Calculus Examples

$x \sqrt[3]{x - 6}$

Step 1

Write $x \sqrt[3]{x - 6}$ as a function.

$f (x) = x \sqrt[3]{x - 6}$

Step 2

Find the first derivative of the function.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x - 6}$ as ${(x - 6)}^{\frac{1}{3}}$ .

$\frac{d}{d x} [x {(x - 6)}^{\frac{1}{3}}]$

Step 2.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = {(x - 6)}^{\frac{1}{3}}$ .

$x \frac{d}{d x} [{(x - 6)}^{\frac{1}{3}}] + {(x - 6)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 2.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{3}}$ and $g (x) = x - 6$ .

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Step 2.3.1

To apply the Chain Rule, set $u$ as $x - 6$ .

$x (\frac{d}{d u} [u^{\frac{1}{3}}] \frac{d}{d x} [x - 6]) + {(x - 6)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 2.3.2

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

$x (\frac{1}{3} u^{\frac{1}{3} - 1} \frac{d}{d x} [x - 6]) + {(x - 6)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 2.3.3

Replace all occurrences of $u$ with $x - 6$ .

$x (\frac{1}{3} {(x - 6)}^{\frac{1}{3} - 1} \frac{d}{d x} [x - 6]) + {(x - 6)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 2.4

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

$x (\frac{1}{3} {(x - 6)}^{\frac{1}{3} - 1 \cdot \frac{3}{3}} \frac{d}{d x} [x - 6]) + {(x - 6)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 2.5

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

$x (\frac{1}{3} {(x - 6)}^{\frac{1}{3} + \frac{- 1 \cdot 3}{3}} \frac{d}{d x} [x - 6]) + {(x - 6)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 2.6

Combine the numerators over the common denominator.

$x (\frac{1}{3} {(x - 6)}^{\frac{1 - 1 \cdot 3}{3}} \frac{d}{d x} [x - 6]) + {(x - 6)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 2.7

Simplify the numerator.

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Step 2.7.1

Multiply $- 1$ by $3$ .

$x (\frac{1}{3} {(x - 6)}^{\frac{1 - 3}{3}} \frac{d}{d x} [x - 6]) + {(x - 6)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 2.7.2

Subtract $3$ from $1$ .

$x (\frac{1}{3} {(x - 6)}^{\frac{- 2}{3}} \frac{d}{d x} [x - 6]) + {(x - 6)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 2.8

Combine fractions.

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Step 2.8.1

Move the negative in front of the fraction.

$x (\frac{1}{3} {(x - 6)}^{- \frac{2}{3}} \frac{d}{d x} [x - 6]) + {(x - 6)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 2.8.2

Combine $\frac{1}{3}$ and ${(x - 6)}^{- \frac{2}{3}}$ .

$x (\frac{{(x - 6)}^{- \frac{2}{3}}}{3} \frac{d}{d x} [x - 6]) + {(x - 6)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 2.8.3

Move ${(x - 6)}^{- \frac{2}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$x (\frac{1}{3 {(x - 6)}^{\frac{2}{3}}} \frac{d}{d x} [x - 6]) + {(x - 6)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 2.8.4

Combine $\frac{1}{3 {(x - 6)}^{\frac{2}{3}}}$ and $x$ .

$\frac{x}{3 {(x - 6)}^{\frac{2}{3}}} \frac{d}{d x} [x - 6] + {(x - 6)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 2.9

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

$\frac{x}{3 {(x - 6)}^{\frac{2}{3}}} (\frac{d}{d x} [x] + \frac{d}{d x} [- 6]) + {(x - 6)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 2.10

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

$\frac{x}{3 {(x - 6)}^{\frac{2}{3}}} (1 + \frac{d}{d x} [- 6]) + {(x - 6)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 2.11

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

$\frac{x}{3 {(x - 6)}^{\frac{2}{3}}} (1 + 0) + {(x - 6)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 2.12

Simplify the expression.

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Step 2.12.1

Add $1$ and $0$ .

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

Step 2.12.2

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

$\frac{x}{3 {(x - 6)}^{\frac{2}{3}}} + {(x - 6)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 2.13

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

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

Step 2.14

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

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

Step 2.15

To write ${(x - 6)}^{\frac{1}{3}}$ as a fraction with a common denominator, multiply by $\frac{3 {(x - 6)}^{\frac{2}{3}}}{3 {(x - 6)}^{\frac{2}{3}}}$ .

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

Step 2.16

Combine ${(x - 6)}^{\frac{1}{3}}$ and $\frac{3 {(x - 6)}^{\frac{2}{3}}}{3 {(x - 6)}^{\frac{2}{3}}}$ .

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

Step 2.17

Combine the numerators over the common denominator.

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

Step 2.18

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

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Step 2.18.1

Move ${(x - 6)}^{\frac{2}{3}}$ .

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

Step 2.18.2

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

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

Step 2.18.3

Combine the numerators over the common denominator.

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

Step 2.18.4

Add $2$ and $1$ .

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

Step 2.18.5

Divide $3$ by $3$ .

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

Step 2.19

Simplify ${(x - 6)}^{1} \cdot 3$ .

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

Step 2.20

Move $3$ to the left of $x - 6$ .

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

Step 2.21

Simplify.

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Step 2.21.1

Apply the distributive property.

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

Step 2.21.2

Simplify the numerator.

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Step 2.21.2.1

Multiply $3$ by $- 6$ .

$\frac{x + 3 x - 18}{3 {(x - 6)}^{\frac{2}{3}}}$

Step 2.21.2.2

Add $x$ and $3 x$ .

$\frac{4 x - 18}{3 {(x - 6)}^{\frac{2}{3}}}$

Step 2.21.3

Factor $2$ out of $4 x - 18$ .

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Step 2.21.3.1

Factor $2$ out of $4 x$ .

$\frac{2 (2 x) - 18}{3 {(x - 6)}^{\frac{2}{3}}}$

Step 2.21.3.2

Factor $2$ out of $- 18$ .

$\frac{2 (2 x) + 2 (- 9)}{3 {(x - 6)}^{\frac{2}{3}}}$

Step 2.21.3.3

Factor $2$ out of $2 (2 x) + 2 (- 9)$ .

$\frac{2 (2 x - 9)}{3 {(x - 6)}^{\frac{2}{3}}}$

Step 3

Find the second derivative of the function.

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Step 3.1

Since $\frac{2}{3}$ is constant with respect to $x$ , the derivative of $\frac{2 (2 x - 9)}{3 {(x - 6)}^{\frac{2}{3}}}$ with respect to $x$ is $\frac{2}{3} \frac{d}{d x} [\frac{2 x - 9}{{(x - 6)}^{\frac{2}{3}}}]$ .

$f'' (x) = \frac{2}{3} \cdot \frac{d}{d x} (\frac{2 x - 9}{{(x - 6)}^{\frac{2}{3}}})$

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 $f (x) = 2 x - 9$ and $g (x) = {(x - 6)}^{\frac{2}{3}}$ .

$f'' (x) = \frac{2}{3} \cdot \frac{{(x - 6)}^{\frac{2}{3}} \frac{d}{d x} (2 x - 9) - (2 x - 9) \frac{d}{d x} {(x - 6)}^{\frac{2}{3}}}{{({(x - 6)}^{\frac{2}{3}})}^{2}}$

Step 3.3

Differentiate.

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Step 3.3.1

Multiply the exponents in ${({(x - 6)}^{\frac{2}{3}})}^{2}$ .

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Step 3.3.1.1

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

$f'' (x) = \frac{2}{3} \cdot \frac{{(x - 6)}^{\frac{2}{3}} \frac{d}{d x} (2 x - 9) - (2 x - 9) \frac{d}{d x} {(x - 6)}^{\frac{2}{3}}}{{(x - 6)}^{\frac{2}{3} \cdot 2}}$

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 $2$ .

$f'' (x) = \frac{2}{3} \cdot \frac{{(x - 6)}^{\frac{2}{3}} \frac{d}{d x} (2 x - 9) - (2 x - 9) \frac{d}{d x} {(x - 6)}^{\frac{2}{3}}}{{(x - 6)}^{\frac{2 \cdot 2}{3}}}$

Step 3.3.1.2.2

Multiply $2$ by $2$ .

$f'' (x) = \frac{2}{3} \cdot \frac{{(x - 6)}^{\frac{2}{3}} \frac{d}{d x} (2 x - 9) - (2 x - 9) \frac{d}{d x} {(x - 6)}^{\frac{2}{3}}}{{(x - 6)}^{\frac{4}{3}}}$

Step 3.3.2

By the Sum Rule, the derivative of $2 x - 9$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [- 9]$ .

$f'' (x) = \frac{2}{3} \cdot \frac{{(x - 6)}^{\frac{2}{3}} (\frac{d}{d x} (2 x) + \frac{d}{d x} (- 9)) - (2 x - 9) \frac{d}{d x} {(x - 6)}^{\frac{2}{3}}}{{(x - 6)}^{\frac{4}{3}}}$

Step 3.3.3

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

$f'' (x) = \frac{2}{3} \cdot \frac{{(x - 6)}^{\frac{2}{3}} (2 \frac{d}{d x} (x) + \frac{d}{d x} (- 9)) - (2 x - 9) \frac{d}{d x} {(x - 6)}^{\frac{2}{3}}}{{(x - 6)}^{\frac{4}{3}}}$

Step 3.3.4

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

$f'' (x) = \frac{2}{3} \cdot \frac{{(x - 6)}^{\frac{2}{3}} (2 \cdot 1 + \frac{d}{d x} (- 9)) - (2 x - 9) \frac{d}{d x} {(x - 6)}^{\frac{2}{3}}}{{(x - 6)}^{\frac{4}{3}}}$

Step 3.3.5

Multiply $2$ by $1$ .

$f'' (x) = \frac{2}{3} \cdot \frac{{(x - 6)}^{\frac{2}{3}} (2 + \frac{d}{d x} (- 9)) - (2 x - 9) \frac{d}{d x} {(x - 6)}^{\frac{2}{3}}}{{(x - 6)}^{\frac{4}{3}}}$

Step 3.3.6

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9$ with respect to $x$ is $0$ .

$f'' (x) = \frac{2}{3} \cdot \frac{{(x - 6)}^{\frac{2}{3}} (2 + 0) - (2 x - 9) \frac{d}{d x} {(x - 6)}^{\frac{2}{3}}}{{(x - 6)}^{\frac{4}{3}}}$

Step 3.3.7

Simplify the expression.

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Step 3.3.7.1

Add $2$ and $0$ .

$f'' (x) = \frac{2}{3} \cdot \frac{{(x - 6)}^{\frac{2}{3}} \cdot 2 - (2 x - 9) \frac{d}{d x} {(x - 6)}^{\frac{2}{3}}}{{(x - 6)}^{\frac{4}{3}}}$

Step 3.3.7.2

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

$f'' (x) = \frac{2}{3} \cdot \frac{2 \cdot {(x - 6)}^{\frac{2}{3}} - (2 x - 9) \frac{d}{d x} {(x - 6)}^{\frac{2}{3}}}{{(x - 6)}^{\frac{4}{3}}}$

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 $g (x) = x - 6$ .

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Step 3.4.1

To apply the Chain Rule, set $u$ as $x - 6$ .

$f'' (x) = \frac{2}{3} \cdot \frac{2 {(x - 6)}^{\frac{2}{3}} - (2 x - 9) (\frac{d}{d u} (u^{\frac{2}{3}}) \frac{d}{d x} (x - 6))}{{(x - 6)}^{\frac{4}{3}}}$

Step 3.4.2

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

$f'' (x) = \frac{2}{3} \cdot \frac{2 {(x - 6)}^{\frac{2}{3}} - (2 x - 9) (\frac{2}{3} u^{\frac{2}{3} - 1} \frac{d}{d x} (x - 6))}{{(x - 6)}^{\frac{4}{3}}}$

Step 3.4.3

Replace all occurrences of $u$ with $x - 6$ .

$f'' (x) = \frac{2}{3} \cdot \frac{2 {(x - 6)}^{\frac{2}{3}} - (2 x - 9) (\frac{2}{3} \cdot {(x - 6)}^{\frac{2}{3} - 1} \frac{d}{d x} (x - 6))}{{(x - 6)}^{\frac{4}{3}}}$

Step 3.5

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

$f'' (x) = \frac{2}{3} \cdot \frac{2 {(x - 6)}^{\frac{2}{3}} - (2 x - 9) (\frac{2}{3} \cdot {(x - 6)}^{\frac{2}{3} - 1 \cdot \frac{3}{3}} \frac{d}{d x} (x - 6))}{{(x - 6)}^{\frac{4}{3}}}$

Step 3.6

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

$f'' (x) = \frac{2}{3} \cdot \frac{2 {(x - 6)}^{\frac{2}{3}} - (2 x - 9) (\frac{2}{3} \cdot {(x - 6)}^{\frac{2}{3} + \frac{- 1 \cdot 3}{3}} \frac{d}{d x} (x - 6))}{{(x - 6)}^{\frac{4}{3}}}$

Step 3.7

Combine the numerators over the common denominator.

$f'' (x) = \frac{2}{3} \cdot \frac{2 {(x - 6)}^{\frac{2}{3}} - (2 x - 9) (\frac{2}{3} \cdot {(x - 6)}^{\frac{2 - 1 \cdot 3}{3}} \frac{d}{d x} (x - 6))}{{(x - 6)}^{\frac{4}{3}}}$

Step 3.8

Simplify the numerator.

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Step 3.8.1

Multiply $- 1$ by $3$ .

$f'' (x) = \frac{2}{3} \cdot \frac{2 {(x - 6)}^{\frac{2}{3}} - (2 x - 9) (\frac{2}{3} \cdot {(x - 6)}^{\frac{2 - 3}{3}} \frac{d}{d x} (x - 6))}{{(x - 6)}^{\frac{4}{3}}}$

Step 3.8.2

Subtract $3$ from $2$ .

$f'' (x) = \frac{2}{3} \cdot \frac{2 {(x - 6)}^{\frac{2}{3}} - (2 x - 9) (\frac{2}{3} \cdot {(x - 6)}^{\frac{- 1}{3}} \frac{d}{d x} (x - 6))}{{(x - 6)}^{\frac{4}{3}}}$

Step 3.9

Combine fractions.

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Step 3.9.1

Move the negative in front of the fraction.

$f'' (x) = \frac{2}{3} \cdot \frac{2 {(x - 6)}^{\frac{2}{3}} - (2 x - 9) (\frac{2}{3} \cdot {(x - 6)}^{- \frac{1}{3}} \frac{d}{d x} (x - 6))}{{(x - 6)}^{\frac{4}{3}}}$

Step 3.9.2

Combine $\frac{2}{3}$ and ${(x - 6)}^{- \frac{1}{3}}$ .

$f'' (x) = \frac{2}{3} \cdot \frac{2 {(x - 6)}^{\frac{2}{3}} - (2 x - 9) (\frac{2 {(x - 6)}^{- \frac{1}{3}}}{3} \cdot \frac{d}{d x} (x - 6))}{{(x - 6)}^{\frac{4}{3}}}$

Step 3.9.3

Move ${(x - 6)}^{- \frac{1}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (x) = \frac{2}{3} \cdot \frac{2 {(x - 6)}^{\frac{2}{3}} - (2 x - 9) (\frac{2}{3 {(x - 6)}^{\frac{1}{3}}} \cdot \frac{d}{d x} (x - 6))}{{(x - 6)}^{\frac{4}{3}}}$

Step 3.10

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

$f'' (x) = \frac{2}{3} \cdot \frac{2 {(x - 6)}^{\frac{2}{3}} - (2 x - 9) (\frac{2}{3 {(x - 6)}^{\frac{1}{3}}} \cdot (\frac{d}{d x} (x) + \frac{d}{d x} (- 6)))}{{(x - 6)}^{\frac{4}{3}}}$

Step 3.11

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

$f'' (x) = \frac{2}{3} \cdot \frac{2 {(x - 6)}^{\frac{2}{3}} - (2 x - 9) (\frac{2}{3 {(x - 6)}^{\frac{1}{3}}} \cdot (1 + \frac{d}{d x} (- 6)))}{{(x - 6)}^{\frac{4}{3}}}$

Step 3.12

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

$f'' (x) = \frac{2}{3} \cdot \frac{2 {(x - 6)}^{\frac{2}{3}} - (2 x - 9) (\frac{2}{3 {(x - 6)}^{\frac{1}{3}}} \cdot (1 + 0))}{{(x - 6)}^{\frac{4}{3}}}$

Step 3.13

Combine fractions.

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Step 3.13.1

Add $1$ and $0$ .

$f'' (x) = \frac{2}{3} \cdot \frac{2 {(x - 6)}^{\frac{2}{3}} - (2 x - 9) (\frac{2}{3 {(x - 6)}^{\frac{1}{3}}} \cdot 1)}{{(x - 6)}^{\frac{4}{3}}}$

Step 3.13.2

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

$f'' (x) = \frac{2}{3} \cdot \frac{2 {(x - 6)}^{\frac{2}{3}} - (2 x - 9) \frac{2}{3 {(x - 6)}^{\frac{1}{3}}}}{{(x - 6)}^{\frac{4}{3}}}$

Step 3.13.3

Multiply $\frac{2}{3}$ by $\frac{2 {(x - 6)}^{\frac{2}{3}} - (2 x - 9) \frac{2}{3 {(x - 6)}^{\frac{1}{3}}}}{{(x - 6)}^{\frac{4}{3}}}$ .

$f'' (x) = \frac{2 (2 {(x - 6)}^{\frac{2}{3}} - (2 x - 9) \frac{2}{3 {(x - 6)}^{\frac{1}{3}}})}{3 {(x - 6)}^{\frac{4}{3}}}$

Step 3.14

Simplify.

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Step 3.14.1

Apply the distributive property.

$f'' (x) = \frac{2 (2 {(x - 6)}^{\frac{2}{3}} + (- (2 x) + 9) (\frac{2}{3 {(x - 6)}^{\frac{1}{3}}}))}{3 {(x - 6)}^{\frac{4}{3}}}$

Step 3.14.2

Apply the distributive property.

$f'' (x) = \frac{2 (2 {(x - 6)}^{\frac{2}{3}}) + 2 ((- (2 x) + 9) (\frac{2}{3 {(x - 6)}^{\frac{1}{3}}}))}{3 {(x - 6)}^{\frac{4}{3}}}$

Step 3.14.3

Simplify the numerator.

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Step 3.14.3.1

Multiply $2$ by $2$ .

$f'' (x) = \frac{4 {(x - 6)}^{\frac{2}{3}} + 2 ((- (2 x) + 9) (\frac{2}{3 {(x - 6)}^{\frac{1}{3}}}))}{3 {(x - 6)}^{\frac{4}{3}}}$

Step 3.14.3.2

Simplify each term.

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Step 3.14.3.2.1

Multiply $2$ by $- 1$ .

$f'' (x) = \frac{4 {(x - 6)}^{\frac{2}{3}} + 2 ((- 2 x + 9) (\frac{2}{3 {(x - 6)}^{\frac{1}{3}}}))}{3 {(x - 6)}^{\frac{4}{3}}}$

Step 3.14.3.2.2

Multiply $- 1$ by $- 9$ .

$f'' (x) = \frac{4 {(x - 6)}^{\frac{2}{3}} + 2 ((- 2 x + 9) (\frac{2}{3 {(x - 6)}^{\frac{1}{3}}}))}{3 {(x - 6)}^{\frac{4}{3}}}$

Step 3.14.3.3

Multiply $- 2 x + 9$ by $\frac{2}{3 {(x - 6)}^{\frac{1}{3}}}$ .

$f'' (x) = \frac{4 {(x - 6)}^{\frac{2}{3}} + 2 (\frac{(- 2 x + 9) \cdot 2}{3 {(x - 6)}^{\frac{1}{3}}})}{3 {(x - 6)}^{\frac{4}{3}}}$

Step 3.14.3.4

Move $2$ to the left of $- 2 x + 9$ .

$f'' (x) = \frac{4 {(x - 6)}^{\frac{2}{3}} + 2 (\frac{2 \cdot (- 2 x + 9)}{3 {(x - 6)}^{\frac{1}{3}}})}{3 {(x - 6)}^{\frac{4}{3}}}$

Step 3.14.3.5

Multiply $2 \frac{2 (- 2 x + 9)}{3 {(x - 6)}^{\frac{1}{3}}}$ .

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Step 3.14.3.5.1

Combine $2$ and $\frac{2 (- 2 x + 9)}{3 {(x - 6)}^{\frac{1}{3}}}$ .

$f'' (x) = \frac{4 {(x - 6)}^{\frac{2}{3}} + \frac{2 (2 (- 2 x + 9))}{3 {(x - 6)}^{\frac{1}{3}}}}{3 {(x - 6)}^{\frac{4}{3}}}$

Step 3.14.3.5.2

Multiply $2$ by $2$ .

$f'' (x) = \frac{4 {(x - 6)}^{\frac{2}{3}} + \frac{4 (- 2 x + 9)}{3 {(x - 6)}^{\frac{1}{3}}}}{3 {(x - 6)}^{\frac{4}{3}}}$

Step 3.14.3.6

To write $4 {(x - 6)}^{\frac{2}{3}}$ as a fraction with a common denominator, multiply by $\frac{3 {(x - 6)}^{\frac{1}{3}}}{3 {(x - 6)}^{\frac{1}{3}}}$ .

$f'' (x) = \frac{4 {(x - 6)}^{\frac{2}{3}} \cdot \frac{3 {(x - 6)}^{\frac{1}{3}}}{3 {(x - 6)}^{\frac{1}{3}}} + \frac{4 (- 2 x + 9)}{3 {(x - 6)}^{\frac{1}{3}}}}{3 {(x - 6)}^{\frac{4}{3}}}$

Step 3.14.3.7

Combine $4 {(x - 6)}^{\frac{2}{3}}$ and $\frac{3 {(x - 6)}^{\frac{1}{3}}}{3 {(x - 6)}^{\frac{1}{3}}}$ .

$f'' (x) = \frac{\frac{4 {(x - 6)}^{\frac{2}{3}} (3 {(x - 6)}^{\frac{1}{3}})}{3 {(x - 6)}^{\frac{1}{3}}} + \frac{4 (- 2 x + 9)}{3 {(x - 6)}^{\frac{1}{3}}}}{3 {(x - 6)}^{\frac{4}{3}}}$

Step 3.14.3.8

Combine the numerators over the common denominator.

$f'' (x) = \frac{\frac{4 {(x - 6)}^{\frac{2}{3}} (3 {(x - 6)}^{\frac{1}{3}}) + 4 (- 2 x + 9)}{3 {(x - 6)}^{\frac{1}{3}}}}{3 {(x - 6)}^{\frac{4}{3}}}$

Step 3.14.3.9

Rewrite $\frac{4 {(x - 6)}^{\frac{2}{3}} (3 {(x - 6)}^{\frac{1}{3}}) + 4 (- 2 x + 9)}{3 {(x - 6)}^{\frac{1}{3}}}$ in a factored form.

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Step 3.14.3.9.1

Factor $4$ out of $4 {(x - 6)}^{\frac{2}{3}} (3 {(x - 6)}^{\frac{1}{3}}) + 4 (- 2 x + 9)$ .

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Step 3.14.3.9.1.1

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

$f'' (x) = \frac{\frac{4 ({(x - 6)}^{\frac{2}{3}} (3 {(x - 6)}^{\frac{1}{3}})) + 4 (- 2 x + 9)}{3 {(x - 6)}^{\frac{1}{3}}}}{3 {(x - 6)}^{\frac{4}{3}}}$

Step 3.14.3.9.1.2

Factor $4$ out of $4 ({(x - 6)}^{\frac{2}{3}} (3 {(x - 6)}^{\frac{1}{3}})) + 4 (- 2 x + 9)$ .

$f'' (x) = \frac{\frac{4 ({(x - 6)}^{\frac{2}{3}} (3 {(x - 6)}^{\frac{1}{3}}) - 2 x + 9)}{3 {(x - 6)}^{\frac{1}{3}}}}{3 {(x - 6)}^{\frac{4}{3}}}$

Step 3.14.3.9.2

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{\frac{4 (3 {(x - 6)}^{\frac{2}{3}} {(x - 6)}^{\frac{1}{3}} - 2 x + 9)}{3 {(x - 6)}^{\frac{1}{3}}}}{3 {(x - 6)}^{\frac{4}{3}}}$

Step 3.14.3.9.3

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

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Step 3.14.3.9.3.1

Move ${(x - 6)}^{\frac{1}{3}}$ .

$f'' (x) = \frac{\frac{4 (3 ({(x - 6)}^{\frac{1}{3}} {(x - 6)}^{\frac{2}{3}}) - 2 x + 9)}{3 {(x - 6)}^{\frac{1}{3}}}}{3 {(x - 6)}^{\frac{4}{3}}}$

Step 3.14.3.9.3.2

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

$f'' (x) = \frac{\frac{4 (3 {(x - 6)}^{\frac{1}{3} + \frac{2}{3}} - 2 x + 9)}{3 {(x - 6)}^{\frac{1}{3}}}}{3 {(x - 6)}^{\frac{4}{3}}}$

Step 3.14.3.9.3.3

Combine the numerators over the common denominator.

$f'' (x) = \frac{\frac{4 (3 {(x - 6)}^{\frac{1 + 2}{3}} - 2 x + 9)}{3 {(x - 6)}^{\frac{1}{3}}}}{3 {(x - 6)}^{\frac{4}{3}}}$

Step 3.14.3.9.3.4

Add $1$ and $2$ .

$f'' (x) = \frac{\frac{4 (3 {(x - 6)}^{\frac{3}{3}} - 2 x + 9)}{3 {(x - 6)}^{\frac{1}{3}}}}{3 {(x - 6)}^{\frac{4}{3}}}$

Step 3.14.3.9.3.5

Divide $3$ by $3$ .

$f'' (x) = \frac{\frac{4 (3 (x - 6) - 2 x + 9)}{3 {(x - 6)}^{\frac{1}{3}}}}{3 {(x - 6)}^{\frac{4}{3}}}$

Step 3.14.3.9.4

Simplify $3 {(x - 6)}^{1}$ .

$f'' (x) = \frac{\frac{4 (3 (x - 6) - 2 x + 9)}{3 {(x - 6)}^{\frac{1}{3}}}}{3 {(x - 6)}^{\frac{4}{3}}}$

Step 3.14.3.9.5

Apply the distributive property.

$f'' (x) = \frac{\frac{4 (3 x + 3 \cdot - 6 - 2 x + 9)}{3 {(x - 6)}^{\frac{1}{3}}}}{3 {(x - 6)}^{\frac{4}{3}}}$

Step 3.14.3.9.6

Multiply $3$ by $- 6$ .

$f'' (x) = \frac{\frac{4 (3 x - 18 - 2 x + 9)}{3 {(x - 6)}^{\frac{1}{3}}}}{3 {(x - 6)}^{\frac{4}{3}}}$

Step 3.14.3.9.7

Subtract $2 x$ from $3 x$ .

$f'' (x) = \frac{\frac{4 (x - 18 + 9)}{3 {(x - 6)}^{\frac{1}{3}}}}{3 {(x - 6)}^{\frac{4}{3}}}$

Step 3.14.3.9.8

Add $- 18$ and $9$ .

$f'' (x) = \frac{\frac{4 (x - 9)}{3 {(x - 6)}^{\frac{1}{3}}}}{3 {(x - 6)}^{\frac{4}{3}}}$

Step 3.14.4

Combine terms.

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Step 3.14.4.1

Rewrite $\frac{\frac{4 (x - 9)}{3 {(x - 6)}^{\frac{1}{3}}}}{3 {(x - 6)}^{\frac{4}{3}}}$ as a product.

$f'' (x) = \frac{4 (x - 9)}{3 {(x - 6)}^{\frac{1}{3}}} \cdot \frac{1}{3 {(x - 6)}^{\frac{4}{3}}}$

Step 3.14.4.2

Multiply $\frac{4 (x - 9)}{3 {(x - 6)}^{\frac{1}{3}}}$ by $\frac{1}{3 {(x - 6)}^{\frac{4}{3}}}$ .

$f'' (x) = \frac{4 (x - 9)}{3 {(x - 6)}^{\frac{1}{3}} (3 {(x - 6)}^{\frac{4}{3}})}$

Step 3.14.4.3

Multiply $3$ by $3$ .

$f'' (x) = \frac{4 (x - 9)}{9 {(x - 6)}^{\frac{1}{3}} {(x - 6)}^{\frac{4}{3}}}$

Step 3.14.4.4

Multiply ${(x - 6)}^{\frac{1}{3}}$ by ${(x - 6)}^{\frac{4}{3}}$ by adding the exponents.

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Step 3.14.4.4.1

Move ${(x - 6)}^{\frac{4}{3}}$ .

$f'' (x) = \frac{4 (x - 9)}{9 ({(x - 6)}^{\frac{4}{3}} {(x - 6)}^{\frac{1}{3}})}$

Step 3.14.4.4.2

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

$f'' (x) = \frac{4 (x - 9)}{9 {(x - 6)}^{\frac{4}{3} + \frac{1}{3}}}$

Step 3.14.4.4.3

Combine the numerators over the common denominator.

$f'' (x) = \frac{4 (x - 9)}{9 {(x - 6)}^{\frac{4 + 1}{3}}}$

Step 3.14.4.4.4

Add $4$ and $1$ .

$f'' (x) = \frac{4 (x - 9)}{9 {(x - 6)}^{\frac{5}{3}}}$

Step 4

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

$\frac{2 (2 x - 9)}{3 {(x - 6)}^{\frac{2}{3}}} = 0$

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

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x - 6}$ as ${(x - 6)}^{\frac{1}{3}}$ .

$\frac{d}{d x} [x {(x - 6)}^{\frac{1}{3}}]$

Step 5.1.2

$x \frac{d}{d x} [{(x - 6)}^{\frac{1}{3}}] + {(x - 6)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 5.1.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{3}}$ and $g (x) = x - 6$ .

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Step 5.1.3.1

To apply the Chain Rule, set $u$ as $x - 6$ .

$x (\frac{d}{d u} [u^{\frac{1}{3}}] \frac{d}{d x} [x - 6]) + {(x - 6)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 5.1.3.2

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

$x (\frac{1}{3} u^{\frac{1}{3} - 1} \frac{d}{d x} [x - 6]) + {(x - 6)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 5.1.3.3

Replace all occurrences of $u$ with $x - 6$ .

$x (\frac{1}{3} {(x - 6)}^{\frac{1}{3} - 1} \frac{d}{d x} [x - 6]) + {(x - 6)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 5.1.4

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

$x (\frac{1}{3} {(x - 6)}^{\frac{1}{3} - 1 \cdot \frac{3}{3}} \frac{d}{d x} [x - 6]) + {(x - 6)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 5.1.5

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

$x (\frac{1}{3} {(x - 6)}^{\frac{1}{3} + \frac{- 1 \cdot 3}{3}} \frac{d}{d x} [x - 6]) + {(x - 6)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 5.1.6

Combine the numerators over the common denominator.

$x (\frac{1}{3} {(x - 6)}^{\frac{1 - 1 \cdot 3}{3}} \frac{d}{d x} [x - 6]) + {(x - 6)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 5.1.7

Simplify the numerator.

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Step 5.1.7.1

Multiply $- 1$ by $3$ .

$x (\frac{1}{3} {(x - 6)}^{\frac{1 - 3}{3}} \frac{d}{d x} [x - 6]) + {(x - 6)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 5.1.7.2

Subtract $3$ from $1$ .

$x (\frac{1}{3} {(x - 6)}^{\frac{- 2}{3}} \frac{d}{d x} [x - 6]) + {(x - 6)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 5.1.8

Combine fractions.

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Step 5.1.8.1

Move the negative in front of the fraction.

$x (\frac{1}{3} {(x - 6)}^{- \frac{2}{3}} \frac{d}{d x} [x - 6]) + {(x - 6)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 5.1.8.2

Combine $\frac{1}{3}$ and ${(x - 6)}^{- \frac{2}{3}}$ .

$x (\frac{{(x - 6)}^{- \frac{2}{3}}}{3} \frac{d}{d x} [x - 6]) + {(x - 6)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 5.1.8.3

Move ${(x - 6)}^{- \frac{2}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$x (\frac{1}{3 {(x - 6)}^{\frac{2}{3}}} \frac{d}{d x} [x - 6]) + {(x - 6)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 5.1.8.4

Combine $\frac{1}{3 {(x - 6)}^{\frac{2}{3}}}$ and $x$ .

$\frac{x}{3 {(x - 6)}^{\frac{2}{3}}} \frac{d}{d x} [x - 6] + {(x - 6)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 5.1.9

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

$\frac{x}{3 {(x - 6)}^{\frac{2}{3}}} (\frac{d}{d x} [x] + \frac{d}{d x} [- 6]) + {(x - 6)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 5.1.10

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

$\frac{x}{3 {(x - 6)}^{\frac{2}{3}}} (1 + \frac{d}{d x} [- 6]) + {(x - 6)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 5.1.11

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

$\frac{x}{3 {(x - 6)}^{\frac{2}{3}}} (1 + 0) + {(x - 6)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 5.1.12

Simplify the expression.

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Step 5.1.12.1

Add $1$ and $0$ .

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

Step 5.1.12.2

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

$\frac{x}{3 {(x - 6)}^{\frac{2}{3}}} + {(x - 6)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 5.1.13

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

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

Step 5.1.14

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

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

Step 5.1.15

To write ${(x - 6)}^{\frac{1}{3}}$ as a fraction with a common denominator, multiply by $\frac{3 {(x - 6)}^{\frac{2}{3}}}{3 {(x - 6)}^{\frac{2}{3}}}$ .

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

Step 5.1.16

Combine ${(x - 6)}^{\frac{1}{3}}$ and $\frac{3 {(x - 6)}^{\frac{2}{3}}}{3 {(x - 6)}^{\frac{2}{3}}}$ .

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

Step 5.1.17

Combine the numerators over the common denominator.

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

Step 5.1.18

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

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Step 5.1.18.1

Move ${(x - 6)}^{\frac{2}{3}}$ .

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

Step 5.1.18.2

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

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

Step 5.1.18.3

Combine the numerators over the common denominator.

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

Step 5.1.18.4

Add $2$ and $1$ .

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

Step 5.1.18.5

Divide $3$ by $3$ .

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

Step 5.1.19

Simplify ${(x - 6)}^{1} \cdot 3$ .

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

Step 5.1.20

Move $3$ to the left of $x - 6$ .

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

Step 5.1.21

Simplify.

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Step 5.1.21.1

Apply the distributive property.

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

Step 5.1.21.2

Simplify the numerator.

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Step 5.1.21.2.1

Multiply $3$ by $- 6$ .

$\frac{x + 3 x - 18}{3 {(x - 6)}^{\frac{2}{3}}}$

Step 5.1.21.2.2

Add $x$ and $3 x$ .

$\frac{4 x - 18}{3 {(x - 6)}^{\frac{2}{3}}}$

Step 5.1.21.3

Factor $2$ out of $4 x - 18$ .

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Step 5.1.21.3.1

Factor $2$ out of $4 x$ .

$\frac{2 (2 x) - 18}{3 {(x - 6)}^{\frac{2}{3}}}$

Step 5.1.21.3.2

Factor $2$ out of $- 18$ .

$\frac{2 (2 x) + 2 (- 9)}{3 {(x - 6)}^{\frac{2}{3}}}$

Step 5.1.21.3.3

Factor $2$ out of $2 (2 x) + 2 (- 9)$ .

$f' (x) = \frac{2 (2 x - 9)}{3 {(x - 6)}^{\frac{2}{3}}}$

Step 5.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{2 (2 x - 9)}{3 {(x - 6)}^{\frac{2}{3}}}$ .

$\frac{2 (2 x - 9)}{3 {(x - 6)}^{\frac{2}{3}}}$

Step 6

Set the first derivative equal to $0$ then solve the equation $\frac{2 (2 x - 9)}{3 {(x - 6)}^{\frac{2}{3}}} = 0$ .

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Step 6.1

Set the first derivative equal to $0$ .

$\frac{2 (2 x - 9)}{3 {(x - 6)}^{\frac{2}{3}}} = 0$

Step 6.2

Set the numerator equal to zero.

$2 (2 x - 9) = 0$

Step 6.3

Solve the equation for $x$ .

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Step 6.3.1

Divide each term in $2 (2 x - 9) = 0$ by $2$ and simplify.

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Step 6.3.1.1

Divide each term in $2 (2 x - 9) = 0$ by $2$ .

$\frac{2 (2 x - 9)}{2} = \frac{0}{2}$

Step 6.3.1.2

Simplify the left side.

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Step 6.3.1.2.1

Cancel the common factor of $2$ .

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Step 6.3.1.2.1.1

Cancel the common factor.

$\frac{2 (2 x - 9)}{2} = \frac{0}{2}$

Step 6.3.1.2.1.2

Divide $2 x - 9$ by $1$ .

$2 x - 9 = \frac{0}{2}$

Step 6.3.1.3

Simplify the right side.

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Step 6.3.1.3.1

Divide $0$ by $2$ .

$2 x - 9 = 0$

Step 6.3.2

Add $9$ to both sides of the equation.

$2 x = 9$

Step 6.3.3

Divide each term in $2 x = 9$ by $2$ and simplify.

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Step 6.3.3.1

Divide each term in $2 x = 9$ by $2$ .

$\frac{2 x}{2} = \frac{9}{2}$

Step 6.3.3.2

Simplify the left side.

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Step 6.3.3.2.1

Cancel the common factor of $2$ .

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Step 6.3.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{9}{2}$

Step 6.3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{9}{2}$

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.

$\frac{2 (2 x - 9)}{3 \sqrt[3]{{(x - 6)}^{2}}}$

Step 7.2

Set the denominator in $\frac{2 (2 x - 9)}{3 \sqrt[3]{{(x - 6)}^{2}}}$ equal to $0$ to find where the expression is undefined.

$3 \sqrt[3]{{(x - 6)}^{2}} = 0$

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.

${(3 \sqrt[3]{{(x - 6)}^{2}})}^{3} = 0^{3}$

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

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

Step 7.3.2.2

Simplify the left side.

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Step 7.3.2.2.1

Simplify ${(3 {(x - 6)}^{\frac{2}{3}})}^{3}$ .

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Step 7.3.2.2.1.1

Apply the product rule to $3 {(x - 6)}^{\frac{2}{3}}$ .

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

Step 7.3.2.2.1.2

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

$27 {({(x - 6)}^{\frac{2}{3}})}^{3} = 0^{3}$

Step 7.3.2.2.1.3

Multiply the exponents in ${({(x - 6)}^{\frac{2}{3}})}^{3}$ .

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Step 7.3.2.2.1.3.1

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

$27 {(x - 6)}^{\frac{2}{3} \cdot 3} = 0^{3}$

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.

$27 {(x - 6)}^{\frac{2}{3} \cdot 3} = 0^{3}$

Step 7.3.2.2.1.3.2.2

Rewrite the expression.

$27 {(x - 6)}^{2} = 0^{3}$

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 $0$ .

$27 {(x - 6)}^{2} = 0$

Step 7.3.3

Solve for $x$ .

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Step 7.3.3.1

Divide each term in $27 {(x - 6)}^{2} = 0$ by $27$ and simplify.

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Step 7.3.3.1.1

Divide each term in $27 {(x - 6)}^{2} = 0$ by $27$ .

$\frac{27 {(x - 6)}^{2}}{27} = \frac{0}{27}$

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.

$\frac{27 {(x - 6)}^{2}}{27} = \frac{0}{27}$

Step 7.3.3.1.2.1.2

Divide ${(x - 6)}^{2}$ by $1$ .

${(x - 6)}^{2} = \frac{0}{27}$

Step 7.3.3.1.3

Simplify the right side.

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Step 7.3.3.1.3.1

Divide $0$ by $27$ .

${(x - 6)}^{2} = 0$

Step 7.3.3.2

Set the $x - 6$ equal to $0$ .

$x - 6 = 0$

Step 7.3.3.3

Add $6$ to both sides of the equation.

$x = 6$

Step 8

Critical points to evaluate.

$x = \frac{9}{2}, 6$

Step 9

Evaluate the second derivative at $x = \frac{9}{2}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

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

Step 10

Evaluate the second derivative.

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Step 10.1

Factor $9$ out of $4 ((\frac{9}{2}) - 9)$ .

$\frac{9 (4 (\frac{1}{2} - 1))}{9 {((\frac{9}{2}) - 6)}^{\frac{5}{3}}}$

Step 10.2

Cancel the common factors.

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Step 10.2.1

Factor $9$ out of $9 {((\frac{9}{2}) - 6)}^{\frac{5}{3}}$ .

$\frac{9 (4 (\frac{1}{2} - 1))}{9 ({((\frac{9}{2}) - 6)}^{\frac{5}{3}})}$

Step 10.2.2

Cancel the common factor.

$\frac{9 (4 (\frac{1}{2} - 1))}{9 {((\frac{9}{2}) - 6)}^{\frac{5}{3}}}$

Step 10.2.3

Rewrite the expression.

$\frac{4 (\frac{1}{2} - 1)}{{((\frac{9}{2}) - 6)}^{\frac{5}{3}}}$

Step 10.3

Simplify the numerator.

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Step 10.3.1

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

$\frac{4 (\frac{1}{2} - 1 \cdot \frac{2}{2})}{{(\frac{9}{2} - 6)}^{\frac{5}{3}}}$

Step 10.3.2

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

$\frac{4 (\frac{1}{2} + \frac{- 1 \cdot 2}{2})}{{(\frac{9}{2} - 6)}^{\frac{5}{3}}}$

Step 10.3.3

Combine the numerators over the common denominator.

$\frac{4 \frac{1 - 1 \cdot 2}{2}}{{(\frac{9}{2} - 6)}^{\frac{5}{3}}}$

Step 10.3.4

Simplify the numerator.

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Step 10.3.4.1

Multiply $- 1$ by $2$ .

$\frac{4 \frac{1 - 2}{2}}{{(\frac{9}{2} - 6)}^{\frac{5}{3}}}$

Step 10.3.4.2

Subtract $2$ from $1$ .

$\frac{4 (\frac{- 1}{2})}{{(\frac{9}{2} - 6)}^{\frac{5}{3}}}$

Step 10.3.5

Move the negative in front of the fraction.

$\frac{4 \cdot - 1 \frac{1}{2}}{{(\frac{9}{2} - 6)}^{\frac{5}{3}}}$

Step 10.3.6

Combine exponents.

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Step 10.3.6.1

Factor out negative.

$\frac{- (4 (\frac{1}{2}))}{{(\frac{9}{2} - 6)}^{\frac{5}{3}}}$

Step 10.3.6.2

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

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

Step 10.3.7

Divide $4$ by $2$ .

$\frac{- 1 \cdot 2}{{(\frac{9}{2} - 6)}^{\frac{5}{3}}}$

Step 10.4

Simplify the denominator.

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Step 10.4.1

To write $- 6$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{- 1 \cdot 2}{{(\frac{9}{2} - 6 \cdot \frac{2}{2})}^{\frac{5}{3}}}$

Step 10.4.2

Combine $- 6$ and $\frac{2}{2}$ .

$\frac{- 1 \cdot 2}{{(\frac{9}{2} + \frac{- 6 \cdot 2}{2})}^{\frac{5}{3}}}$

Step 10.4.3

Combine the numerators over the common denominator.

$\frac{- 1 \cdot 2}{{(\frac{9 - 6 \cdot 2}{2})}^{\frac{5}{3}}}$

Step 10.4.4

Simplify the numerator.

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Step 10.4.4.1

Multiply $- 6$ by $2$ .

$\frac{- 1 \cdot 2}{{(\frac{9 - 12}{2})}^{\frac{5}{3}}}$

Step 10.4.4.2

Subtract $12$ from $9$ .

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

Step 10.4.5

Move the negative in front of the fraction.

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

Step 10.4.6

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

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Step 10.4.6.1

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

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

Step 10.4.6.2

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

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

Step 10.4.7

Rewrite $- 1$ as ${(- 1)}^{3}$ .

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

Step 10.4.8

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

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

Step 10.4.9

Cancel the common factor of $3$ .

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Step 10.4.9.1

Cancel the common factor.

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

Step 10.4.9.2

Rewrite the expression.

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

Step 10.4.10

Raise $- 1$ to the power of $5$ .

$\frac{- 1 \cdot 2}{- \frac{3^{\frac{5}{3}}}{2^{\frac{5}{3}}}}$

Step 10.5

Multiply $- 1$ by $2$ .

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

Step 10.6

Multiply the numerator by the reciprocal of the denominator.

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

Step 10.7

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

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Step 10.7.1

Multiply $- 1$ by $- 2$ .

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

Step 10.7.2

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

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

Step 10.7.3

Multiply $2$ by $2^{\frac{5}{3}}$ by adding the exponents.

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Step 10.7.3.1

Multiply $2$ by $2^{\frac{5}{3}}$ .

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Step 10.7.3.1.1

Raise $2$ to the power of $1$ .

$\frac{2^{1} \cdot 2^{\frac{5}{3}}}{3^{\frac{5}{3}}}$

Step 10.7.3.1.2

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

$\frac{2^{1 + \frac{5}{3}}}{3^{\frac{5}{3}}}$

Step 10.7.3.2

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

$\frac{2^{\frac{3}{3} + \frac{5}{3}}}{3^{\frac{5}{3}}}$

Step 10.7.3.3

Combine the numerators over the common denominator.

$\frac{2^{\frac{3 + 5}{3}}}{3^{\frac{5}{3}}}$

Step 10.7.3.4

Add $3$ and $5$ .

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

Step 11

$x = \frac{9}{2}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = \frac{9}{2}$ is a local minimum

Step 12

Find the y-value when $x = \frac{9}{2}$ .

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Step 12.1

Replace the variable $x$ with $\frac{9}{2}$ in the expression.

$f (\frac{9}{2}) = (\frac{9}{2}) \sqrt[3]{(\frac{9}{2}) - 6}$

Step 12.2

Simplify the result.

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Step 12.2.1

To write $- 6$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (\frac{9}{2}) = \frac{9}{2} \cdot \sqrt[3]{\frac{9}{2} - 6 \cdot \frac{2}{2}}$

Step 12.2.2

Combine $- 6$ and $\frac{2}{2}$ .

$f (\frac{9}{2}) = \frac{9}{2} \cdot \sqrt[3]{\frac{9}{2} + \frac{- 6 \cdot 2}{2}}$

Step 12.2.3

Combine the numerators over the common denominator.

$f (\frac{9}{2}) = \frac{9}{2} \cdot \sqrt[3]{\frac{9 - 6 \cdot 2}{2}}$

Step 12.2.4

Simplify the numerator.

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Step 12.2.4.1

Multiply $- 6$ by $2$ .

$f (\frac{9}{2}) = \frac{9}{2} \cdot \sqrt[3]{\frac{9 - 12}{2}}$

Step 12.2.4.2

Subtract $12$ from $9$ .

$f (\frac{9}{2}) = \frac{9}{2} \cdot \sqrt[3]{\frac{- 3}{2}}$

Step 12.2.5

Move the negative in front of the fraction.

$f (\frac{9}{2}) = \frac{9}{2} \cdot \sqrt[3]{- \frac{3}{2}}$

Step 12.2.6

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

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Step 12.2.6.1

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$f (\frac{9}{2}) = \frac{9}{2} \cdot \sqrt[3]{{(- 1)}^{3} (\frac{3}{2})}$

Step 12.2.6.2

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$f (\frac{9}{2}) = \frac{9}{2} \cdot \sqrt[3]{{({(- 1)}^{3})}^{3} (\frac{3}{2})}$

Step 12.2.7

Pull terms out from under the radical.

$f (\frac{9}{2}) = \frac{9}{2} \cdot ({(- 1)}^{3} \sqrt[3]{\frac{3}{2}})$

Step 12.2.8

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

$f (\frac{9}{2}) = \frac{9}{2} \cdot (- \sqrt[3]{\frac{3}{2}})$

Step 12.2.9

Rewrite $\sqrt[3]{\frac{3}{2}}$ as $\frac{\sqrt[3]{3}}{\sqrt[3]{2}}$ .

$f (\frac{9}{2}) = \frac{9}{2} \cdot (- \frac{\sqrt[3]{3}}{\sqrt[3]{2}})$

Step 12.2.10

Multiply $\frac{\sqrt[3]{3}}{\sqrt[3]{2}}$ by $\frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}}$ .

$f (\frac{9}{2}) = \frac{9}{2} \cdot (- (\frac{\sqrt[3]{3}}{\sqrt[3]{2}} \cdot \frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}}))$

Step 12.2.11

Combine and simplify the denominator.

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Step 12.2.11.1

Multiply $\frac{\sqrt[3]{3}}{\sqrt[3]{2}}$ by $\frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}}$ .

$f (\frac{9}{2}) = \frac{9}{2} \cdot (- \frac{\sqrt[3]{3} {\sqrt[3]{2}}^{2}}{\sqrt[3]{2} {\sqrt[3]{2}}^{2}})$

Step 12.2.11.2

Raise $\sqrt[3]{2}$ to the power of $1$ .

$f (\frac{9}{2}) = \frac{9}{2} \cdot (- \frac{\sqrt[3]{3} {\sqrt[3]{2}}^{2}}{\sqrt[3]{2} {\sqrt[3]{2}}^{2}})$

Step 12.2.11.3

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

$f (\frac{9}{2}) = \frac{9}{2} \cdot (- \frac{\sqrt[3]{3} {\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{1 + 2}})$

Step 12.2.11.4

Add $1$ and $2$ .

$f (\frac{9}{2}) = \frac{9}{2} \cdot (- \frac{\sqrt[3]{3} {\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{3}})$

Step 12.2.11.5

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

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Step 12.2.11.5.1

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

$f (\frac{9}{2}) = \frac{9}{2} \cdot (- \frac{\sqrt[3]{3} {\sqrt[3]{2}}^{2}}{{(2^{\frac{1}{3}})}^{3}})$

Step 12.2.11.5.2

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

$f (\frac{9}{2}) = \frac{9}{2} \cdot (- \frac{\sqrt[3]{3} {\sqrt[3]{2}}^{2}}{2^{\frac{1}{3} \cdot 3}})$

Step 12.2.11.5.3

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

$f (\frac{9}{2}) = \frac{9}{2} \cdot (- \frac{\sqrt[3]{3} {\sqrt[3]{2}}^{2}}{2^{\frac{3}{3}}})$

Step 12.2.11.5.4

Cancel the common factor of $3$ .

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Step 12.2.11.5.4.1

Cancel the common factor.

$f (\frac{9}{2}) = \frac{9}{2} \cdot (- \frac{\sqrt[3]{3} {\sqrt[3]{2}}^{2}}{2^{\frac{3}{3}}})$

Step 12.2.11.5.4.2

Rewrite the expression.

$f (\frac{9}{2}) = \frac{9}{2} \cdot (- \frac{\sqrt[3]{3} {\sqrt[3]{2}}^{2}}{2})$

Step 12.2.11.5.5

Evaluate the exponent.

$f (\frac{9}{2}) = \frac{9}{2} \cdot (- \frac{\sqrt[3]{3} {\sqrt[3]{2}}^{2}}{2})$

Step 12.2.12

Simplify the numerator.

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Step 12.2.12.1

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

$f (\frac{9}{2}) = \frac{9}{2} \cdot (- \frac{\sqrt[3]{3} \sqrt[3]{2^{2}}}{2})$

Step 12.2.12.2

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

$f (\frac{9}{2}) = \frac{9}{2} \cdot (- \frac{\sqrt[3]{3} \sqrt[3]{4}}{2})$

Step 12.2.13

Simplify the numerator.

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Step 12.2.13.1

Combine using the product rule for radicals.

$f (\frac{9}{2}) = \frac{9}{2} \cdot (- \frac{\sqrt[3]{3 \cdot 4}}{2})$

Step 12.2.13.2

Multiply $3$ by $4$ .

$f (\frac{9}{2}) = \frac{9}{2} \cdot (- \frac{\sqrt[3]{12}}{2})$

Step 12.2.14

Multiply $\frac{9}{2} (- \frac{\sqrt[3]{12}}{2})$ .

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Step 12.2.14.1

Multiply $\frac{9}{2}$ by $\frac{\sqrt[3]{12}}{2}$ .

$f (\frac{9}{2}) = - \frac{9 \sqrt[3]{12}}{2 \cdot 2}$

Step 12.2.14.2

Multiply $2$ by $2$ .

$f (\frac{9}{2}) = - \frac{9 \sqrt[3]{12}}{4}$

Step 12.2.15

The final answer is $- \frac{9 \sqrt[3]{12}}{4}$ .

$y = - \frac{9 \sqrt[3]{12}}{4}$

Step 13

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.

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

Step 14

Evaluate the second derivative.

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Step 14.1

Simplify the expression.

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Step 14.1.1

Subtract $6$ from $6$ .

$\frac{4 ((6) - 9)}{9 \cdot 0^{\frac{5}{3}}}$

Step 14.1.2

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

$\frac{4 ((6) - 9)}{9 \cdot {(0^{3})}^{\frac{5}{3}}}$

Step 14.1.3

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

$\frac{4 ((6) - 9)}{9 \cdot 0^{3 (\frac{5}{3})}}$

Step 14.2

Cancel the common factor of $3$ .

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Step 14.2.1

Cancel the common factor.

$\frac{4 ((6) - 9)}{9 \cdot 0^{3 (\frac{5}{3})}}$

Step 14.2.2

Rewrite the expression.

$\frac{4 ((6) - 9)}{9 \cdot 0^{5}}$

Step 14.3

Simplify the expression.

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Step 14.3.1

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

$\frac{4 ((6) - 9)}{9 \cdot 0}$

Step 14.3.2

Multiply $9$ by $0$ .

$\frac{4 ((6) - 9)}{0}$

Step 14.3.3

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

Undefined

$\frac{4 ((6) - 9)}{0}$

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.

$(- \infty, \frac{9}{2}) \cup (\frac{9}{2}, 6) \cup (6, \infty)$

Step 15.2

Substitute any number, such as $0$ , from the interval $(- \infty, \frac{9}{2})$ in the first derivative $\frac{2 (2 x - 9)}{3 {(x - 6)}^{\frac{2}{3}}}$ 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.

$f' (0) = \frac{2 (2 (0) - 9)}{3 {((0) - 6)}^{\frac{2}{3}}}$

Step 15.2.2

Simplify the result.

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Step 15.2.2.1

Factor $3$ out of $2 (2 (0) - 9)$ .

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

Step 15.2.2.2

Cancel the common factors.

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Step 15.2.2.2.1

Factor $3$ out of $3 {((0) - 6)}^{\frac{2}{3}}$ .

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

Step 15.2.2.2.2

Cancel the common factor.

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

Step 15.2.2.2.3

Rewrite the expression.

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

Step 15.2.2.3

Simplify the numerator.

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Step 15.2.2.3.1

Multiply $2$ by $0$ .

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

Step 15.2.2.3.2

Subtract $3$ from $0$ .

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

Step 15.2.2.4

Simplify the expression.

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Step 15.2.2.4.1

Subtract $6$ from $0$ .

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

Step 15.2.2.4.2

Multiply $2$ by $- 3$ .

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

Step 15.2.2.5

Move ${(- 6)}^{\frac{2}{3}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

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

Step 15.2.2.6

Multiply $- 6$ by ${(- 6)}^{- \frac{2}{3}}$ by adding the exponents.

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Step 15.2.2.6.1

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

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Step 15.2.2.6.1.1

Raise $- 6$ to the power of $1$ .

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

Step 15.2.2.6.1.2

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

$f' (0) = {(- 6)}^{1 - \frac{2}{3}}$

Step 15.2.2.6.2

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

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

Step 15.2.2.6.3

Combine the numerators over the common denominator.

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

Step 15.2.2.6.4

Subtract $2$ from $3$ .

$f' (0) = {(- 6)}^{\frac{1}{3}}$

Step 15.2.2.7

The final answer is ${(- 6)}^{\frac{1}{3}}$ .

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

Step 15.3

Substitute any number, such as $5$ , from the interval $(\frac{9}{2}, 6)$ in the first derivative $\frac{2 (2 x - 9)}{3 {(x - 6)}^{\frac{2}{3}}}$ to check if the result is negative or positive.

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Step 15.3.1

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

$f' (5) = \frac{2 (2 (5) - 9)}{3 {((5) - 6)}^{\frac{2}{3}}}$

Step 15.3.2

Simplify the result.

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Step 15.3.2.1

Simplify the numerator.

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Step 15.3.2.1.1

Multiply $2$ by $5$ .

$f' (5) = \frac{2 (10 - 9)}{3 {(5 - 6)}^{\frac{2}{3}}}$

Step 15.3.2.1.2

Subtract $9$ from $10$ .

$f' (5) = \frac{2 \cdot 1}{3 {(5 - 6)}^{\frac{2}{3}}}$

Step 15.3.2.2

Simplify the denominator.

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Step 15.3.2.2.1

Subtract $6$ from $5$ .

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

Step 15.3.2.2.2

Rewrite $- 1$ as ${(- 1)}^{3}$ .

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

Step 15.3.2.2.3

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

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

Step 15.3.2.2.4

Cancel the common factor of $3$ .

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Step 15.3.2.2.4.1

Cancel the common factor.

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

Step 15.3.2.2.4.2

Rewrite the expression.

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

Step 15.3.2.2.5

Raise $- 1$ to the power of $2$ .

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

Step 15.3.2.3

Simplify.

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Step 15.3.2.3.1

Multiply $2$ by $1$ .

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

Step 15.3.2.3.2

Multiply $3$ by $1$ .

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

Step 15.3.2.4

The final answer is $\frac{2}{3}$ .

$\frac{2}{3}$

Step 15.4

Substitute any number, such as $8$ , from the interval $(6, \infty)$ in the first derivative $\frac{2 (2 x - 9)}{3 {(x - 6)}^{\frac{2}{3}}}$ to check if the result is negative or positive.

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Step 15.4.1

Replace the variable $x$ with $8$ in the expression.

$f' (8) = \frac{2 (2 (8) - 9)}{3 {((8) - 6)}^{\frac{2}{3}}}$

Step 15.4.2

Simplify the result.

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Step 15.4.2.1

Simplify the numerator.

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Step 15.4.2.1.1

Multiply $2$ by $8$ .

$f' (8) = \frac{2 (16 - 9)}{3 {(8 - 6)}^{\frac{2}{3}}}$

Step 15.4.2.1.2

Subtract $9$ from $16$ .

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

Step 15.4.2.2

Simplify the expression.

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Step 15.4.2.2.1

Subtract $6$ from $8$ .

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

Step 15.4.2.2.2

Multiply $2$ by $7$ .

$f' (8) = \frac{14}{3 \cdot 2^{\frac{2}{3}}}$

Step 15.4.2.3

The final answer is $\frac{14}{3 \cdot 2^{\frac{2}{3}}}$ .

$\frac{14}{3 \cdot 2^{\frac{2}{3}}}$

Step 15.5

Since the first derivative changed signs from negative to positive around $x = \frac{9}{2}$ , then $x = \frac{9}{2}$ is a local minimum.

$x = \frac{9}{2}$ is a local minimum

Step 15.6

Since the first derivative did not change signs around $x = 6$ , this is not a local maximum or minimum.

Not a local maximum or minimum

Step 15.7

These are the local extrema for $f (x) = x \sqrt[3]{x - 6}$ .

$x = \frac{9}{2}$ is a local minimum

Step 16