Calculus Examples

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

Step 1

Find the first derivative of the function.

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Step 1.1

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

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

Step 1.2

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

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

Step 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 $g (x) = x - 2$ .

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Step 1.3.1

To apply the Chain Rule, set $u$ as $x - 2$ .

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

Step 1.3.2

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

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

Step 1.3.3

Replace all occurrences of $u$ with $x - 2$ .

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

Step 1.4

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

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

Step 1.5

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

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

Step 1.6

Combine the numerators over the common denominator.

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

Step 1.7

Simplify the numerator.

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Step 1.7.1

Multiply $- 1$ by $3$ .

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

Step 1.7.2

Subtract $3$ from $1$ .

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

Step 1.8

Combine fractions.

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Step 1.8.1

Move the negative in front of the fraction.

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

Step 1.8.2

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

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

Step 1.8.3

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

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

Step 1.8.4

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

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

Step 1.9

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

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

Step 1.10

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

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

Step 1.11

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

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

Step 1.12

Simplify the expression.

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Step 1.12.1

Add $1$ and $0$ .

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

Step 1.12.2

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

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

Step 1.13

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

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

Step 1.14

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

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

Step 1.15

Combine $\frac{x^{2}}{3 {(x - 2)}^{\frac{2}{3}}}$ and $2 {(x - 2)}^{\frac{1}{3}} x$ using a common denominator.

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Step 1.15.1

Move ${(x - 2)}^{\frac{1}{3}}$ .

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

Step 1.15.2

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

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

Step 1.15.3

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

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

Step 1.15.4

Combine the numerators over the common denominator.

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

Step 1.16

Multiply $3$ by $2$ .

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

Step 1.17

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

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Step 1.17.1

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

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

Step 1.17.2

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

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

Step 1.17.3

Combine the numerators over the common denominator.

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

Step 1.17.4

Add $2$ and $1$ .

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

Step 1.17.5

Divide $3$ by $3$ .

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

Step 1.18

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

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

Step 1.19

Simplify.

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Step 1.19.1

Apply the distributive property.

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

Step 1.19.2

Simplify the numerator.

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Step 1.19.2.1

Simplify each term.

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Step 1.19.2.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 1.19.2.1.1.1

Move $x$ .

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

Step 1.19.2.1.1.2

Multiply $x$ by $x$ .

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

Step 1.19.2.1.2

Multiply $- 2$ by $6$ .

$\frac{x^{2} + 6 x^{2} - 12 x}{3 {(x - 2)}^{\frac{2}{3}}}$

Step 1.19.2.2

Add $x^{2}$ and $6 x^{2}$ .

$\frac{7 x^{2} - 12 x}{3 {(x - 2)}^{\frac{2}{3}}}$

Step 1.19.3

Factor $x$ out of $7 x^{2} - 12 x$ .

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Step 1.19.3.1

Factor $x$ out of $7 x^{2}$ .

$\frac{x (7 x) - 12 x}{3 {(x - 2)}^{\frac{2}{3}}}$

Step 1.19.3.2

Factor $x$ out of $- 12 x$ .

$\frac{x (7 x) + x \cdot - 12}{3 {(x - 2)}^{\frac{2}{3}}}$

Step 1.19.3.3

Factor $x$ out of $x (7 x) + x \cdot - 12$ .

$\frac{x (7 x - 12)}{3 {(x - 2)}^{\frac{2}{3}}}$

Step 2

Find the second derivative of the function.

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Step 2.1

Since $\frac{1}{3}$ is constant with respect to $x$ , the derivative of $\frac{x (7 x - 12)}{3 {(x - 2)}^{\frac{2}{3}}}$ with respect to $x$ is $\frac{1}{3} \frac{d}{d x} [\frac{x (7 x - 12)}{{(x - 2)}^{\frac{2}{3}}}]$ .

$f'' (x) = \frac{1}{3} \cdot \frac{d}{d x} (\frac{x (7 x - 12)}{{(x - 2)}^{\frac{2}{3}}})$

Step 2.2

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 $f (x) = x (7 x - 12)$ and $g (x) = {(x - 2)}^{\frac{2}{3}}$ .

$f'' (x) = \frac{1}{3} \cdot \frac{{(x - 2)}^{\frac{2}{3}} \frac{d}{d x} (x (7 x - 12)) - x (7 x - 12) \frac{d}{d x} {(x - 2)}^{\frac{2}{3}}}{{({(x - 2)}^{\frac{2}{3}})}^{2}}$

Step 2.3

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

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Step 2.3.1

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

$f'' (x) = \frac{1}{3} \cdot \frac{{(x - 2)}^{\frac{2}{3}} \frac{d}{d x} (x (7 x - 12)) - x (7 x - 12) \frac{d}{d x} {(x - 2)}^{\frac{2}{3}}}{{(x - 2)}^{\frac{2}{3} \cdot 2}}$

Step 2.3.2

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

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Step 2.3.2.1

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

$f'' (x) = \frac{1}{3} \cdot \frac{{(x - 2)}^{\frac{2}{3}} \frac{d}{d x} (x (7 x - 12)) - x (7 x - 12) \frac{d}{d x} {(x - 2)}^{\frac{2}{3}}}{{(x - 2)}^{\frac{2 \cdot 2}{3}}}$

Step 2.3.2.2

Multiply $2$ by $2$ .

$f'' (x) = \frac{1}{3} \cdot \frac{{(x - 2)}^{\frac{2}{3}} \frac{d}{d x} (x (7 x - 12)) - x (7 x - 12) \frac{d}{d x} {(x - 2)}^{\frac{2}{3}}}{{(x - 2)}^{\frac{4}{3}}}$

Step 2.4

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 $g (x) = 7 x - 12$ .

$f'' (x) = \frac{1}{3} \cdot \frac{{(x - 2)}^{\frac{2}{3}} (x \frac{d}{d x} (7 x - 12) + (7 x - 12) \frac{d}{d x} (x)) - x (7 x - 12) \frac{d}{d x} {(x - 2)}^{\frac{2}{3}}}{{(x - 2)}^{\frac{4}{3}}}$

Step 2.5

Differentiate.

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Step 2.5.1

By the Sum Rule, the derivative of $7 x - 12$ with respect to $x$ is $\frac{d}{d x} [7 x] + \frac{d}{d x} [- 12]$ .

$f'' (x) = \frac{1}{3} \cdot \frac{{(x - 2)}^{\frac{2}{3}} (x (\frac{d}{d x} (7 x) + \frac{d}{d x} (- 12)) + (7 x - 12) \frac{d}{d x} (x)) - x (7 x - 12) \frac{d}{d x} {(x - 2)}^{\frac{2}{3}}}{{(x - 2)}^{\frac{4}{3}}}$

Step 2.5.2

Since $7$ is constant with respect to $x$ , the derivative of $7 x$ with respect to $x$ is $7 \frac{d}{d x} [x]$ .

$f'' (x) = \frac{1}{3} \cdot \frac{{(x - 2)}^{\frac{2}{3}} (x (7 \frac{d}{d x} (x) + \frac{d}{d x} (- 12)) + (7 x - 12) \frac{d}{d x} (x)) - x (7 x - 12) \frac{d}{d x} {(x - 2)}^{\frac{2}{3}}}{{(x - 2)}^{\frac{4}{3}}}$

Step 2.5.3

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

$f'' (x) = \frac{1}{3} \cdot \frac{{(x - 2)}^{\frac{2}{3}} (x (7 \cdot 1 + \frac{d}{d x} (- 12)) + (7 x - 12) \frac{d}{d x} (x)) - x (7 x - 12) \frac{d}{d x} {(x - 2)}^{\frac{2}{3}}}{{(x - 2)}^{\frac{4}{3}}}$

Step 2.5.4

Multiply $7$ by $1$ .

$f'' (x) = \frac{1}{3} \cdot \frac{{(x - 2)}^{\frac{2}{3}} (x (7 + \frac{d}{d x} (- 12)) + (7 x - 12) \frac{d}{d x} (x)) - x (7 x - 12) \frac{d}{d x} {(x - 2)}^{\frac{2}{3}}}{{(x - 2)}^{\frac{4}{3}}}$

Step 2.5.5

Since $- 12$ is constant with respect to $x$ , the derivative of $- 12$ with respect to $x$ is $0$ .

$f'' (x) = \frac{1}{3} \cdot \frac{{(x - 2)}^{\frac{2}{3}} (x (7 + 0) + (7 x - 12) \frac{d}{d x} (x)) - x (7 x - 12) \frac{d}{d x} {(x - 2)}^{\frac{2}{3}}}{{(x - 2)}^{\frac{4}{3}}}$

Step 2.5.6

Simplify the expression.

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Step 2.5.6.1

Add $7$ and $0$ .

$f'' (x) = \frac{1}{3} \cdot \frac{{(x - 2)}^{\frac{2}{3}} (x \cdot 7 + (7 x - 12) \frac{d}{d x} (x)) - x (7 x - 12) \frac{d}{d x} {(x - 2)}^{\frac{2}{3}}}{{(x - 2)}^{\frac{4}{3}}}$

Step 2.5.6.2

Move $7$ to the left of $x$ .

$f'' (x) = \frac{1}{3} \cdot \frac{{(x - 2)}^{\frac{2}{3}} (7 \cdot x + (7 x - 12) \frac{d}{d x} (x)) - x (7 x - 12) \frac{d}{d x} {(x - 2)}^{\frac{2}{3}}}{{(x - 2)}^{\frac{4}{3}}}$

Step 2.5.7

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

$f'' (x) = \frac{1}{3} \cdot \frac{{(x - 2)}^{\frac{2}{3}} (7 x + (7 x - 12) \cdot 1) - x (7 x - 12) \frac{d}{d x} {(x - 2)}^{\frac{2}{3}}}{{(x - 2)}^{\frac{4}{3}}}$

Step 2.5.8

Simplify by adding terms.

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Step 2.5.8.1

Multiply $7 x - 12$ by $1$ .

$f'' (x) = \frac{1}{3} \cdot \frac{{(x - 2)}^{\frac{2}{3}} (7 x + 7 x - 12) - x (7 x - 12) \frac{d}{d x} {(x - 2)}^{\frac{2}{3}}}{{(x - 2)}^{\frac{4}{3}}}$

Step 2.5.8.2

Add $7 x$ and $7 x$ .

$f'' (x) = \frac{1}{3} \cdot \frac{{(x - 2)}^{\frac{2}{3}} (14 x - 12) - x (7 x - 12) \frac{d}{d x} {(x - 2)}^{\frac{2}{3}}}{{(x - 2)}^{\frac{4}{3}}}$

Step 2.6

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

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Step 2.6.1

To apply the Chain Rule, set $u$ as $x - 2$ .

$f'' (x) = \frac{1}{3} \cdot \frac{{(x - 2)}^{\frac{2}{3}} (14 x - 12) - x (7 x - 12) (\frac{d}{d u} (u^{\frac{2}{3}}) \frac{d}{d x} (x - 2))}{{(x - 2)}^{\frac{4}{3}}}$

Step 2.6.2

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

$f'' (x) = \frac{1}{3} \cdot \frac{{(x - 2)}^{\frac{2}{3}} (14 x - 12) - x (7 x - 12) (\frac{2}{3} u^{\frac{2}{3} - 1} \frac{d}{d x} (x - 2))}{{(x - 2)}^{\frac{4}{3}}}$

Step 2.6.3

Replace all occurrences of $u$ with $x - 2$ .

$f'' (x) = \frac{1}{3} \cdot \frac{{(x - 2)}^{\frac{2}{3}} (14 x - 12) - x (7 x - 12) (\frac{2}{3} \cdot {(x - 2)}^{\frac{2}{3} - 1} \frac{d}{d x} (x - 2))}{{(x - 2)}^{\frac{4}{3}}}$

Step 2.7

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

$f'' (x) = \frac{1}{3} \cdot \frac{{(x - 2)}^{\frac{2}{3}} (14 x - 12) - x (7 x - 12) (\frac{2}{3} \cdot {(x - 2)}^{\frac{2}{3} - 1 \cdot \frac{3}{3}} \frac{d}{d x} (x - 2))}{{(x - 2)}^{\frac{4}{3}}}$

Step 2.8

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

$f'' (x) = \frac{1}{3} \cdot \frac{{(x - 2)}^{\frac{2}{3}} (14 x - 12) - x (7 x - 12) (\frac{2}{3} \cdot {(x - 2)}^{\frac{2}{3} + \frac{- 1 \cdot 3}{3}} \frac{d}{d x} (x - 2))}{{(x - 2)}^{\frac{4}{3}}}$

Step 2.9

Combine the numerators over the common denominator.

$f'' (x) = \frac{1}{3} \cdot \frac{{(x - 2)}^{\frac{2}{3}} (14 x - 12) - x (7 x - 12) (\frac{2}{3} \cdot {(x - 2)}^{\frac{2 - 1 \cdot 3}{3}} \frac{d}{d x} (x - 2))}{{(x - 2)}^{\frac{4}{3}}}$

Step 2.10

Simplify the numerator.

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Step 2.10.1

Multiply $- 1$ by $3$ .

$f'' (x) = \frac{1}{3} \cdot \frac{{(x - 2)}^{\frac{2}{3}} (14 x - 12) - x (7 x - 12) (\frac{2}{3} \cdot {(x - 2)}^{\frac{2 - 3}{3}} \frac{d}{d x} (x - 2))}{{(x - 2)}^{\frac{4}{3}}}$

Step 2.10.2

Subtract $3$ from $2$ .

$f'' (x) = \frac{1}{3} \cdot \frac{{(x - 2)}^{\frac{2}{3}} (14 x - 12) - x (7 x - 12) (\frac{2}{3} \cdot {(x - 2)}^{\frac{- 1}{3}} \frac{d}{d x} (x - 2))}{{(x - 2)}^{\frac{4}{3}}}$

Step 2.11

Combine fractions.

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Step 2.11.1

Move the negative in front of the fraction.

$f'' (x) = \frac{1}{3} \cdot \frac{{(x - 2)}^{\frac{2}{3}} (14 x - 12) - x (7 x - 12) (\frac{2}{3} \cdot {(x - 2)}^{- \frac{1}{3}} \frac{d}{d x} (x - 2))}{{(x - 2)}^{\frac{4}{3}}}$

Step 2.11.2

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

$f'' (x) = \frac{1}{3} \cdot \frac{{(x - 2)}^{\frac{2}{3}} (14 x - 12) - x (7 x - 12) (\frac{2 {(x - 2)}^{- \frac{1}{3}}}{3} \cdot \frac{d}{d x} (x - 2))}{{(x - 2)}^{\frac{4}{3}}}$

Step 2.11.3

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

$f'' (x) = \frac{1}{3} \cdot \frac{{(x - 2)}^{\frac{2}{3}} (14 x - 12) - x (7 x - 12) (\frac{2}{3 {(x - 2)}^{\frac{1}{3}}} \cdot \frac{d}{d x} (x - 2))}{{(x - 2)}^{\frac{4}{3}}}$

Step 2.11.4

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

$f'' (x) = \frac{1}{3} \cdot \frac{{(x - 2)}^{\frac{2}{3}} (14 x - 12) - \frac{2 x}{3 {(x - 2)}^{\frac{1}{3}}} \cdot (7 x - 12) \frac{d}{d x} (x - 2)}{{(x - 2)}^{\frac{4}{3}}}$

Step 2.12

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

$f'' (x) = \frac{1}{3} \cdot \frac{{(x - 2)}^{\frac{2}{3}} (14 x - 12) - \frac{2 x}{3 {(x - 2)}^{\frac{1}{3}}} \cdot ((7 x - 12) (\frac{d}{d x} (x) + \frac{d}{d x} (- 2)))}{{(x - 2)}^{\frac{4}{3}}}$

Step 2.13

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

$f'' (x) = \frac{1}{3} \cdot \frac{{(x - 2)}^{\frac{2}{3}} (14 x - 12) - \frac{2 x}{3 {(x - 2)}^{\frac{1}{3}}} \cdot ((7 x - 12) (1 + \frac{d}{d x} (- 2)))}{{(x - 2)}^{\frac{4}{3}}}$

Step 2.14

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$f'' (x) = \frac{1}{3} \cdot \frac{{(x - 2)}^{\frac{2}{3}} (14 x - 12) - \frac{2 x}{3 {(x - 2)}^{\frac{1}{3}}} \cdot ((7 x - 12) (1 + 0))}{{(x - 2)}^{\frac{4}{3}}}$

Step 2.15

Combine fractions.

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Step 2.15.1

Add $1$ and $0$ .

$f'' (x) = \frac{1}{3} \cdot \frac{{(x - 2)}^{\frac{2}{3}} (14 x - 12) - \frac{2 x}{3 {(x - 2)}^{\frac{1}{3}}} \cdot (7 x - 12) \cdot 1}{{(x - 2)}^{\frac{4}{3}}}$

Step 2.15.2

Multiply $- 1$ by $1$ .

$f'' (x) = \frac{1}{3} \cdot \frac{{(x - 2)}^{\frac{2}{3}} (14 x - 12) - \frac{2 x}{3 {(x - 2)}^{\frac{1}{3}}} \cdot (7 x - 12)}{{(x - 2)}^{\frac{4}{3}}}$

Step 2.15.3

Multiply $\frac{1}{3}$ by $\frac{{(x - 2)}^{\frac{2}{3}} (14 x - 12) - \frac{2 x}{3 {(x - 2)}^{\frac{1}{3}}} (7 x - 12)}{{(x - 2)}^{\frac{4}{3}}}$ .

$f'' (x) = \frac{{(x - 2)}^{\frac{2}{3}} (14 x - 12) - \frac{2 x}{3 {(x - 2)}^{\frac{1}{3}}} \cdot (7 x - 12)}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16

Simplify.

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Step 2.16.1

Simplify the numerator.

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Step 2.16.1.1

Apply the distributive property.

$f'' (x) = \frac{{(x - 2)}^{\frac{2}{3}} (14 x) + {(x - 2)}^{\frac{2}{3}} \cdot - 12 - \frac{2 x}{3 {(x - 2)}^{\frac{1}{3}}} \cdot (7 x - 12)}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.1.2

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{14 {(x - 2)}^{\frac{2}{3}} x + {(x - 2)}^{\frac{2}{3}} \cdot - 12 - \frac{2 x}{3 {(x - 2)}^{\frac{1}{3}}} \cdot (7 x - 12)}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.1.3

Move $- 12$ to the left of ${(x - 2)}^{\frac{2}{3}}$ .

$f'' (x) = \frac{14 {(x - 2)}^{\frac{2}{3}} x - 12 \cdot {(x - 2)}^{\frac{2}{3}} - \frac{2 x}{3 {(x - 2)}^{\frac{1}{3}}} \cdot (7 x - 12)}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.1.4

Apply the distributive property.

$f'' (x) = \frac{14 {(x - 2)}^{\frac{2}{3}} x - 12 {(x - 2)}^{\frac{2}{3}} - \frac{2 x}{3 {(x - 2)}^{\frac{1}{3}}} \cdot (7 x) - \frac{2 x}{3 {(x - 2)}^{\frac{1}{3}}} \cdot - 12}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.1.5

Multiply $- \frac{2 x}{3 {(x - 2)}^{\frac{1}{3}}} (7 x)$ .

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Step 2.16.1.5.1

Multiply $7$ by $- 1$ .

$f'' (x) = \frac{14 {(x - 2)}^{\frac{2}{3}} x - 12 {(x - 2)}^{\frac{2}{3}} - 7 \frac{2 x}{3 {(x - 2)}^{\frac{1}{3}}} \cdot x - \frac{2 x}{3 {(x - 2)}^{\frac{1}{3}}} \cdot - 12}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.1.5.2

Combine $- 7$ and $\frac{2 x}{3 {(x - 2)}^{\frac{1}{3}}}$ .

$f'' (x) = \frac{14 {(x - 2)}^{\frac{2}{3}} x - 12 {(x - 2)}^{\frac{2}{3}} + \frac{- 7 (2 x)}{3 {(x - 2)}^{\frac{1}{3}}} \cdot x - \frac{2 x}{3 {(x - 2)}^{\frac{1}{3}}} \cdot - 12}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.1.5.3

Multiply $2$ by $- 7$ .

$f'' (x) = \frac{14 {(x - 2)}^{\frac{2}{3}} x - 12 {(x - 2)}^{\frac{2}{3}} + \frac{- 14 x}{3 {(x - 2)}^{\frac{1}{3}}} \cdot x - \frac{2 x}{3 {(x - 2)}^{\frac{1}{3}}} \cdot - 12}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.1.5.4

Combine $\frac{- 14 x}{3 {(x - 2)}^{\frac{1}{3}}}$ and $x$ .

$f'' (x) = \frac{14 {(x - 2)}^{\frac{2}{3}} x - 12 {(x - 2)}^{\frac{2}{3}} + \frac{- 14 x \cdot x}{3 {(x - 2)}^{\frac{1}{3}}} - \frac{2 x}{3 {(x - 2)}^{\frac{1}{3}}} \cdot - 12}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.1.5.5

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

$f'' (x) = \frac{14 {(x - 2)}^{\frac{2}{3}} x - 12 {(x - 2)}^{\frac{2}{3}} + \frac{- 14 (x \cdot x)}{3 {(x - 2)}^{\frac{1}{3}}} - \frac{2 x}{3 {(x - 2)}^{\frac{1}{3}}} \cdot - 12}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.1.5.6

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

Step 2.16.1.5.7

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

$f'' (x) = \frac{14 {(x - 2)}^{\frac{2}{3}} x - 12 {(x - 2)}^{\frac{2}{3}} + \frac{- 14 x^{1 + 1}}{3 {(x - 2)}^{\frac{1}{3}}} - \frac{2 x}{3 {(x - 2)}^{\frac{1}{3}}} \cdot - 12}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.1.5.8

Add $1$ and $1$ .

$f'' (x) = \frac{14 {(x - 2)}^{\frac{2}{3}} x - 12 {(x - 2)}^{\frac{2}{3}} + \frac{- 14 x^{2}}{3 {(x - 2)}^{\frac{1}{3}}} - \frac{2 x}{3 {(x - 2)}^{\frac{1}{3}}} \cdot - 12}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.1.6

Cancel the common factor of $3$ .

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Step 2.16.1.6.1

Move the leading negative in $- \frac{2 x}{3 {(x - 2)}^{\frac{1}{3}}}$ into the numerator.

$f'' (x) = \frac{14 {(x - 2)}^{\frac{2}{3}} x - 12 {(x - 2)}^{\frac{2}{3}} + \frac{- 14 x^{2}}{3 {(x - 2)}^{\frac{1}{3}}} + \frac{- 2 x}{3 {(x - 2)}^{\frac{1}{3}}} \cdot - 12}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.1.6.2

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

$f'' (x) = \frac{14 {(x - 2)}^{\frac{2}{3}} x - 12 {(x - 2)}^{\frac{2}{3}} + \frac{- 14 x^{2}}{3 {(x - 2)}^{\frac{1}{3}}} + \frac{- 2 x}{3 ({(x - 2)}^{\frac{1}{3}})} \cdot - 12}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.1.6.3

Factor $3$ out of $- 12$ .

$f'' (x) = \frac{14 {(x - 2)}^{\frac{2}{3}} x - 12 {(x - 2)}^{\frac{2}{3}} + \frac{- 14 x^{2}}{3 {(x - 2)}^{\frac{1}{3}}} + \frac{- 2 x}{3 ({(x - 2)}^{\frac{1}{3}})} \cdot (3 (- 4))}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.1.6.4

Cancel the common factor.

$f'' (x) = \frac{14 {(x - 2)}^{\frac{2}{3}} x - 12 {(x - 2)}^{\frac{2}{3}} + \frac{- 14 x^{2}}{3 {(x - 2)}^{\frac{1}{3}}} + \frac{- 2 x}{3 {(x - 2)}^{\frac{1}{3}}} \cdot (3 \cdot - 4)}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.1.6.5

Rewrite the expression.

$f'' (x) = \frac{14 {(x - 2)}^{\frac{2}{3}} x - 12 {(x - 2)}^{\frac{2}{3}} + \frac{- 14 x^{2}}{3 {(x - 2)}^{\frac{1}{3}}} + \frac{- 2 x}{{(x - 2)}^{\frac{1}{3}}} \cdot - 4}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.1.7

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

$f'' (x) = \frac{14 {(x - 2)}^{\frac{2}{3}} x - 12 {(x - 2)}^{\frac{2}{3}} + \frac{- 14 x^{2}}{3 {(x - 2)}^{\frac{1}{3}}} + \frac{- 2 x \cdot - 4}{{(x - 2)}^{\frac{1}{3}}}}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.1.8

Multiply $- 4$ by $- 2$ .

$f'' (x) = \frac{14 {(x - 2)}^{\frac{2}{3}} x - 12 {(x - 2)}^{\frac{2}{3}} + \frac{- 14 x^{2}}{3 {(x - 2)}^{\frac{1}{3}}} + \frac{8 x}{{(x - 2)}^{\frac{1}{3}}}}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.1.9

Move the negative in front of the fraction.

$f'' (x) = \frac{14 {(x - 2)}^{\frac{2}{3}} x - 12 {(x - 2)}^{\frac{2}{3}} - \frac{14 x^{2}}{3 {(x - 2)}^{\frac{1}{3}}} + \frac{8 x}{{(x - 2)}^{\frac{1}{3}}}}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.1.10

Subtract $\frac{14 x^{2}}{3 {(x - 2)}^{\frac{1}{3}}}$ from $14 {(x - 2)}^{\frac{2}{3}} x$ .

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Step 2.16.1.10.1

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

$f'' (x) = \frac{14 x {(x - 2)}^{\frac{2}{3}} - \frac{14 x^{2}}{3 {(x - 2)}^{\frac{1}{3}}} - 12 {(x - 2)}^{\frac{2}{3}} + \frac{8 x}{{(x - 2)}^{\frac{1}{3}}}}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.1.10.2

To write $14 x {(x - 2)}^{\frac{2}{3}}$ as a fraction with a common denominator, multiply by $\frac{3 {(x - 2)}^{\frac{1}{3}}}{3 {(x - 2)}^{\frac{1}{3}}}$ .

$f'' (x) = \frac{14 x {(x - 2)}^{\frac{2}{3}} \cdot \frac{3 {(x - 2)}^{\frac{1}{3}}}{3 {(x - 2)}^{\frac{1}{3}}} - \frac{14 x^{2}}{3 {(x - 2)}^{\frac{1}{3}}} - 12 {(x - 2)}^{\frac{2}{3}} + \frac{8 x}{{(x - 2)}^{\frac{1}{3}}}}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.1.10.3

Combine $14 x {(x - 2)}^{\frac{2}{3}}$ and $\frac{3 {(x - 2)}^{\frac{1}{3}}}{3 {(x - 2)}^{\frac{1}{3}}}$ .

$f'' (x) = \frac{\frac{14 x {(x - 2)}^{\frac{2}{3}} (3 {(x - 2)}^{\frac{1}{3}})}{3 {(x - 2)}^{\frac{1}{3}}} - \frac{14 x^{2}}{3 {(x - 2)}^{\frac{1}{3}}} - 12 {(x - 2)}^{\frac{2}{3}} + \frac{8 x}{{(x - 2)}^{\frac{1}{3}}}}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.1.10.4

Combine the numerators over the common denominator.

$f'' (x) = \frac{\frac{14 x {(x - 2)}^{\frac{2}{3}} (3 {(x - 2)}^{\frac{1}{3}}) - 14 x^{2}}{3 {(x - 2)}^{\frac{1}{3}}} - 12 {(x - 2)}^{\frac{2}{3}} + \frac{8 x}{{(x - 2)}^{\frac{1}{3}}}}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.1.11

To write $- 12 {(x - 2)}^{\frac{2}{3}}$ as a fraction with a common denominator, multiply by $\frac{3 {(x - 2)}^{\frac{1}{3}}}{3 {(x - 2)}^{\frac{1}{3}}}$ .

$f'' (x) = \frac{\frac{14 x {(x - 2)}^{\frac{2}{3}} (3 {(x - 2)}^{\frac{1}{3}}) - 14 x^{2}}{3 {(x - 2)}^{\frac{1}{3}}} - 12 {(x - 2)}^{\frac{2}{3}} \cdot \frac{3 {(x - 2)}^{\frac{1}{3}}}{3 {(x - 2)}^{\frac{1}{3}}} + \frac{8 x}{{(x - 2)}^{\frac{1}{3}}}}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.1.12

Combine $- 12 {(x - 2)}^{\frac{2}{3}}$ and $\frac{3 {(x - 2)}^{\frac{1}{3}}}{3 {(x - 2)}^{\frac{1}{3}}}$ .

$f'' (x) = \frac{\frac{14 x {(x - 2)}^{\frac{2}{3}} (3 {(x - 2)}^{\frac{1}{3}}) - 14 x^{2}}{3 {(x - 2)}^{\frac{1}{3}}} + \frac{- 12 {(x - 2)}^{\frac{2}{3}} (3 {(x - 2)}^{\frac{1}{3}})}{3 {(x - 2)}^{\frac{1}{3}}} + \frac{8 x}{{(x - 2)}^{\frac{1}{3}}}}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.1.13

Combine the numerators over the common denominator.

$f'' (x) = \frac{\frac{14 x {(x - 2)}^{\frac{2}{3}} (3 {(x - 2)}^{\frac{1}{3}}) - 14 x^{2} - 12 {(x - 2)}^{\frac{2}{3}} (3 {(x - 2)}^{\frac{1}{3}})}{3 {(x - 2)}^{\frac{1}{3}}} + \frac{8 x}{{(x - 2)}^{\frac{1}{3}}}}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.1.14

Simplify the numerator.

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Step 2.16.1.14.1

Factor $2$ out of $14 x {(x - 2)}^{\frac{2}{3}} (3 {(x - 2)}^{\frac{1}{3}}) - 14 x^{2} - 12 {(x - 2)}^{\frac{2}{3}} (3 {(x - 2)}^{\frac{1}{3}})$ .

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Step 2.16.1.14.1.1

Factor $2$ out of $14 x {(x - 2)}^{\frac{2}{3}} (3 {(x - 2)}^{\frac{1}{3}})$ .

$f'' (x) = \frac{\frac{2 (7 x {(x - 2)}^{\frac{2}{3}} (3 {(x - 2)}^{\frac{1}{3}})) - 14 x^{2} - 12 {(x - 2)}^{\frac{2}{3}} (3 {(x - 2)}^{\frac{1}{3}})}{3 {(x - 2)}^{\frac{1}{3}}} + \frac{8 x}{{(x - 2)}^{\frac{1}{3}}}}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.1.14.1.2

Factor $2$ out of $- 14 x^{2}$ .

$f'' (x) = \frac{\frac{2 (7 x {(x - 2)}^{\frac{2}{3}} (3 {(x - 2)}^{\frac{1}{3}})) + 2 (- 7 x^{2}) - 12 {(x - 2)}^{\frac{2}{3}} (3 {(x - 2)}^{\frac{1}{3}})}{3 {(x - 2)}^{\frac{1}{3}}} + \frac{8 x}{{(x - 2)}^{\frac{1}{3}}}}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.1.14.1.3

Factor $2$ out of $- 12 {(x - 2)}^{\frac{2}{3}} (3 {(x - 2)}^{\frac{1}{3}})$ .

$f'' (x) = \frac{\frac{2 (7 x {(x - 2)}^{\frac{2}{3}} (3 {(x - 2)}^{\frac{1}{3}})) + 2 (- 7 x^{2}) + 2 (- 6 {(x - 2)}^{\frac{2}{3}} (3 {(x - 2)}^{\frac{1}{3}}))}{3 {(x - 2)}^{\frac{1}{3}}} + \frac{8 x}{{(x - 2)}^{\frac{1}{3}}}}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.1.14.1.4

Factor $2$ out of $2 (7 x {(x - 2)}^{\frac{2}{3}} (3 {(x - 2)}^{\frac{1}{3}})) + 2 (- 7 x^{2})$ .

$f'' (x) = \frac{\frac{2 (7 x {(x - 2)}^{\frac{2}{3}} (3 {(x - 2)}^{\frac{1}{3}}) - 7 x^{2}) + 2 (- 6 {(x - 2)}^{\frac{2}{3}} (3 {(x - 2)}^{\frac{1}{3}}))}{3 {(x - 2)}^{\frac{1}{3}}} + \frac{8 x}{{(x - 2)}^{\frac{1}{3}}}}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.1.14.1.5

Factor $2$ out of $2 (7 x {(x - 2)}^{\frac{2}{3}} (3 {(x - 2)}^{\frac{1}{3}}) - 7 x^{2}) + 2 (- 6 {(x - 2)}^{\frac{2}{3}} (3 {(x - 2)}^{\frac{1}{3}}))$ .

$f'' (x) = \frac{\frac{2 (7 x {(x - 2)}^{\frac{2}{3}} (3 {(x - 2)}^{\frac{1}{3}}) - 7 x^{2} - 6 {(x - 2)}^{\frac{2}{3}} (3 {(x - 2)}^{\frac{1}{3}}))}{3 {(x - 2)}^{\frac{1}{3}}} + \frac{8 x}{{(x - 2)}^{\frac{1}{3}}}}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.1.14.2

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

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Step 2.16.1.14.2.1

Move ${(x - 2)}^{\frac{1}{3}}$ .

$f'' (x) = \frac{\frac{2 (7 x ({(x - 2)}^{\frac{1}{3}} {(x - 2)}^{\frac{2}{3}}) \cdot 3 - 7 x^{2} - 6 {(x - 2)}^{\frac{2}{3}} (3 {(x - 2)}^{\frac{1}{3}}))}{3 {(x - 2)}^{\frac{1}{3}}} + \frac{8 x}{{(x - 2)}^{\frac{1}{3}}}}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.1.14.2.2

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

$f'' (x) = \frac{\frac{2 (7 x {(x - 2)}^{\frac{1}{3} + \frac{2}{3}} \cdot 3 - 7 x^{2} - 6 {(x - 2)}^{\frac{2}{3}} (3 {(x - 2)}^{\frac{1}{3}}))}{3 {(x - 2)}^{\frac{1}{3}}} + \frac{8 x}{{(x - 2)}^{\frac{1}{3}}}}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.1.14.2.3

Combine the numerators over the common denominator.

$f'' (x) = \frac{\frac{2 (7 x {(x - 2)}^{\frac{1 + 2}{3}} \cdot 3 - 7 x^{2} - 6 {(x - 2)}^{\frac{2}{3}} (3 {(x - 2)}^{\frac{1}{3}}))}{3 {(x - 2)}^{\frac{1}{3}}} + \frac{8 x}{{(x - 2)}^{\frac{1}{3}}}}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.1.14.2.4

Add $1$ and $2$ .

$f'' (x) = \frac{\frac{2 (7 x {(x - 2)}^{\frac{3}{3}} \cdot 3 - 7 x^{2} - 6 {(x - 2)}^{\frac{2}{3}} (3 {(x - 2)}^{\frac{1}{3}}))}{3 {(x - 2)}^{\frac{1}{3}}} + \frac{8 x}{{(x - 2)}^{\frac{1}{3}}}}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.1.14.2.5

Divide $3$ by $3$ .

$f'' (x) = \frac{\frac{2 (7 x (x - 2) \cdot 3 - 7 x^{2} - 6 {(x - 2)}^{\frac{2}{3}} (3 {(x - 2)}^{\frac{1}{3}}))}{3 {(x - 2)}^{\frac{1}{3}}} + \frac{8 x}{{(x - 2)}^{\frac{1}{3}}}}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.1.14.3

Simplify $7 x {(x - 2)}^{1} \cdot 3$ .

Step 2.16.1.14.4

Apply the distributive property.

$f'' (x) = \frac{\frac{2 ((7 x \cdot x + 7 x \cdot - 2) \cdot 3 - 7 x^{2} - 6 {(x - 2)}^{\frac{2}{3}} (3 {(x - 2)}^{\frac{1}{3}}))}{3 {(x - 2)}^{\frac{1}{3}}} + \frac{8 x}{{(x - 2)}^{\frac{1}{3}}}}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.1.14.5

Multiply $x$ by $x$ by adding the exponents.

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Step 2.16.1.14.5.1

Move $x$ .

$f'' (x) = \frac{\frac{2 ((7 (x \cdot x) + 7 x \cdot - 2) \cdot 3 - 7 x^{2} - 6 {(x - 2)}^{\frac{2}{3}} (3 {(x - 2)}^{\frac{1}{3}}))}{3 {(x - 2)}^{\frac{1}{3}}} + \frac{8 x}{{(x - 2)}^{\frac{1}{3}}}}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.1.14.5.2

Multiply $x$ by $x$ .

$f'' (x) = \frac{\frac{2 ((7 x^{2} + 7 x \cdot - 2) \cdot 3 - 7 x^{2} - 6 {(x - 2)}^{\frac{2}{3}} (3 {(x - 2)}^{\frac{1}{3}}))}{3 {(x - 2)}^{\frac{1}{3}}} + \frac{8 x}{{(x - 2)}^{\frac{1}{3}}}}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.1.14.6

Multiply $- 2$ by $7$ .

$f'' (x) = \frac{\frac{2 ((7 x^{2} - 14 x) \cdot 3 - 7 x^{2} - 6 {(x - 2)}^{\frac{2}{3}} (3 {(x - 2)}^{\frac{1}{3}}))}{3 {(x - 2)}^{\frac{1}{3}}} + \frac{8 x}{{(x - 2)}^{\frac{1}{3}}}}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.1.14.7

Apply the distributive property.

$f'' (x) = \frac{\frac{2 (7 x^{2} \cdot 3 - 14 x \cdot 3 - 7 x^{2} - 6 {(x - 2)}^{\frac{2}{3}} (3 {(x - 2)}^{\frac{1}{3}}))}{3 {(x - 2)}^{\frac{1}{3}}} + \frac{8 x}{{(x - 2)}^{\frac{1}{3}}}}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.1.14.8

Multiply $3$ by $7$ .

$f'' (x) = \frac{\frac{2 (21 x^{2} - 14 x \cdot 3 - 7 x^{2} - 6 {(x - 2)}^{\frac{2}{3}} (3 {(x - 2)}^{\frac{1}{3}}))}{3 {(x - 2)}^{\frac{1}{3}}} + \frac{8 x}{{(x - 2)}^{\frac{1}{3}}}}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.1.14.9

Multiply $3$ by $- 14$ .

$f'' (x) = \frac{\frac{2 (21 x^{2} - 42 x - 7 x^{2} - 6 {(x - 2)}^{\frac{2}{3}} (3 {(x - 2)}^{\frac{1}{3}}))}{3 {(x - 2)}^{\frac{1}{3}}} + \frac{8 x}{{(x - 2)}^{\frac{1}{3}}}}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.1.14.10

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{\frac{2 (21 x^{2} - 42 x - 7 x^{2} - 6 \cdot (3 {(x - 2)}^{\frac{2}{3}} {(x - 2)}^{\frac{1}{3}}))}{3 {(x - 2)}^{\frac{1}{3}}} + \frac{8 x}{{(x - 2)}^{\frac{1}{3}}}}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.1.14.11

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

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Step 2.16.1.14.11.1

Move ${(x - 2)}^{\frac{1}{3}}$ .

$f'' (x) = \frac{\frac{2 (21 x^{2} - 42 x - 7 x^{2} - 6 \cdot (3 ({(x - 2)}^{\frac{1}{3}} {(x - 2)}^{\frac{2}{3}})))}{3 {(x - 2)}^{\frac{1}{3}}} + \frac{8 x}{{(x - 2)}^{\frac{1}{3}}}}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.1.14.11.2

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

$f'' (x) = \frac{\frac{2 (21 x^{2} - 42 x - 7 x^{2} - 6 \cdot (3 {(x - 2)}^{\frac{1}{3} + \frac{2}{3}}))}{3 {(x - 2)}^{\frac{1}{3}}} + \frac{8 x}{{(x - 2)}^{\frac{1}{3}}}}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.1.14.11.3

Combine the numerators over the common denominator.

$f'' (x) = \frac{\frac{2 (21 x^{2} - 42 x - 7 x^{2} - 6 \cdot (3 {(x - 2)}^{\frac{1 + 2}{3}}))}{3 {(x - 2)}^{\frac{1}{3}}} + \frac{8 x}{{(x - 2)}^{\frac{1}{3}}}}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.1.14.11.4

Add $1$ and $2$ .

$f'' (x) = \frac{\frac{2 (21 x^{2} - 42 x - 7 x^{2} - 6 \cdot (3 {(x - 2)}^{\frac{3}{3}}))}{3 {(x - 2)}^{\frac{1}{3}}} + \frac{8 x}{{(x - 2)}^{\frac{1}{3}}}}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.1.14.11.5

Divide $3$ by $3$ .

$f'' (x) = \frac{\frac{2 (21 x^{2} - 42 x - 7 x^{2} - 6 \cdot (3 (x - 2)))}{3 {(x - 2)}^{\frac{1}{3}}} + \frac{8 x}{{(x - 2)}^{\frac{1}{3}}}}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.1.14.12

Simplify $- 6 \cdot 3 {(x - 2)}^{1}$ .

$f'' (x) = \frac{\frac{2 (21 x^{2} - 42 x - 7 x^{2} - 6 \cdot (3 (x - 2)))}{3 {(x - 2)}^{\frac{1}{3}}} + \frac{8 x}{{(x - 2)}^{\frac{1}{3}}}}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.1.14.13

Multiply $- 6$ by $3$ .

$f'' (x) = \frac{\frac{2 (21 x^{2} - 42 x - 7 x^{2} - 18 (x - 2))}{3 {(x - 2)}^{\frac{1}{3}}} + \frac{8 x}{{(x - 2)}^{\frac{1}{3}}}}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.1.14.14

Apply the distributive property.

$f'' (x) = \frac{\frac{2 (21 x^{2} - 42 x - 7 x^{2} - 18 x - 18 \cdot - 2)}{3 {(x - 2)}^{\frac{1}{3}}} + \frac{8 x}{{(x - 2)}^{\frac{1}{3}}}}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.1.14.15

Multiply $- 18$ by $- 2$ .

$f'' (x) = \frac{\frac{2 (21 x^{2} - 42 x - 7 x^{2} - 18 x + 36)}{3 {(x - 2)}^{\frac{1}{3}}} + \frac{8 x}{{(x - 2)}^{\frac{1}{3}}}}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.1.14.16

Subtract $7 x^{2}$ from $21 x^{2}$ .

$f'' (x) = \frac{\frac{2 (14 x^{2} - 42 x - 18 x + 36)}{3 {(x - 2)}^{\frac{1}{3}}} + \frac{8 x}{{(x - 2)}^{\frac{1}{3}}}}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.1.14.17

Subtract $18 x$ from $- 42 x$ .

$f'' (x) = \frac{\frac{2 (14 x^{2} - 60 x + 36)}{3 {(x - 2)}^{\frac{1}{3}}} + \frac{8 x}{{(x - 2)}^{\frac{1}{3}}}}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.1.14.18

Factor $2$ out of $14 x^{2} - 60 x + 36$ .

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Step 2.16.1.14.18.1

Factor $2$ out of $14 x^{2}$ .

$f'' (x) = \frac{\frac{2 (2 (7 x^{2}) - 60 x + 36)}{3 {(x - 2)}^{\frac{1}{3}}} + \frac{8 x}{{(x - 2)}^{\frac{1}{3}}}}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.1.14.18.2

Factor $2$ out of $- 60 x$ .

$f'' (x) = \frac{\frac{2 (2 (7 x^{2}) + 2 (- 30 x) + 36)}{3 {(x - 2)}^{\frac{1}{3}}} + \frac{8 x}{{(x - 2)}^{\frac{1}{3}}}}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.1.14.18.3

Factor $2$ out of $36$ .

$f'' (x) = \frac{\frac{2 (2 (7 x^{2}) + 2 (- 30 x) + 2 (18))}{3 {(x - 2)}^{\frac{1}{3}}} + \frac{8 x}{{(x - 2)}^{\frac{1}{3}}}}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.1.14.18.4

Factor $2$ out of $2 (7 x^{2}) + 2 (- 30 x)$ .

$f'' (x) = \frac{\frac{2 (2 (7 x^{2} - 30 x) + 2 (18))}{3 {(x - 2)}^{\frac{1}{3}}} + \frac{8 x}{{(x - 2)}^{\frac{1}{3}}}}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.1.14.18.5

Factor $2$ out of $2 (7 x^{2} - 30 x) + 2 (18)$ .

$f'' (x) = \frac{\frac{2 (2 (7 x^{2} - 30 x + 18))}{3 {(x - 2)}^{\frac{1}{3}}} + \frac{8 x}{{(x - 2)}^{\frac{1}{3}}}}{3 {(x - 2)}^{\frac{4}{3}}}$

$f'' (x) = \frac{\frac{2 \cdot (2 (7 x^{2} - 30 x + 18))}{3 {(x - 2)}^{\frac{1}{3}}} + \frac{8 x}{{(x - 2)}^{\frac{1}{3}}}}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.1.14.19

Multiply $2$ by $2$ .

$f'' (x) = \frac{\frac{4 (7 x^{2} - 30 x + 18)}{3 {(x - 2)}^{\frac{1}{3}}} + \frac{8 x}{{(x - 2)}^{\frac{1}{3}}}}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.1.15

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

$f'' (x) = \frac{\frac{4 (7 x^{2} - 30 x + 18)}{3 {(x - 2)}^{\frac{1}{3}}} + \frac{8 x}{{(x - 2)}^{\frac{1}{3}}} \cdot \frac{3}{3}}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.1.16

Write each expression with a common denominator of $3 {(x - 2)}^{\frac{1}{3}}$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.16.1.16.1

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

$f'' (x) = \frac{\frac{4 (7 x^{2} - 30 x + 18)}{3 {(x - 2)}^{\frac{1}{3}}} + \frac{8 x \cdot 3}{{(x - 2)}^{\frac{1}{3}} \cdot 3}}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.1.16.2

Reorder the factors of ${(x - 2)}^{\frac{1}{3}} \cdot 3$ .

$f'' (x) = \frac{\frac{4 (7 x^{2} - 30 x + 18)}{3 {(x - 2)}^{\frac{1}{3}}} + \frac{8 x \cdot 3}{3 {(x - 2)}^{\frac{1}{3}}}}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.1.17

Combine the numerators over the common denominator.

$f'' (x) = \frac{\frac{4 (7 x^{2} - 30 x + 18) + 8 x \cdot 3}{3 {(x - 2)}^{\frac{1}{3}}}}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.1.18

Simplify the numerator.

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Step 2.16.1.18.1

Factor $4$ out of $4 (7 x^{2} - 30 x + 18) + 8 x \cdot 3$ .

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Step 2.16.1.18.1.1

Factor $4$ out of $8 x \cdot 3$ .

$f'' (x) = \frac{\frac{4 (7 x^{2} - 30 x + 18) + 4 (2 x \cdot 3)}{3 {(x - 2)}^{\frac{1}{3}}}}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.1.18.1.2

Factor $4$ out of $4 (7 x^{2} - 30 x + 18) + 4 (2 x \cdot 3)$ .

$f'' (x) = \frac{\frac{4 (7 x^{2} - 30 x + 18 + 2 x \cdot 3)}{3 {(x - 2)}^{\frac{1}{3}}}}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.1.18.2

Multiply $3$ by $2$ .

$f'' (x) = \frac{\frac{4 (7 x^{2} - 30 x + 18 + 6 x)}{3 {(x - 2)}^{\frac{1}{3}}}}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.1.18.3

Add $- 30 x$ and $6 x$ .

$f'' (x) = \frac{\frac{4 (7 x^{2} - 24 x + 18)}{3 {(x - 2)}^{\frac{1}{3}}}}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.2

Combine terms.

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Step 2.16.2.1

Rewrite $\frac{\frac{4 (7 x^{2} - 24 x + 18)}{3 {(x - 2)}^{\frac{1}{3}}}}{3 {(x - 2)}^{\frac{4}{3}}}$ as a product.

$f'' (x) = \frac{4 (7 x^{2} - 24 x + 18)}{3 {(x - 2)}^{\frac{1}{3}}} \cdot \frac{1}{3 {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.2.2

Multiply $\frac{4 (7 x^{2} - 24 x + 18)}{3 {(x - 2)}^{\frac{1}{3}}}$ by $\frac{1}{3 {(x - 2)}^{\frac{4}{3}}}$ .

$f'' (x) = \frac{4 (7 x^{2} - 24 x + 18)}{3 {(x - 2)}^{\frac{1}{3}} (3 {(x - 2)}^{\frac{4}{3}})}$

Step 2.16.2.3

Multiply $3$ by $3$ .

$f'' (x) = \frac{4 (7 x^{2} - 24 x + 18)}{9 {(x - 2)}^{\frac{1}{3}} {(x - 2)}^{\frac{4}{3}}}$

Step 2.16.2.4

Multiply ${(x - 2)}^{\frac{1}{3}}$ by ${(x - 2)}^{\frac{4}{3}}$ by adding the exponents.

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Step 2.16.2.4.1

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

$f'' (x) = \frac{4 (7 x^{2} - 24 x + 18)}{9 ({(x - 2)}^{\frac{4}{3}} {(x - 2)}^{\frac{1}{3}})}$

Step 2.16.2.4.2

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

$f'' (x) = \frac{4 (7 x^{2} - 24 x + 18)}{9 {(x - 2)}^{\frac{4}{3} + \frac{1}{3}}}$

Step 2.16.2.4.3

Combine the numerators over the common denominator.

$f'' (x) = \frac{4 (7 x^{2} - 24 x + 18)}{9 {(x - 2)}^{\frac{4 + 1}{3}}}$

Step 2.16.2.4.4

Add $4$ and $1$ .

$f'' (x) = \frac{4 (7 x^{2} - 24 x + 18)}{9 {(x - 2)}^{\frac{5}{3}}}$

Step 3

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

$\frac{x (7 x - 12)}{3 {(x - 2)}^{\frac{2}{3}}} = 0$

Step 4

Find the first derivative.

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Step 4.1

Find the first derivative.

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Step 4.1.1

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

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

Step 4.1.2

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

Step 4.1.3

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 $g (x) = x - 2$ .

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Step 4.1.3.1

To apply the Chain Rule, set $u$ as $x - 2$ .

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

Step 4.1.3.2

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

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

Step 4.1.3.3

Replace all occurrences of $u$ with $x - 2$ .

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

Step 4.1.4

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

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

Step 4.1.5

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

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

Step 4.1.6

Combine the numerators over the common denominator.

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

Step 4.1.7

Simplify the numerator.

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Step 4.1.7.1

Multiply $- 1$ by $3$ .

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

Step 4.1.7.2

Subtract $3$ from $1$ .

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

Step 4.1.8

Combine fractions.

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Step 4.1.8.1

Move the negative in front of the fraction.

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

Step 4.1.8.2

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

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

Step 4.1.8.3

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

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

Step 4.1.8.4

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

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

Step 4.1.9

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

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

Step 4.1.10

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

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

Step 4.1.11

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

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

Step 4.1.12

Simplify the expression.

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Step 4.1.12.1

Add $1$ and $0$ .

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

Step 4.1.12.2

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

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

Step 4.1.13

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

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

Step 4.1.14

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

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

Step 4.1.15

Combine $\frac{x^{2}}{3 {(x - 2)}^{\frac{2}{3}}}$ and $2 {(x - 2)}^{\frac{1}{3}} x$ using a common denominator.

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Step 4.1.15.1

Move ${(x - 2)}^{\frac{1}{3}}$ .

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

Step 4.1.15.2

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

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

Step 4.1.15.3

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

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

Step 4.1.15.4

Combine the numerators over the common denominator.

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

Step 4.1.16

Multiply $3$ by $2$ .

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

Step 4.1.17

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

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Step 4.1.17.1

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

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

Step 4.1.17.2

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

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

Step 4.1.17.3

Combine the numerators over the common denominator.

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

Step 4.1.17.4

Add $2$ and $1$ .

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

Step 4.1.17.5

Divide $3$ by $3$ .

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

Step 4.1.18

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

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

Step 4.1.19

Simplify.

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Step 4.1.19.1

Apply the distributive property.

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

Step 4.1.19.2

Simplify the numerator.

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Step 4.1.19.2.1

Simplify each term.

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Step 4.1.19.2.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 4.1.19.2.1.1.1

Move $x$ .

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

Step 4.1.19.2.1.1.2

Multiply $x$ by $x$ .

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

Step 4.1.19.2.1.2

Multiply $- 2$ by $6$ .

$\frac{x^{2} + 6 x^{2} - 12 x}{3 {(x - 2)}^{\frac{2}{3}}}$

Step 4.1.19.2.2

Add $x^{2}$ and $6 x^{2}$ .

$\frac{7 x^{2} - 12 x}{3 {(x - 2)}^{\frac{2}{3}}}$

Step 4.1.19.3

Factor $x$ out of $7 x^{2} - 12 x$ .

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Step 4.1.19.3.1

Factor $x$ out of $7 x^{2}$ .

$\frac{x (7 x) - 12 x}{3 {(x - 2)}^{\frac{2}{3}}}$

Step 4.1.19.3.2

Factor $x$ out of $- 12 x$ .

$\frac{x (7 x) + x \cdot - 12}{3 {(x - 2)}^{\frac{2}{3}}}$

Step 4.1.19.3.3

Factor $x$ out of $x (7 x) + x \cdot - 12$ .

$f' (x) = \frac{x (7 x - 12)}{3 {(x - 2)}^{\frac{2}{3}}}$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{x (7 x - 12)}{3 {(x - 2)}^{\frac{2}{3}}}$ .

$\frac{x (7 x - 12)}{3 {(x - 2)}^{\frac{2}{3}}}$

Step 5

Set the first derivative equal to $0$ then solve the equation $\frac{x (7 x - 12)}{3 {(x - 2)}^{\frac{2}{3}}} = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$\frac{x (7 x - 12)}{3 {(x - 2)}^{\frac{2}{3}}} = 0$

Step 5.2

Set the numerator equal to zero.

$x (7 x - 12) = 0$

Step 5.3

Solve the equation for $x$ .

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Step 5.3.1

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

$x = 0$

$7 x - 12 = 0$

Step 5.3.2

Set $x$ equal to $0$ .

$x = 0$

Step 5.3.3

Set $7 x - 12$ equal to $0$ and solve for $x$ .

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Step 5.3.3.1

Set $7 x - 12$ equal to $0$ .

$7 x - 12 = 0$

Step 5.3.3.2

Solve $7 x - 12 = 0$ for $x$ .

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Step 5.3.3.2.1

Add $12$ to both sides of the equation.

$7 x = 12$

Step 5.3.3.2.2

Divide each term in $7 x = 12$ by $7$ and simplify.

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Step 5.3.3.2.2.1

Divide each term in $7 x = 12$ by $7$ .

$\frac{7 x}{7} = \frac{12}{7}$

Step 5.3.3.2.2.2

Simplify the left side.

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Step 5.3.3.2.2.2.1

Cancel the common factor of $7$ .

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Step 5.3.3.2.2.2.1.1

Cancel the common factor.

$\frac{7 x}{7} = \frac{12}{7}$

Step 5.3.3.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{12}{7}$

Step 5.3.4

The final solution is all the values that make $x (7 x - 12) = 0$ true.

$x = 0, \frac{12}{7}$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

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

$\frac{x (7 x - 12)}{3 \sqrt[3]{{(x - 2)}^{2}}}$

Step 6.2

Set the denominator in $\frac{x (7 x - 12)}{3 \sqrt[3]{{(x - 2)}^{2}}}$ equal to $0$ to find where the expression is undefined.

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

Step 6.3

Solve for $x$ .

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Step 6.3.1

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

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

Step 6.3.2

Simplify each side of the equation.

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Step 6.3.2.1

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

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

Step 6.3.2.2

Simplify the left side.

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Step 6.3.2.2.1

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

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Step 6.3.2.2.1.1

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

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

Step 6.3.2.2.1.2

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

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

Step 6.3.2.2.1.3

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

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Step 6.3.2.2.1.3.1

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

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

Step 6.3.2.2.1.3.2

Cancel the common factor of $3$ .

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Step 6.3.2.2.1.3.2.1

Cancel the common factor.

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

Step 6.3.2.2.1.3.2.2

Rewrite the expression.

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

Step 6.3.2.3

Simplify the right side.

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Step 6.3.2.3.1

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

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

Step 6.3.3

Solve for $x$ .

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Step 6.3.3.1

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

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Step 6.3.3.1.1

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

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

Step 6.3.3.1.2

Simplify the left side.

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Step 6.3.3.1.2.1

Cancel the common factor of $27$ .

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Step 6.3.3.1.2.1.1

Cancel the common factor.

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

Step 6.3.3.1.2.1.2

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

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

Step 6.3.3.1.3

Simplify the right side.

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Step 6.3.3.1.3.1

Divide $0$ by $27$ .

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

Step 6.3.3.2

Set the $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 6.3.3.3

Add $2$ to both sides of the equation.

$x = 2$

Step 7

Critical points to evaluate.

$x = 0, \frac{12}{7}, 2$

Step 8

Evaluate the second derivative at $x = 0$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{4 (7 {(0)}^{2} - 24 \cdot 0 + 18)}{9 {((0) - 2)}^{\frac{5}{3}}}$

Step 9

Evaluate the second derivative.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

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

$\frac{4 (7 \cdot 0 - 24 \cdot 0 + 18)}{9 {(0 - 2)}^{\frac{5}{3}}}$

Step 9.1.2

Multiply $7$ by $0$ .

$\frac{4 (0 - 24 \cdot 0 + 18)}{9 {(0 - 2)}^{\frac{5}{3}}}$

Step 9.1.3

Multiply $- 24$ by $0$ .

$\frac{4 (0 + 0 + 18)}{9 {(0 - 2)}^{\frac{5}{3}}}$

Step 9.1.4

Add $0$ and $0$ .

$\frac{4 (0 + 18)}{9 {(0 - 2)}^{\frac{5}{3}}}$

Step 9.1.5

Add $0$ and $18$ .

$\frac{4 \cdot 18}{9 {(0 - 2)}^{\frac{5}{3}}}$

Step 9.2

Simplify with factoring out.

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Step 9.2.1

Subtract $2$ from $0$ .

$\frac{4 \cdot 18}{9 {(- 2)}^{\frac{5}{3}}}$

Step 9.2.2

Multiply $4$ by $18$ .

$\frac{72}{9 {(- 2)}^{\frac{5}{3}}}$

Step 9.2.3

Factor $9$ out of $72$ .

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

Step 9.3

Cancel the common factors.

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Step 9.3.1

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

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

Step 9.3.2

Cancel the common factor.

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

Step 9.3.3

Rewrite the expression.

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

Step 10

$x = 0$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = 0$ is a local maximum

Step 11

Find the y-value when $x = 0$ .

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Step 11.1

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

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

Step 11.2

Simplify the result.

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Step 11.2.1

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

$f (0) = 0 \sqrt[3]{(0) - 2}$

Step 11.2.2

Subtract $2$ from $0$ .

$f (0) = 0 \sqrt[3]{- 2}$

Step 11.2.3

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

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Step 11.2.3.1

Rewrite $- 2$ as $- 1 (2)$ .

$f (0) = 0 \sqrt[3]{- 1 \cdot 2}$

Step 11.2.3.2

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

$f (0) = 0 \sqrt[3]{{(- 1)}^{3} \cdot 2}$

Step 11.2.4

Pull terms out from under the radical.

$f (0) = 0 (- 1 \sqrt[3]{2})$

Step 11.2.5

Rewrite $- 1 \sqrt[3]{2}$ as $- \sqrt[3]{2}$ .

$f (0) = 0 (- \sqrt[3]{2})$

Step 11.2.6

Multiply $0 (- \sqrt[3]{2})$ .

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Step 11.2.6.1

Multiply $- 1$ by $0$ .

$f (0) = 0 \sqrt[3]{2}$

Step 11.2.6.2

Multiply $0$ by $\sqrt[3]{2}$ .

$f (0) = 0$

Step 11.2.7

The final answer is $0$ .

$y = 0$

Step 12

Evaluate the second derivative at $x = \frac{12}{7}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{4 (7 {(\frac{12}{7})}^{2} - 24 (\frac{12}{7}) + 18)}{9 {((\frac{12}{7}) - 2)}^{\frac{5}{3}}}$

Step 13

Evaluate the second derivative.

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Step 13.1

Simplify the numerator.

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Step 13.1.1

Apply the product rule to $\frac{12}{7}$ .

$\frac{4 (7 \frac{12^{2}}{7^{2}} - 24 (\frac{12}{7}) + 18)}{9 {(\frac{12}{7} - 2)}^{\frac{5}{3}}}$

Step 13.1.2

Raise $12$ to the power of $2$ .

$\frac{4 (7 \frac{144}{7^{2}} - 24 (\frac{12}{7}) + 18)}{9 {(\frac{12}{7} - 2)}^{\frac{5}{3}}}$

Step 13.1.3

Raise $7$ to the power of $2$ .

$\frac{4 (7 (\frac{144}{49}) - 24 (\frac{12}{7}) + 18)}{9 {(\frac{12}{7} - 2)}^{\frac{5}{3}}}$

Step 13.1.4

Cancel the common factor of $7$ .

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Step 13.1.4.1

Factor $7$ out of $49$ .

$\frac{4 (7 \frac{144}{7 (7)} - 24 (\frac{12}{7}) + 18)}{9 {(\frac{12}{7} - 2)}^{\frac{5}{3}}}$

Step 13.1.4.2

Cancel the common factor.

$\frac{4 (7 \frac{144}{7 \cdot 7} - 24 (\frac{12}{7}) + 18)}{9 {(\frac{12}{7} - 2)}^{\frac{5}{3}}}$

Step 13.1.4.3

Rewrite the expression.

$\frac{4 (\frac{144}{7} - 24 (\frac{12}{7}) + 18)}{9 {(\frac{12}{7} - 2)}^{\frac{5}{3}}}$

Step 13.1.5

Multiply $- 24 (\frac{12}{7})$ .

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Step 13.1.5.1

Combine $- 24$ and $\frac{12}{7}$ .

$\frac{4 (\frac{144}{7} + \frac{- 24 \cdot 12}{7} + 18)}{9 {(\frac{12}{7} - 2)}^{\frac{5}{3}}}$

Step 13.1.5.2

Multiply $- 24$ by $12$ .

$\frac{4 (\frac{144}{7} + \frac{- 288}{7} + 18)}{9 {(\frac{12}{7} - 2)}^{\frac{5}{3}}}$

Step 13.1.6

Move the negative in front of the fraction.

$\frac{4 (\frac{144}{7} - \frac{288}{7} + 18)}{9 {(\frac{12}{7} - 2)}^{\frac{5}{3}}}$

Step 13.1.7

Combine the numerators over the common denominator.

$\frac{4 (\frac{144 - 288}{7} + 18)}{9 {(\frac{12}{7} - 2)}^{\frac{5}{3}}}$

Step 13.1.8

Subtract $288$ from $144$ .

$\frac{4 (\frac{- 144}{7} + 18)}{9 {(\frac{12}{7} - 2)}^{\frac{5}{3}}}$

Step 13.1.9

To write $18$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$\frac{4 (\frac{- 144}{7} + 18 \cdot \frac{7}{7})}{9 {(\frac{12}{7} - 2)}^{\frac{5}{3}}}$

Step 13.1.10

Combine $18$ and $\frac{7}{7}$ .

$\frac{4 (\frac{- 144}{7} + \frac{18 \cdot 7}{7})}{9 {(\frac{12}{7} - 2)}^{\frac{5}{3}}}$

Step 13.1.11

Combine the numerators over the common denominator.

$\frac{4 \frac{- 144 + 18 \cdot 7}{7}}{9 {(\frac{12}{7} - 2)}^{\frac{5}{3}}}$

Step 13.1.12

Simplify the numerator.

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Step 13.1.12.1

Multiply $18$ by $7$ .

$\frac{4 \frac{- 144 + 126}{7}}{9 {(\frac{12}{7} - 2)}^{\frac{5}{3}}}$

Step 13.1.12.2

Add $- 144$ and $126$ .

$\frac{4 (\frac{- 18}{7})}{9 {(\frac{12}{7} - 2)}^{\frac{5}{3}}}$

Step 13.1.13

Move the negative in front of the fraction.

$\frac{4 \cdot - 1 \frac{18}{7}}{9 {(\frac{12}{7} - 2)}^{\frac{5}{3}}}$

Step 13.1.14

Combine exponents.

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Step 13.1.14.1

Factor out negative.

$\frac{- (4 (\frac{18}{7}))}{9 {(\frac{12}{7} - 2)}^{\frac{5}{3}}}$

Step 13.1.14.2

Combine $4$ and $\frac{18}{7}$ .

$\frac{- \frac{4 \cdot 18}{7}}{9 {(\frac{12}{7} - 2)}^{\frac{5}{3}}}$

Step 13.1.14.3

Multiply $4$ by $18$ .

$\frac{- \frac{72}{7}}{9 {(\frac{12}{7} - 2)}^{\frac{5}{3}}}$

Step 13.2

Simplify the denominator.

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Step 13.2.1

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$\frac{- \frac{72}{7}}{9 {(\frac{12}{7} - 2 \cdot \frac{7}{7})}^{\frac{5}{3}}}$

Step 13.2.2

Combine $- 2$ and $\frac{7}{7}$ .

$\frac{- \frac{72}{7}}{9 {(\frac{12}{7} + \frac{- 2 \cdot 7}{7})}^{\frac{5}{3}}}$

Step 13.2.3

Combine the numerators over the common denominator.

$\frac{- \frac{72}{7}}{9 {(\frac{12 - 2 \cdot 7}{7})}^{\frac{5}{3}}}$

Step 13.2.4

Simplify the numerator.

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Step 13.2.4.1

Multiply $- 2$ by $7$ .

$\frac{- \frac{72}{7}}{9 {(\frac{12 - 14}{7})}^{\frac{5}{3}}}$

Step 13.2.4.2

Subtract $14$ from $12$ .

$\frac{- \frac{72}{7}}{9 {(\frac{- 2}{7})}^{\frac{5}{3}}}$

Step 13.2.5

Move the negative in front of the fraction.

$\frac{- \frac{72}{7}}{9 {(- \frac{2}{7})}^{\frac{5}{3}}}$

Step 13.2.6

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

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Step 13.2.6.1

Apply the product rule to $- \frac{2}{7}$ .

$\frac{- \frac{72}{7}}{9 ({(- 1)}^{\frac{5}{3}} {(\frac{2}{7})}^{\frac{5}{3}})}$

Step 13.2.6.2

Apply the product rule to $\frac{2}{7}$ .

$\frac{- \frac{72}{7}}{9 ({(- 1)}^{\frac{5}{3}} \frac{2^{\frac{5}{3}}}{7^{\frac{5}{3}}})}$

Step 13.2.7

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

$\frac{- \frac{72}{7}}{9 ({({(- 1)}^{3})}^{\frac{5}{3}} \frac{2^{\frac{5}{3}}}{7^{\frac{5}{3}}})}$

Step 13.2.8

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

$\frac{- \frac{72}{7}}{9 ({(- 1)}^{3 (\frac{5}{3})} \frac{2^{\frac{5}{3}}}{7^{\frac{5}{3}}})}$

Step 13.2.9

Cancel the common factor of $3$ .

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Step 13.2.9.1

Cancel the common factor.

$\frac{- \frac{72}{7}}{9 ({(- 1)}^{3 (\frac{5}{3})} \frac{2^{\frac{5}{3}}}{7^{\frac{5}{3}}})}$

Step 13.2.9.2

Rewrite the expression.

$\frac{- \frac{72}{7}}{9 ({(- 1)}^{5} \frac{2^{\frac{5}{3}}}{7^{\frac{5}{3}}})}$

Step 13.2.10

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

$\frac{- \frac{72}{7}}{9 (- \frac{2^{\frac{5}{3}}}{7^{\frac{5}{3}}})}$

Step 13.3

Simplify the denominator.

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Step 13.3.1

Multiply $- 1$ by $9$ .

$\frac{- \frac{72}{7}}{- 9 \frac{2^{\frac{5}{3}}}{7^{\frac{5}{3}}}}$

Step 13.3.2

Combine $- 9$ and $\frac{2^{\frac{5}{3}}}{7^{\frac{5}{3}}}$ .

$\frac{- \frac{72}{7}}{\frac{- 9 \cdot 2^{\frac{5}{3}}}{7^{\frac{5}{3}}}}$

Step 13.4

Reduce the expression by cancelling the common factors.

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Step 13.4.1

Move the negative in front of the fraction.

$\frac{- \frac{72}{7}}{- \frac{9 \cdot 2^{\frac{5}{3}}}{7^{\frac{5}{3}}}}$

Step 13.4.2

Dividing two negative values results in a positive value.

$\frac{\frac{72}{7}}{\frac{9 \cdot 2^{\frac{5}{3}}}{7^{\frac{5}{3}}}}$

Step 13.5

Multiply the numerator by the reciprocal of the denominator.

$\frac{72}{7} \cdot \frac{7^{\frac{5}{3}}}{9 \cdot 2^{\frac{5}{3}}}$

Step 13.6

Combine.

$\frac{72 \cdot 7^{\frac{5}{3}}}{7 (9 \cdot 2^{\frac{5}{3}})}$

Step 13.7

Move $7^{1}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$\frac{72 \cdot 7^{\frac{5}{3}} \cdot 7^{- 1}}{9 \cdot 2^{\frac{5}{3}}}$

Step 13.8

Multiply $7^{\frac{5}{3}}$ by $7^{- 1}$ by adding the exponents.

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Step 13.8.1

Move $7^{- 1}$ .

$\frac{72 \cdot (7^{- 1} \cdot 7^{\frac{5}{3}})}{9 \cdot 2^{\frac{5}{3}}}$

Step 13.8.2

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

$\frac{72 \cdot 7^{- 1 + \frac{5}{3}}}{9 \cdot 2^{\frac{5}{3}}}$

Step 13.8.3

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

$\frac{72 \cdot 7^{- 1 \cdot \frac{3}{3} + \frac{5}{3}}}{9 \cdot 2^{\frac{5}{3}}}$

Step 13.8.4

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

$\frac{72 \cdot 7^{\frac{- 1 \cdot 3}{3} + \frac{5}{3}}}{9 \cdot 2^{\frac{5}{3}}}$

Step 13.8.5

Combine the numerators over the common denominator.

$\frac{72 \cdot 7^{\frac{- 1 \cdot 3 + 5}{3}}}{9 \cdot 2^{\frac{5}{3}}}$

Step 13.8.6

Simplify the numerator.

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Step 13.8.6.1

Multiply $- 1$ by $3$ .

$\frac{72 \cdot 7^{\frac{- 3 + 5}{3}}}{9 \cdot 2^{\frac{5}{3}}}$

Step 13.8.6.2

Add $- 3$ and $5$ .

$\frac{72 \cdot 7^{\frac{2}{3}}}{9 \cdot 2^{\frac{5}{3}}}$

Step 13.9

Factor $9$ out of $72 \cdot 7^{\frac{2}{3}}$ .

$\frac{9 (8 \cdot 7^{\frac{2}{3}})}{9 \cdot 2^{\frac{5}{3}}}$

Step 13.10

Cancel the common factors.

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Step 13.10.1

Factor $9$ out of $9 \cdot 2^{\frac{5}{3}}$ .

$\frac{9 (8 \cdot 7^{\frac{2}{3}})}{9 (2^{\frac{5}{3}})}$

Step 13.10.2

Cancel the common factor.

$\frac{9 (8 \cdot 7^{\frac{2}{3}})}{9 \cdot 2^{\frac{5}{3}}}$

Step 13.10.3

Rewrite the expression.

$\frac{8 \cdot 7^{\frac{2}{3}}}{2^{\frac{5}{3}}}$

Step 14

$x = \frac{12}{7}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = \frac{12}{7}$ is a local minimum

Step 15

Find the y-value when $x = \frac{12}{7}$ .

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Step 15.1

Replace the variable $x$ with $\frac{12}{7}$ in the expression.

$f (\frac{12}{7}) = {(\frac{12}{7})}^{2} \sqrt[3]{(\frac{12}{7}) - 2}$

Step 15.2

Simplify the result.

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Step 15.2.1

Apply the product rule to $\frac{12}{7}$ .

$f (\frac{12}{7}) = \frac{12^{2}}{7^{2}} \cdot \sqrt[3]{(\frac{12}{7}) - 2}$

Step 15.2.2

Raise $12$ to the power of $2$ .

$f (\frac{12}{7}) = \frac{144}{7^{2}} \cdot \sqrt[3]{(\frac{12}{7}) - 2}$

Step 15.2.3

Raise $7$ to the power of $2$ .

$f (\frac{12}{7}) = \frac{144}{49} \cdot \sqrt[3]{(\frac{12}{7}) - 2}$

Step 15.2.4

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$f (\frac{12}{7}) = \frac{144}{49} \cdot \sqrt[3]{\frac{12}{7} - 2 \cdot \frac{7}{7}}$

Step 15.2.5

Combine $- 2$ and $\frac{7}{7}$ .

$f (\frac{12}{7}) = \frac{144}{49} \cdot \sqrt[3]{\frac{12}{7} + \frac{- 2 \cdot 7}{7}}$

Step 15.2.6

Combine the numerators over the common denominator.

$f (\frac{12}{7}) = \frac{144}{49} \cdot \sqrt[3]{\frac{12 - 2 \cdot 7}{7}}$

Step 15.2.7

Simplify the numerator.

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Step 15.2.7.1

Multiply $- 2$ by $7$ .

$f (\frac{12}{7}) = \frac{144}{49} \cdot \sqrt[3]{\frac{12 - 14}{7}}$

Step 15.2.7.2

Subtract $14$ from $12$ .

$f (\frac{12}{7}) = \frac{144}{49} \cdot \sqrt[3]{\frac{- 2}{7}}$

Step 15.2.8

Move the negative in front of the fraction.

$f (\frac{12}{7}) = \frac{144}{49} \cdot \sqrt[3]{- \frac{2}{7}}$

Step 15.2.9

Rewrite $- \frac{2}{7}$ as ${({(- 1)}^{3})}^{3} \frac{2}{7}$ .

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Step 15.2.9.1

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

$f (\frac{12}{7}) = \frac{144}{49} \cdot \sqrt[3]{{(- 1)}^{3} (\frac{2}{7})}$

Step 15.2.9.2

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

$f (\frac{12}{7}) = \frac{144}{49} \cdot \sqrt[3]{{({(- 1)}^{3})}^{3} (\frac{2}{7})}$

Step 15.2.10

Pull terms out from under the radical.

$f (\frac{12}{7}) = \frac{144}{49} \cdot ({(- 1)}^{3} \sqrt[3]{\frac{2}{7}})$

Step 15.2.11

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

$f (\frac{12}{7}) = \frac{144}{49} \cdot (- \sqrt[3]{\frac{2}{7}})$

Step 15.2.12

Rewrite $\sqrt[3]{\frac{2}{7}}$ as $\frac{\sqrt[3]{2}}{\sqrt[3]{7}}$ .

$f (\frac{12}{7}) = \frac{144}{49} \cdot (- \frac{\sqrt[3]{2}}{\sqrt[3]{7}})$

Step 15.2.13

Multiply $\frac{\sqrt[3]{2}}{\sqrt[3]{7}}$ by $\frac{{\sqrt[3]{7}}^{2}}{{\sqrt[3]{7}}^{2}}$ .

$f (\frac{12}{7}) = \frac{144}{49} \cdot (- (\frac{\sqrt[3]{2}}{\sqrt[3]{7}} \cdot \frac{{\sqrt[3]{7}}^{2}}{{\sqrt[3]{7}}^{2}}))$

Step 15.2.14

Combine and simplify the denominator.

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Step 15.2.14.1

Multiply $\frac{\sqrt[3]{2}}{\sqrt[3]{7}}$ by $\frac{{\sqrt[3]{7}}^{2}}{{\sqrt[3]{7}}^{2}}$ .

$f (\frac{12}{7}) = \frac{144}{49} \cdot (- \frac{\sqrt[3]{2} {\sqrt[3]{7}}^{2}}{\sqrt[3]{7} {\sqrt[3]{7}}^{2}})$

Step 15.2.14.2

Raise $\sqrt[3]{7}$ to the power of $1$ .

$f (\frac{12}{7}) = \frac{144}{49} \cdot (- \frac{\sqrt[3]{2} {\sqrt[3]{7}}^{2}}{\sqrt[3]{7} {\sqrt[3]{7}}^{2}})$

Step 15.2.14.3

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

$f (\frac{12}{7}) = \frac{144}{49} \cdot (- \frac{\sqrt[3]{2} {\sqrt[3]{7}}^{2}}{{\sqrt[3]{7}}^{1 + 2}})$

Step 15.2.14.4

Add $1$ and $2$ .

$f (\frac{12}{7}) = \frac{144}{49} \cdot (- \frac{\sqrt[3]{2} {\sqrt[3]{7}}^{2}}{{\sqrt[3]{7}}^{3}})$

Step 15.2.14.5

Rewrite ${\sqrt[3]{7}}^{3}$ as $7$ .

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Step 15.2.14.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{7}$ as $7^{\frac{1}{3}}$ .

$f (\frac{12}{7}) = \frac{144}{49} \cdot (- \frac{\sqrt[3]{2} {\sqrt[3]{7}}^{2}}{{(7^{\frac{1}{3}})}^{3}})$

Step 15.2.14.5.2

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

$f (\frac{12}{7}) = \frac{144}{49} \cdot (- \frac{\sqrt[3]{2} {\sqrt[3]{7}}^{2}}{7^{\frac{1}{3} \cdot 3}})$

Step 15.2.14.5.3

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

$f (\frac{12}{7}) = \frac{144}{49} \cdot (- \frac{\sqrt[3]{2} {\sqrt[3]{7}}^{2}}{7^{\frac{3}{3}}})$

Step 15.2.14.5.4

Cancel the common factor of $3$ .

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Step 15.2.14.5.4.1

Cancel the common factor.

$f (\frac{12}{7}) = \frac{144}{49} \cdot (- \frac{\sqrt[3]{2} {\sqrt[3]{7}}^{2}}{7^{\frac{3}{3}}})$

Step 15.2.14.5.4.2

Rewrite the expression.

$f (\frac{12}{7}) = \frac{144}{49} \cdot (- \frac{\sqrt[3]{2} {\sqrt[3]{7}}^{2}}{7})$

Step 15.2.14.5.5

Evaluate the exponent.

$f (\frac{12}{7}) = \frac{144}{49} \cdot (- \frac{\sqrt[3]{2} {\sqrt[3]{7}}^{2}}{7})$

Step 15.2.15

Simplify the numerator.

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Step 15.2.15.1

Rewrite ${\sqrt[3]{7}}^{2}$ as $\sqrt[3]{7^{2}}$ .

$f (\frac{12}{7}) = \frac{144}{49} \cdot (- \frac{\sqrt[3]{2} \sqrt[3]{7^{2}}}{7})$

Step 15.2.15.2

Raise $7$ to the power of $2$ .

$f (\frac{12}{7}) = \frac{144}{49} \cdot (- \frac{\sqrt[3]{2} \sqrt[3]{49}}{7})$

Step 15.2.16

Simplify the numerator.

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Step 15.2.16.1

Combine using the product rule for radicals.

$f (\frac{12}{7}) = \frac{144}{49} \cdot (- \frac{\sqrt[3]{2 \cdot 49}}{7})$

Step 15.2.16.2

Multiply $2$ by $49$ .

$f (\frac{12}{7}) = \frac{144}{49} \cdot (- \frac{\sqrt[3]{98}}{7})$

Step 15.2.17

Multiply $\frac{144}{49} (- \frac{\sqrt[3]{98}}{7})$ .

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Step 15.2.17.1

Multiply $\frac{144}{49}$ by $\frac{\sqrt[3]{98}}{7}$ .

$f (\frac{12}{7}) = - \frac{144 \sqrt[3]{98}}{49 \cdot 7}$

Step 15.2.17.2

Multiply $49$ by $7$ .

$f (\frac{12}{7}) = - \frac{144 \sqrt[3]{98}}{343}$

Step 15.2.18

The final answer is $- \frac{144 \sqrt[3]{98}}{343}$ .

$y = - \frac{144 \sqrt[3]{98}}{343}$

Step 16

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

$\frac{4 (7 {(2)}^{2} - 24 \cdot 2 + 18)}{9 {((2) - 2)}^{\frac{5}{3}}}$

Step 17

Evaluate the second derivative.

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Step 17.1

Simplify the expression.

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Step 17.1.1

Subtract $2$ from $2$ .

$\frac{4 (7 {(2)}^{2} - 24 \cdot 2 + 18)}{9 \cdot 0^{\frac{5}{3}}}$

Step 17.1.2

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

$\frac{4 (7 {(2)}^{2} - 24 \cdot 2 + 18)}{9 \cdot {(0^{3})}^{\frac{5}{3}}}$

Step 17.1.3

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

$\frac{4 (7 {(2)}^{2} - 24 \cdot 2 + 18)}{9 \cdot 0^{3 (\frac{5}{3})}}$

Step 17.2

Cancel the common factor of $3$ .

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Step 17.2.1

Cancel the common factor.

$\frac{4 (7 {(2)}^{2} - 24 \cdot 2 + 18)}{9 \cdot 0^{3 (\frac{5}{3})}}$

Step 17.2.2

Rewrite the expression.

$\frac{4 (7 {(2)}^{2} - 24 \cdot 2 + 18)}{9 \cdot 0^{5}}$

Step 17.3

Simplify the expression.

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Step 17.3.1

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

$\frac{4 (7 {(2)}^{2} - 24 \cdot 2 + 18)}{9 \cdot 0}$

Step 17.3.2

Multiply $9$ by $0$ .

$\frac{4 (7 {(2)}^{2} - 24 \cdot 2 + 18)}{0}$

Step 17.3.3

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

Undefined

$\frac{4 (7 {(2)}^{2} - 24 \cdot 2 + 18)}{0}$

Step 17.4

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

Undefined

Step 18

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

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Step 18.1

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

$(- \infty, 0) \cup (0, \frac{12}{7}) \cup (\frac{12}{7}, 2) \cup (2, \infty)$

Step 18.2

Substitute any number, such as $- 2$ , from the interval $(- \infty, 0)$ in the first derivative $\frac{x (7 x - 12)}{3 {(x - 2)}^{\frac{2}{3}}}$ to check if the result is negative or positive.

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Step 18.2.1

Replace the variable $x$ with $- 2$ in the expression.

$f' (- 2) = \frac{(- 2) (7 (- 2) - 12)}{3 {((- 2) - 2)}^{\frac{2}{3}}}$

Step 18.2.2

Simplify the result.

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Step 18.2.2.1

Simplify the numerator.

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Step 18.2.2.1.1

Multiply $7$ by $- 2$ .

$f' (- 2) = \frac{- 2 (- 14 - 12)}{3 {(- 2 - 2)}^{\frac{2}{3}}}$

Step 18.2.2.1.2

Subtract $12$ from $- 14$ .

$f' (- 2) = \frac{- 2 \cdot - 26}{3 {(- 2 - 2)}^{\frac{2}{3}}}$

Step 18.2.2.2

Simplify the expression.

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Step 18.2.2.2.1

Subtract $2$ from $- 2$ .

$f' (- 2) = \frac{- 2 \cdot - 26}{3 {(- 4)}^{\frac{2}{3}}}$

Step 18.2.2.2.2

Multiply $- 2$ by $- 26$ .

$f' (- 2) = \frac{52}{3 {(- 4)}^{\frac{2}{3}}}$

Step 18.2.2.3

The final answer is $\frac{52}{3 {(- 4)}^{\frac{2}{3}}}$ .

$\frac{52}{3 {(- 4)}^{\frac{2}{3}}}$

Step 18.3

Substitute any number, such as $1$ , from the interval $(0, \frac{12}{7})$ in the first derivative $\frac{x (7 x - 12)}{3 {(x - 2)}^{\frac{2}{3}}}$ to check if the result is negative or positive.

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Step 18.3.1

Replace the variable $x$ with $1$ in the expression.

$f' (1) = \frac{(1) (7 (1) - 12)}{3 {((1) - 2)}^{\frac{2}{3}}}$

Step 18.3.2

Simplify the result.

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Step 18.3.2.1

Multiply $7 (1) - 12$ by $1$ .

$f' (1) = \frac{7 (1) - 12}{3 {(1 - 2)}^{\frac{2}{3}}}$

Step 18.3.2.2

Simplify the denominator.

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Step 18.3.2.2.1

Subtract $2$ from $1$ .

$f' (1) = \frac{7 (1) - 12}{3 {(- 1)}^{\frac{2}{3}}}$

Step 18.3.2.2.2

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

$f' (1) = \frac{7 (1) - 12}{3 {({(- 1)}^{3})}^{\frac{2}{3}}}$

Step 18.3.2.2.3

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

$f' (1) = \frac{7 (1) - 12}{3 {(- 1)}^{3 (\frac{2}{3})}}$

Step 18.3.2.2.4

Cancel the common factor of $3$ .

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Step 18.3.2.2.4.1

Cancel the common factor.

$f' (1) = \frac{7 (1) - 12}{3 {(- 1)}^{3 (\frac{2}{3})}}$

Step 18.3.2.2.4.2

Rewrite the expression.

$f' (1) = \frac{7 (1) - 12}{3 {(- 1)}^{2}}$

Step 18.3.2.2.5

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

$f' (1) = \frac{7 (1) - 12}{3 \cdot 1}$

Step 18.3.2.3

Simplify the numerator.

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Step 18.3.2.3.1

Multiply $7$ by $1$ .

$f' (1) = \frac{7 - 12}{3 \cdot 1}$

Step 18.3.2.3.2

Subtract $12$ from $7$ .

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

Step 18.3.2.4

Simplify the expression.

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Step 18.3.2.4.1

Multiply $3$ by $1$ .

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

Step 18.3.2.4.2

Move the negative in front of the fraction.

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

Step 18.3.2.5

The final answer is $- \frac{5}{3}$ .

$- \frac{5}{3}$

Step 18.4

Substitute any number, such as $1.86$ , from the interval $(\frac{12}{7}, 2)$ in the first derivative $\frac{x (7 x - 12)}{3 {(x - 2)}^{\frac{2}{3}}}$ to check if the result is negative or positive.

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Step 18.4.1

Replace the variable $x$ with $1.86$ in the expression.

$f' (1.86) = \frac{(1.86) (7 (1.86) - 12)}{3 {((1.86) - 2)}^{\frac{2}{3}}}$

Step 18.4.2

Simplify the result.

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Step 18.4.2.1

Multiply $1.86$ by $7 (1.86) - 12$ .

$f' (1.86) = \frac{1.8972}{3 {(1.86 - 2)}^{\frac{2}{3}}}$

Step 18.4.2.2

Subtract $2$ from $1.86$ .

$f' (1.86) = \frac{1.8972}{3 {(- 0.14)}^{\frac{2}{3}}}$

Step 18.4.2.3

The final answer is $\frac{1.8972}{3 {(- 0.14)}^{\frac{2}{3}}}$ .

$\frac{1.8972}{3 {(- 0.14)}^{\frac{2}{3}}}$

Step 18.5

Substitute any number, such as $4$ , from the interval $(2, \infty)$ in the first derivative $\frac{x (7 x - 12)}{3 {(x - 2)}^{\frac{2}{3}}}$ to check if the result is negative or positive.

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Step 18.5.1

Replace the variable $x$ with $4$ in the expression.

$f' (4) = \frac{(4) (7 (4) - 12)}{3 {((4) - 2)}^{\frac{2}{3}}}$

Step 18.5.2

Simplify the result.

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Step 18.5.2.1

Simplify the numerator.

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Step 18.5.2.1.1

Multiply $7$ by $4$ .

$f' (4) = \frac{4 (28 - 12)}{3 {(4 - 2)}^{\frac{2}{3}}}$

Step 18.5.2.1.2

Subtract $12$ from $28$ .

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

Step 18.5.2.2

Simplify the expression.

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Step 18.5.2.2.1

Subtract $2$ from $4$ .

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

Step 18.5.2.2.2

Multiply $4$ by $16$ .

$f' (4) = \frac{64}{3 \cdot 2^{\frac{2}{3}}}$

Step 18.5.2.3

The final answer is $\frac{64}{3 \cdot 2^{\frac{2}{3}}}$ .

$\frac{64}{3 \cdot 2^{\frac{2}{3}}}$

Step 18.6

Since the first derivative changed signs from positive to negative around $x = 0$ , then $x = 0$ is a local maximum.

$x = 0$ is a local maximum

Step 18.7

Since the first derivative changed signs from negative to positive around $x = \frac{12}{7}$ , then $x = \frac{12}{7}$ is a local minimum.

$x = \frac{12}{7}$ is a local minimum

Step 18.8

Since the first derivative did not change signs around $x = 2$ , this is not a local maximum or minimum.

Not a local maximum or minimum

Step 18.9

These are the local extrema for $f (x) = x^{2} \sqrt[3]{x - 2}$ .

$x = 0$ is a local maximum

$x = \frac{12}{7}$ is a local minimum

$x = 0$ is a local maximum

$x = \frac{12}{7}$ is a local minimum

Step 19