - x)}^{\frac{1}{3}}}$ by $\frac{1}{3 {(8 - x)}^{\frac{4}{3}}}$ .

$f'' (x) = - \frac{4 (- x + 12)}{3 {(8 - x)}^{\frac{1}{3}} (3 {(8 - x)}^{\frac{4}{3}})}$

Step 3.17.3.4

Multiply $3$ by $3$ .

$f'' (x) = - \frac{4 (- x + 12)}{9 {(8 - x)}^{\frac{1}{3}} {(8 - x)}^{\frac{4}{3}}}$

Step 3.17.3.5

Multiply ${(8 - x)}^{\frac{1}{3}}$ by ${(8 - x)}^{\frac{4}{3}}$ by adding the exponents.

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Step 3.17.3.5.1

Move ${(8 - x)}^{\frac{4}{3}}$ .

$f'' (x) = - \frac{4 (- x + 12)}{9 ({(8 - x)}^{\frac{4}{3}} {(8 - x)}^{\frac{1}{3}})}$

Step 3.17.3.5.2

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

$f'' (x) = - \frac{4 (- x + 12)}{9 {(8 - x)}^{\frac{4}{3} + \frac{1}{3}}}$

Step 3.17.3.5.3

Combine the numerators over the common denominator.

$f'' (x) = - \frac{4 (- x + 12)}{9 {(8 - x)}^{\frac{4 + 1}{3}}}$

Step 3.17.3.5.4

Add $4$ and $1$ .

$f'' (x) = - \frac{4 (- x + 12)}{9 {(8 - x)}^{\frac{5}{3}}}$

Step 3.17.4

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

$f'' (x) = - \frac{4 (- (x) + 12)}{9 {(8 - x)}^{\frac{5}{3}}}$

Step 3.17.5

Rewrite $12$ as $- 1 (- 12)$ .

$f'' (x) = - \frac{4 (- (x) - 1 \cdot - 12)}{9 {(8 - x)}^{\frac{5}{3}}}$

Step 3.17.6

Factor $- 1$ out of $- (x) - 1 (- 12)$ .

$f'' (x) = - \frac{4 (- (x - 12))}{9 {(8 - x)}^{\frac{5}{3}}}$

Step 3.17.7

Rewrite $- (x - 12)$ as $- 1 (x - 12)$ .

$f'' (x) = - \frac{4 (- 1 (x - 12))}{9 {(8 - x)}^{\frac{5}{3}}}$

Step 3.17.8

Move the negative in front of the fraction.

$f'' (x) = \frac{4 (x - 12)}{9 {(8 - x)}^{\frac{5}{3}}}$

Step 3.17.9

Multiply $- 1$ by $- 1$ .

$f'' (x) = 1 (\frac{4 (x - 12)}{9 {(8 - x)}^{\frac{5}{3}}})$

Step 3.17.10

Multiply $\frac{4 (x - 12)}{9 {(8 - x)}^{\frac{5}{3}}}$ by $1$ .

$f'' (x) = \frac{4 (x - 12)}{9 {(8 - x)}^{\frac{5}{3}}}$

Step 4

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

$- \frac{4 (x - 6)}{3 {(8 - x)}^{\frac{2}{3}}} = 0$

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

$x \frac{d}{d x} [{(8 - x)}^{\frac{1}{3}}] + {(8 - x)}^{\frac{1}{3}} \frac{d}{d x} [x]$

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

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Step 5.1.2.1

To apply the Chain Rule, set $u$ as $8 - x$ .

$x (\frac{d}{d u} [u^{\frac{1}{3}}] \frac{d}{d x} [8 - x]) + {(8 - x)}^{\frac{1}{3}} \frac{d}{d x} [x]$

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 $n = \frac{1}{3}$ .

$x (\frac{1}{3} u^{\frac{1}{3} - 1} \frac{d}{d x} [8 - x]) + {(8 - x)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 5.1.2.3

Replace all occurrences of $u$ with $8 - x$ .

$x (\frac{1}{3} {(8 - x)}^{\frac{1}{3} - 1} \frac{d}{d x} [8 - x]) + {(8 - x)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 5.1.3

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

$x (\frac{1}{3} {(8 - x)}^{\frac{1}{3} - 1 \cdot \frac{3}{3}} \frac{d}{d x} [8 - x]) + {(8 - x)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 5.1.4

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

$x (\frac{1}{3} {(8 - x)}^{\frac{1}{3} + \frac{- 1 \cdot 3}{3}} \frac{d}{d x} [8 - x]) + {(8 - x)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 5.1.5

Combine the numerators over the common denominator.

$x (\frac{1}{3} {(8 - x)}^{\frac{1 - 1 \cdot 3}{3}} \frac{d}{d x} [8 - x]) + {(8 - x)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 5.1.6

Simplify the numerator.

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Step 5.1.6.1

Multiply $- 1$ by $3$ .

$x (\frac{1}{3} {(8 - x)}^{\frac{1 - 3}{3}} \frac{d}{d x} [8 - x]) + {(8 - x)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 5.1.6.2

Subtract $3$ from $1$ .

$x (\frac{1}{3} {(8 - x)}^{\frac{- 2}{3}} \frac{d}{d x} [8 - x]) + {(8 - x)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 5.1.7

Combine fractions.

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Step 5.1.7.1

Move the negative in front of the fraction.

$x (\frac{1}{3} {(8 - x)}^{- \frac{2}{3}} \frac{d}{d x} [8 - x]) + {(8 - x)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 5.1.7.2

Combine $\frac{1}{3}$ and ${(8 - x)}^{- \frac{2}{3}}$ .

$x (\frac{{(8 - x)}^{- \frac{2}{3}}}{3} \frac{d}{d x} [8 - x]) + {(8 - x)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 5.1.7.3

Move ${(8 - x)}^{- \frac{2}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$x (\frac{1}{3 {(8 - x)}^{\frac{2}{3}}} \frac{d}{d x} [8 - x]) + {(8 - x)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 5.1.7.4

Combine $\frac{1}{3 {(8 - x)}^{\frac{2}{3}}}$ and $x$ .

$\frac{x}{3 {(8 - x)}^{\frac{2}{3}}} \frac{d}{d x} [8 - x] + {(8 - x)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 5.1.8

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

$\frac{x}{3 {(8 - x)}^{\frac{2}{3}}} (\frac{d}{d x} [8] + \frac{d}{d x} [- x]) + {(8 - x)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 5.1.9

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

$\frac{x}{3 {(8 - x)}^{\frac{2}{3}}} (0 + \frac{d}{d x} [- x]) + {(8 - x)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 5.1.10

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

$\frac{x}{3 {(8 - x)}^{\frac{2}{3}}} \frac{d}{d x} [- x] + {(8 - x)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 5.1.11

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

$\frac{x}{3 {(8 - x)}^{\frac{2}{3}}} (- \frac{d}{d x} [x]) + {(8 - x)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 5.1.12

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

$\frac{x}{3 {(8 - x)}^{\frac{2}{3}}} (- 1 \cdot 1) + {(8 - x)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 5.1.13

Combine fractions.

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Step 5.1.13.1

Multiply $- 1$ by $1$ .

$\frac{x}{3 {(8 - x)}^{\frac{2}{3}}} \cdot - 1 + {(8 - x)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 5.1.13.2

Combine $\frac{x}{3 {(8 - x)}^{\frac{2}{3}}}$ and $- 1$ .

$\frac{x \cdot - 1}{3 {(8 - x)}^{\frac{2}{3}}} + {(8 - x)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 5.1.13.3

Simplify the expression.

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Step 5.1.13.3.1

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

$\frac{- 1 \cdot x}{3 {(8 - x)}^{\frac{2}{3}}} + {(8 - x)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 5.1.13.3.2

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

$\frac{- x}{3 {(8 - x)}^{\frac{2}{3}}} + {(8 - x)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 5.1.13.3.3

Move the negative in front of the fraction.

$- \frac{x}{3 {(8 - x)}^{\frac{2}{3}}} + {(8 - x)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 5.1.14

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

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

Step 5.1.15

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

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

Step 5.1.16

To write ${(8 - x)}^{\frac{1}{3}}$ as a fraction with a common denominator, multiply by $\frac{3 {(8 - x)}^{\frac{2}{3}}}{3 {(8 - x)}^{\frac{2}{3}}}$ .

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

Step 5.1.17

Combine ${(8 - x)}^{\frac{1}{3}}$ and $\frac{3 {(8 - x)}^{\frac{2}{3}}}{3 {(8 - x)}^{\frac{2}{3}}}$ .

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

Step 5.1.18

Combine the numerators over the common denominator.

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

Step 5.1.19

Multiply ${(8 - x)}^{\frac{1}{3}}$ by ${(8 - x)}^{\frac{2}{3}}$ by adding the exponents.

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Step 5.1.19.1

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

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

Step 5.1.19.2

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

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

Step 5.1.19.3

Combine the numerators over the common denominator.

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

Step 5.1.19.4

Add $2$ and $1$ .

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

Step 5.1.19.5

Divide $3$ by $3$ .

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

Step 5.1.20

Simplify ${(8 - x)}^{1} \cdot 3$ .

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

Step 5.1.21

Move $3$ to the left of $8 - x$ .

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

Step 5.1.22

Simplify.

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Step 5.1.22.1

Apply the distributive property.

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

Step 5.1.22.2

Simplify the numerator.

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Step 5.1.22.2.1

Simplify each term.

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Step 5.1.22.2.1.1

Multiply $3$ by $8$ .

$\frac{- x + 24 + 3 (- x)}{3 {(8 - x)}^{\frac{2}{3}}}$

Step 5.1.22.2.1.2

Multiply $- 1$ by $3$ .

$\frac{- x + 24 - 3 x}{3 {(8 - x)}^{\frac{2}{3}}}$

Step 5.1.22.2.2

Subtract $3 x$ from $- x$ .

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

Step 5.1.22.3

Factor $4$ out of $- 4 x + 24$ .

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Step 5.1.22.3.1

Factor $4$ out of $- 4 x$ .

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

Step 5.1.22.3.2

Factor $4$ out of $24$ .

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

Step 5.1.22.3.3

Factor $4$ out of $4 (- x) + 4 (6)$ .

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

Step 5.1.22.4

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

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

Step 5.1.22.5

Rewrite $6$ as $- 1 (- 6)$ .

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

Step 5.1.22.6

Factor $- 1$ out of $- (x) - 1 (- 6)$ .

$\frac{4 (- (x - 6))}{3 {(8 - x)}^{\frac{2}{3}}}$

Step 5.1.22.7

Rewrite $- (x - 6)$ as $- 1 (x - 6)$ .

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

Step 5.1.22.8

Move the negative in front of the fraction.

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

Step 5.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{4 (x - 6)}{3 {(8 - x)}^{\frac{2}{3}}}$ .

$- \frac{4 (x - 6)}{3 {(8 - x)}^{\frac{2}{3}}}$

Step 6

Set the first derivative equal to $0$ then solve the equation $- \frac{4 (x - 6)}{3 {(8 - x)}^{\frac{2}{3}}} = 0$ .

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Step 6.1

Set the first derivative equal to $0$ .

$- \frac{4 (x - 6)}{3 {(8 - x)}^{\frac{2}{3}}} = 0$

Step 6.2

Set the numerator equal to zero.

$4 (x - 6) = 0$

Step 6.3

Solve the equation for $x$ .

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Step 6.3.1

Divide each term in $4 (x - 6) = 0$ by $4$ and simplify.

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Step 6.3.1.1

Divide each term in $4 (x - 6) = 0$ by $4$ .

$\frac{4 (x - 6)}{4} = \frac{0}{4}$

Step 6.3.1.2

Simplify the left side.

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Step 6.3.1.2.1

Cancel the common factor of $4$ .

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Step 6.3.1.2.1.1

Cancel the common factor.

$\frac{4 (x - 6)}{4} = \frac{0}{4}$

Step 6.3.1.2.1.2

Divide $x - 6$ by $1$ .

$x - 6 = \frac{0}{4}$

Step 6.3.1.3

Simplify the right side.

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Step 6.3.1.3.1

Divide $0$ by $4$ .

$x - 6 = 0$

Step 6.3.2

Add $6$ to both sides of the equation.

$x = 6$

Step 7

Find the values where the derivative is undefined.

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Step 7.1

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

$- \frac{4 (x - 6)}{3 \sqrt[3]{{(8 - x)}^{2}}}$

Step 7.2

Set the denominator in $\frac{4 (x - 6)}{3 \sqrt[3]{{(8 - x)}^{2}}}$ equal to $0$ to find where the expression is undefined.

$3 \sqrt[3]{{(8 - x)}^{2}} = 0$

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, cube both sides of the equation.

${(3 \sqrt[3]{{(8 - x)}^{2}})}^{3} = 0^{3}$

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 $\sqrt[3]{{(8 - x)}^{2}}$ as ${(8 - x)}^{\frac{2}{3}}$ .

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

Step 7.3.2.2

Simplify the left side.

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Step 7.3.2.2.1

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

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Step 7.3.2.2.1.1

Apply the product rule to $3 {(8 - x)}^{\frac{2}{3}}$ .

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

Step 7.3.2.2.1.2

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

$27 {({(8 - x)}^{\frac{2}{3}})}^{3} = 0^{3}$

Step 7.3.2.2.1.3

Multiply the exponents in ${({(8 - x)}^{\frac{2}{3}})}^{3}$ .

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Step 7.3.2.2.1.3.1

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

$27 {(8 - x)}^{\frac{2}{3} \cdot 3} = 0^{3}$

Step 7.3.2.2.1.3.2

Cancel the common factor of $3$ .

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Step 7.3.2.2.1.3.2.1

Cancel the common factor.

$27 {(8 - x)}^{\frac{2}{3} \cdot 3} = 0^{3}$

Step 7.3.2.2.1.3.2.2

Rewrite the expression.

$27 {(8 - x)}^{2} = 0^{3}$

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 $0$ .

$27 {(8 - x)}^{2} = 0$

Step 7.3.3

Solve for $x$ .

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Step 7.3.3.1

Divide each term in $27 {(8 - x)}^{2} = 0$ by $27$ and simplify.

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Step 7.3.3.1.1

Divide each term in $27 {(8 - x)}^{2} = 0$ by $27$ .

$\frac{27 {(8 - x)}^{2}}{27} = \frac{0}{27}$

Step 7.3.3.1.2

Simplify the left side.

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Step 7.3.3.1.2.1

Cancel the common factor of $27$ .

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Step 7.3.3.1.2.1.1

Cancel the common factor.

$\frac{27 {(8 - x)}^{2}}{27} = \frac{0}{27}$

Step 7.3.3.1.2.1.2

Divide ${(8 - x)}^{2}$ by $1$ .

${(8 - x)}^{2} = \frac{0}{27}$

Step 7.3.3.1.3

Simplify the right side.

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Step 7.3.3.1.3.1

Divide $0$ by $27$ .

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

Step 7.3.3.2

Set the $8 - x$ equal to $0$ .

$8 - x = 0$

Step 7.3.3.3

Solve for $x$ .

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Step 7.3.3.3.1

Subtract $8$ from both sides of the equation.

$- x = - 8$

Step 7.3.3.3.2

Divide each term in $- x = - 8$ by $- 1$ and simplify.

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Step 7.3.3.3.2.1

Divide each term in $- x = - 8$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 8}{- 1}$

Step 7.3.3.3.2.2

Simplify the left side.

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Step 7.3.3.3.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 8}{- 1}$

Step 7.3.3.3.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 8}{- 1}$

Step 7.3.3.3.2.3

Simplify the right side.

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Step 7.3.3.3.2.3.1

Divide $- 8$ by $- 1$ .

$x = 8$

Step 8

Critical points to evaluate.

$x = 6, 8$

Step 9

Evaluate the second derivative at $x = 6$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{4 ((6) - 12)}{9 {(8 - (6))}^{\frac{5}{3}}}$

Step 10

Evaluate the second derivative.

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Step 10.1

Factor $3$ out of $4 ((6) - 12)$ .

$\frac{3 (4 (2 - 4))}{9 {(8 - (6))}^{\frac{5}{3}}}$

Step 10.2

Cancel the common factors.

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Step 10.2.1

Factor $3$ out of $9 {(8 - (6))}^{\frac{5}{3}}$ .

$\frac{3 (4 (2 - 4))}{3 (3 {(8 - (6))}^{\frac{5}{3}})}$

Step 10.2.2

Cancel the common factor.

$\frac{3 (4 (2 - 4))}{3 (3 {(8 - (6))}^{\frac{5}{3}})}$

Step 10.2.3

Rewrite the expression.

$\frac{4 (2 - 4)}{3 {(8 - (6))}^{\frac{5}{3}}}$

Step 10.3

Subtract $4$ from $2$ .

$\frac{4 \cdot - 2}{3 {(8 - (6))}^{\frac{5}{3}}}$

Step 10.4

Simplify the denominator.

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Step 10.4.1

Multiply $- 1$ by $6$ .

$\frac{4 \cdot - 2}{3 {(8 - 6)}^{\frac{5}{3}}}$

Step 10.4.2

Subtract $6$ from $8$ .

$\frac{4 \cdot - 2}{3 \cdot 2^{\frac{5}{3}}}$

Step 10.5

Simplify the expression.

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Step 10.5.1

Multiply $4$ by $- 2$ .

$\frac{- 8}{3 \cdot 2^{\frac{5}{3}}}$

Step 10.5.2

Move the negative in front of the fraction.

$- \frac{8}{3 \cdot 2^{\frac{5}{3}}}$

Step 11

$x = 6$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = 6$ is a local maximum

Step 12

Find the y-value when $x = 6$ .

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Step 12.1

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

$f (6) = (6) {(8 - (6))}^{\frac{1}{3}}$

Step 12.2

Simplify the result.

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Step 12.2.1

Multiply $- 1$ by $6$ .

$f (6) = 6 {(8 - 6)}^{\frac{1}{3}}$

Step 12.2.2

Subtract $6$ from $8$ .

$f (6) = 6 \cdot 2^{\frac{1}{3}}$

Step 12.2.3

The final answer is $6 \cdot 2^{\frac{1}{3}}$ .

$y = 6 \cdot 2^{\frac{1}{3}}$

Step 13

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.

$\frac{4 ((8) - 12)}{9 {(8 - (8))}^{\frac{5}{3}}}$

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 $8$ .

$\frac{4 ((8) - 12)}{9 {(8 - 8)}^{\frac{5}{3}}}$

Step 14.1.2

Subtract $8$ from $8$ .

$\frac{4 ((8) - 12)}{9 \cdot 0^{\frac{5}{3}}}$

Step 14.1.3

Rewrite $0$ as $0^{3}$ .

$\frac{4 ((8) - 12)}{9 \cdot {(0^{3})}^{\frac{5}{3}}}$

Step 14.1.4

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

$\frac{4 ((8) - 12)}{9 \cdot 0^{3 (\frac{5}{3})}}$

Step 14.2

Cancel the common factor of $3$ .

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Step 14.2.1

Cancel the common factor.

$\frac{4 ((8) - 12)}{9 \cdot 0^{3 (\frac{5}{3})}}$

Step 14.2.2

Rewrite the expression.

$\frac{4 ((8) - 12)}{9 \cdot 0^{5}}$

Step 14.3

Simplify the expression.

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Step 14.3.1

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

$\frac{4 ((8) - 12)}{9 \cdot 0}$

Step 14.3.2

Multiply $9$ by $0$ .

$\frac{4 ((8) - 12)}{0}$

Step 14.3.3

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

Undefined

$\frac{4 ((8) - 12)}{0}$

Step 14.4

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

Undefined

Step 15

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

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Step 15.1

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

$(- \infty, 6) \cup (6, 8) \cup (8, \infty)$

Step 15.2

Substitute any number, such as $0$ , from the interval $(- \infty, 6)$ in the first derivative $- \frac{4 (x - 6)}{3 {(8 - x)}^{\frac{2}{3}}}$ to check if the result is negative or positive.

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Step 15.2.1

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

$f' (0) = - \frac{4 ((0) - 6)}{3 {(8 - (0))}^{\frac{2}{3}}}$

Step 15.2.2

Simplify the result.

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Step 15.2.2.1

Factor $3$ out of $4 ((0) - 6)$ .

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

Step 15.2.2.2

Cancel the common factors.

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Step 15.2.2.2.1

Factor $3$ out of $3 {(8 - (0))}^{\frac{2}{3}}$ .

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

Step 15.2.2.2.2

Cancel the common factor.

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

Step 15.2.2.2.3

Rewrite the expression.

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

Step 15.2.2.3

Subtract $2$ from $0$ .

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

Step 15.2.2.4

Simplify the denominator.

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Step 15.2.2.4.1

Subtract $0$ from $8$ .

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

Step 15.2.2.4.2

Rewrite $8$ as $2^{3}$ .

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

Step 15.2.2.4.3

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

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

Step 15.2.2.4.4

Cancel the common factor of $3$ .

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Step 15.2.2.4.4.1

Cancel the common factor.

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

Step 15.2.2.4.4.2

Rewrite the expression.

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

Step 15.2.2.4.5

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

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

Step 15.2.2.5

Simplify the expression.

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Step 15.2.2.5.1

Multiply $4$ by $- 2$ .

$f' (0) = - \frac{- 8}{4}$

Step 15.2.2.5.2

Divide $- 8$ by $4$ .

$f' (0) = 2$

Step 15.2.2.5.3

Multiply $- 1$ by $- 2$ .

$f' (0) = 2$

Step 15.2.2.6

The final answer is $2$ .

$2$

Step 15.3

Substitute any number, such as $7$ , from the interval $(6, 8)$ in the first derivative $- \frac{4 (x - 6)}{3 {(8 - x)}^{\frac{2}{3}}}$ to check if the result is negative or positive.

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Step 15.3.1

Replace the variable $x$ with $7$ in the expression.

$f' (7) = - \frac{4 ((7) - 6)}{3 {(8 - (7))}^{\frac{2}{3}}}$

Step 15.3.2

Simplify the result.

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Step 15.3.2.1

Subtract $6$ from $7$ .

$f' (7) = - \frac{4 \cdot 1}{3 {(8 - (7))}^{\frac{2}{3}}}$

Step 15.3.2.2

Simplify the denominator.

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Step 15.3.2.2.1

Multiply $- 1$ by $7$ .

$f' (7) = - \frac{4 \cdot 1}{3 {(8 - 7)}^{\frac{2}{3}}}$

Step 15.3.2.2.2

Subtract $7$ from $8$ .

$f' (7) = - \frac{4 \cdot 1}{3 \cdot 1^{\frac{2}{3}}}$

Step 15.3.2.2.3

One to any power is one.

$f' (7) = - \frac{4 \cdot 1}{3 \cdot 1}$

Step 15.3.2.3

Simplify.

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Step 15.3.2.3.1

Multiply $4$ by $1$ .

$f' (7) = - \frac{4}{3 \cdot 1}$

Step 15.3.2.3.2

Multiply $3$ by $1$ .

$f' (7) = - \frac{4}{3}$

Step 15.3.2.4

The final answer is $- \frac{4}{3}$ .

$- \frac{4}{3}$

Step 15.4

Substitute any number, such as $10$ , from the interval $(8, \infty)$ in the first derivative $- \frac{4 (x - 6)}{3 {(8 - x)}^{\frac{2}{3}}}$ to check if the result is negative or positive.

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Step 15.4.1

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

$f' (10) = - \frac{4 ((10) - 6)}{3 {(8 - (10))}^{\frac{2}{3}}}$

Step 15.4.2

Simplify the result.

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Step 15.4.2.1

Subtract $6$ from $10$ .

$f' (10) = - \frac{4 \cdot 4}{3 {(8 - (10))}^{\frac{2}{3}}}$

Step 15.4.2.2

Simplify the denominator.

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Step 15.4.2.2.1

Multiply $- 1$ by $10$ .

$f' (10) = - \frac{4 \cdot 4}{3 {(8 - 10)}^{\frac{2}{3}}}$

Step 15.4.2.2.2

Subtract $10$ from $8$ .

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

Step 15.4.2.3

Multiply $4$ by $4$ .

$f' (10) = - \frac{16}{3 {(- 2)}^{\frac{2}{3}}}$

Step 15.4.2.4

The final answer is $- \frac{16}{3 {(- 2)}^{\frac{2}{3}}}$ .

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

Step 15.5

Since the first derivative changed signs from positive to negative around $x = 6$ , then $x = 6$ is a local maximum.

$x = 6$ is a local maximum

Step 15.6

Since the first derivative did not change signs around $x = 8$ , this is not a local maximum or minimum.

Not a local maximum or minimum

Step 15.7

These are the local extrema for $f (x) = x {(8 - x)}^{\frac{1}{3}}$ .

$x = 6$ is a local maximum

Step 16