Calculus Examples

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

Step 1

Find the first derivative of the function.

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Step 1.1

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 $g (x) = x^{2} - 4$ .

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Step 1.1.1

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

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

Step 1.1.2

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

$\frac{2}{3} u^{\frac{2}{3} - 1} \frac{d}{d x} [x^{2} - 4]$

Step 1.1.3

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

Combine the numerators over the common denominator.

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

Step 1.5

Simplify the numerator.

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Step 1.5.1

Multiply $- 1$ by $3$ .

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

Step 1.5.2

Subtract $3$ from $2$ .

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

Step 1.6

Combine fractions.

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Step 1.6.1

Move the negative in front of the fraction.

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

Step 1.6.2

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

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

Step 1.6.3

Move ${(x^{2} - 4)}^{- \frac{1}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 1.7

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

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

Step 1.8

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

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

Step 1.9

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

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

Step 1.10

Combine fractions.

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Step 1.10.1

Add $2 x$ and $0$ .

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

Step 1.10.2

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

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

Step 1.10.3

Multiply $2$ by $2$ .

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

Step 1.10.4

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

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

Step 2

Find the second derivative of the function.

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Step 2.1

Since $\frac{4}{3}$ is constant with respect to $x$ , the derivative of $\frac{4 x}{3 {(x^{2} - 4)}^{\frac{1}{3}}}$ with respect to $x$ is $\frac{4}{3} \frac{d}{d x} [\frac{x}{{(x^{2} - 4)}^{\frac{1}{3}}}]$ .

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

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

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

Step 2.3

Differentiate using the Power Rule.

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Step 2.3.1

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

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Step 2.3.1.1

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

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

Step 2.3.1.2

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

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{3}}$ and $g (x) = x^{2} - 4$ .

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Step 2.4.1

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

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

Step 2.4.2

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

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

Step 2.4.3

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

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

Combine the numerators over the common denominator.

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

Step 2.8

Simplify the numerator.

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Step 2.8.1

Multiply $- 1$ by $3$ .

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

Step 2.8.2

Subtract $3$ from $1$ .

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

Step 2.9

Combine fractions.

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Step 2.9.1

Move the negative in front of the fraction.

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

Step 2.9.2

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

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

Step 2.9.3

Move ${(x^{2} - 4)}^{- \frac{2}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2.9.4

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

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

Step 2.10

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

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

Step 2.11

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

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

Step 2.12

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

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

Step 2.13

Combine fractions.

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Step 2.13.1

Add $2 x$ and $0$ .

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

Step 2.13.2

Multiply $2$ by $- 1$ .

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

Step 2.13.3

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

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

Step 2.13.4

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

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

Step 2.14

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

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

Step 2.15

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

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

Step 2.16

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

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

Step 2.17

Add $1$ and $1$ .

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

Step 2.18

Move the negative in front of the fraction.

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

Step 2.19

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

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

Step 2.20

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

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

Step 2.21

Combine the numerators over the common denominator.

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

Step 2.22

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

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Step 2.22.1

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

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

Step 2.22.2

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

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

Step 2.22.3

Combine the numerators over the common denominator.

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

Step 2.22.4

Add $2$ and $1$ .

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

Step 2.22.5

Divide $3$ by $3$ .

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

Step 2.23

Simplify ${(x^{2} - 4)}^{1} \cdot 3$ .

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

Step 2.24

Move $3$ to the left of $x^{2} - 4$ .

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

Step 2.25

Rewrite $\frac{\frac{3 (x^{2} - 4) - 2 x^{2}}{3 {(x^{2} - 4)}^{\frac{2}{3}}}}{{(x^{2} - 4)}^{\frac{2}{3}}}$ as a product.

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

Step 2.26

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

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

Step 2.27

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

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Step 2.27.1

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

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

Step 2.27.2

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

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

Step 2.27.3

Combine the numerators over the common denominator.

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

Step 2.27.4

Add $2$ and $2$ .

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

Step 2.28

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

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

Step 2.29

Multiply $3$ by $3$ .

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

Step 2.30

Simplify.

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Step 2.30.1

Apply the distributive property.

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

Step 2.30.2

Apply the distributive property.

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

Step 2.30.3

Simplify the numerator.

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Step 2.30.3.1

Simplify each term.

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Step 2.30.3.1.1

Multiply $3$ by $4$ .

$f'' (x) = \frac{12 x^{2} + 4 (3 \cdot - 4) + 4 (- 2 x^{2})}{9 {(x^{2} - 4)}^{\frac{4}{3}}}$

Step 2.30.3.1.2

Multiply $4 (3 \cdot - 4)$ .

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Step 2.30.3.1.2.1

Multiply $3$ by $- 4$ .

$f'' (x) = \frac{12 x^{2} + 4 \cdot - 12 + 4 (- 2 x^{2})}{9 {(x^{2} - 4)}^{\frac{4}{3}}}$

Step 2.30.3.1.2.2

Multiply $4$ by $- 12$ .

$f'' (x) = \frac{12 x^{2} - 48 + 4 (- 2 x^{2})}{9 {(x^{2} - 4)}^{\frac{4}{3}}}$

Step 2.30.3.1.3

Multiply $- 2$ by $4$ .

$f'' (x) = \frac{12 x^{2} - 48 - 8 x^{2}}{9 {(x^{2} - 4)}^{\frac{4}{3}}}$

Step 2.30.3.2

Subtract $8 x^{2}$ from $12 x^{2}$ .

$f'' (x) = \frac{4 x^{2} - 48}{9 {(x^{2} - 4)}^{\frac{4}{3}}}$

Step 2.30.4

Factor $4$ out of $4 x^{2} - 48$ .

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Step 2.30.4.1

Factor $4$ out of $4 x^{2}$ .

$f'' (x) = \frac{4 (x^{2}) - 48}{9 {(x^{2} - 4)}^{\frac{4}{3}}}$

Step 2.30.4.2

Factor $4$ out of $- 48$ .

$f'' (x) = \frac{4 x^{2} + 4 \cdot - 12}{9 {(x^{2} - 4)}^{\frac{4}{3}}}$

Step 2.30.4.3

Factor $4$ out of $4 x^{2} + 4 \cdot - 12$ .

$f'' (x) = \frac{4 (x^{2} - 12)}{9 {(x^{2} - 4)}^{\frac{4}{3}}}$

Step 3

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

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

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

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 $g (x) = x^{2} - 4$ .

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Step 4.1.1.1

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

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

Step 4.1.1.2

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

$\frac{2}{3} u^{\frac{2}{3} - 1} \frac{d}{d x} [x^{2} - 4]$

Step 4.1.1.3

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

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

Step 4.1.2

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

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

Step 4.1.3

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

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

Step 4.1.4

Combine the numerators over the common denominator.

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

Step 4.1.5

Simplify the numerator.

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Step 4.1.5.1

Multiply $- 1$ by $3$ .

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

Step 4.1.5.2

Subtract $3$ from $2$ .

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

Step 4.1.6

Combine fractions.

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Step 4.1.6.1

Move the negative in front of the fraction.

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

Step 4.1.6.2

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

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

Step 4.1.6.3

Move ${(x^{2} - 4)}^{- \frac{1}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 4.1.7

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

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

Step 4.1.8

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

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

Step 4.1.9

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

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

Step 4.1.10

Combine fractions.

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Step 4.1.10.1

Add $2 x$ and $0$ .

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

Step 4.1.10.2

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

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

Step 4.1.10.3

Multiply $2$ by $2$ .

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

Step 4.1.10.4

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

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

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{4 x}{3 {(x^{2} - 4)}^{\frac{1}{3}}}$ .

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

Step 5

Set the first derivative equal to $0$ then solve the equation $\frac{4 x}{3 {(x^{2} - 4)}^{\frac{1}{3}}} = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

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

Step 5.2

Set the numerator equal to zero.

$4 x = 0$

Step 5.3

Divide each term in $4 x = 0$ by $4$ and simplify.

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Step 5.3.1

Divide each term in $4 x = 0$ by $4$ .

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

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $4$ .

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Step 5.3.2.1.1

Cancel the common factor.

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

Step 5.3.2.1.2

Divide $x$ by $1$ .

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

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Divide $0$ by $4$ .

$x = 0$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

Convert expressions with fractional exponents to radicals.

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Step 6.1.1

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

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

Step 6.1.2

Anything raised to $1$ is the base itself.

$\frac{4 x}{3 \sqrt[3]{x^{2} - 4}}$

Step 6.2

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

$3 \sqrt[3]{x^{2} - 4} = 0$

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.

${(3 \sqrt[3]{x^{2} - 4})}^{3} = 0^{3}$

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 $\sqrt[3]{x^{2} - 4}$ as ${(x^{2} - 4)}^{\frac{1}{3}}$ .

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

Step 6.3.2.2

Simplify the left side.

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Step 6.3.2.2.1

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

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Step 6.3.2.2.1.1

Apply the product rule to $3 {(x^{2} - 4)}^{\frac{1}{3}}$ .

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

Step 6.3.2.2.1.2

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

$27 {({(x^{2} - 4)}^{\frac{1}{3}})}^{3} = 0^{3}$

Step 6.3.2.2.1.3

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

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Step 6.3.2.2.1.3.1

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

$27 {(x^{2} - 4)}^{\frac{1}{3} \cdot 3} = 0^{3}$

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.

$27 {(x^{2} - 4)}^{\frac{1}{3} \cdot 3} = 0^{3}$

Step 6.3.2.2.1.3.2.2

Rewrite the expression.

$27 {(x^{2} - 4)}^{1} = 0^{3}$

Step 6.3.2.2.1.4

Simplify.

$27 (x^{2} - 4) = 0^{3}$

Step 6.3.2.2.1.5

Apply the distributive property.

$27 x^{2} + 27 \cdot - 4 = 0^{3}$

Step 6.3.2.2.1.6

Multiply $27$ by $- 4$ .

$27 x^{2} - 108 = 0^{3}$

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 $0$ .

$27 x^{2} - 108 = 0$

Step 6.3.3

Solve for $x$ .

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Step 6.3.3.1

Add $108$ to both sides of the equation.

$27 x^{2} = 108$

Step 6.3.3.2

Divide each term in $27 x^{2} = 108$ by $27$ and simplify.

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Step 6.3.3.2.1

Divide each term in $27 x^{2} = 108$ by $27$ .

$\frac{27 x^{2}}{27} = \frac{108}{27}$

Step 6.3.3.2.2

Simplify the left side.

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Step 6.3.3.2.2.1

Cancel the common factor of $27$ .

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Step 6.3.3.2.2.1.1

Cancel the common factor.

$\frac{27 x^{2}}{27} = \frac{108}{27}$

Step 6.3.3.2.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{108}{27}$

Step 6.3.3.2.3

Simplify the right side.

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Step 6.3.3.2.3.1

Divide $108$ by $27$ .

$x^{2} = 4$

Step 6.3.3.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{4}$

Step 6.3.3.4

Simplify $\pm \sqrt{4}$ .

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Step 6.3.3.4.1

Rewrite $4$ as $2^{2}$ .

$x = \pm \sqrt{2^{2}}$

Step 6.3.3.4.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 2$

Step 6.3.3.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 6.3.3.5.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 2$

Step 6.3.3.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 2$

Step 6.3.3.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 2, - 2$

Step 6.4

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x = - 2, x = 2$

Step 7

Critical points to evaluate.

$x = 0, - 2, 2$

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.

$\frac{4 ({(0)}^{2} - 12)}{9 {({(0)}^{2} - 4)}^{\frac{4}{3}}}$

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 $0$ .

$\frac{4 (0 - 12)}{9 {(0^{2} - 4)}^{\frac{4}{3}}}$

Step 9.1.2

Subtract $12$ from $0$ .

$\frac{4 \cdot - 12}{9 {(0^{2} - 4)}^{\frac{4}{3}}}$

Step 9.2

Simplify the denominator.

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Step 9.2.1

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

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

Step 9.2.2

Subtract $4$ from $0$ .

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

Step 9.3

Simplify with factoring out.

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Step 9.3.1

Multiply $4$ by $- 12$ .

$\frac{- 48}{9 {(- 4)}^{\frac{4}{3}}}$

Step 9.3.2

Factor $3$ out of $- 48$ .

$\frac{3 (- 16)}{9 {(- 4)}^{\frac{4}{3}}}$

Step 9.4

Cancel the common factors.

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Step 9.4.1

Factor $3$ out of $9 {(- 4)}^{\frac{4}{3}}$ .

$\frac{3 (- 16)}{3 (3 {(- 4)}^{\frac{4}{3}})}$

Step 9.4.2

Cancel the common factor.

$\frac{3 \cdot - 16}{3 (3 {(- 4)}^{\frac{4}{3}})}$

Step 9.4.3

Rewrite the expression.

$\frac{- 16}{3 {(- 4)}^{\frac{4}{3}}}$

Step 9.5

Move the negative in front of the fraction.

$- \frac{16}{3 {(- 4)}^{\frac{4}{3}}}$

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.

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

Step 11.2

Simplify the result.

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Step 11.2.1

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

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

Step 11.2.2

Subtract $4$ from $0$ .

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

Step 11.2.3

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

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

Step 12

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.

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

Step 13

Evaluate the second derivative.

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Step 13.1

Simplify the expression.

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Step 13.1.1

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

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

Step 13.1.2

Subtract $4$ from $4$ .

$\frac{4 ({(- 2)}^{2} - 12)}{9 \cdot 0^{\frac{4}{3}}}$

Step 13.1.3

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

$\frac{4 ({(- 2)}^{2} - 12)}{9 \cdot {(0^{3})}^{\frac{4}{3}}}$

Step 13.1.4

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

$\frac{4 ({(- 2)}^{2} - 12)}{9 \cdot 0^{3 (\frac{4}{3})}}$

Step 13.2

Cancel the common factor of $3$ .

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Step 13.2.1

Cancel the common factor.

$\frac{4 ({(- 2)}^{2} - 12)}{9 \cdot 0^{3 (\frac{4}{3})}}$

Step 13.2.2

Rewrite the expression.

$\frac{4 ({(- 2)}^{2} - 12)}{9 \cdot 0^{4}}$

Step 13.3

Simplify the expression.

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Step 13.3.1

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

$\frac{4 ({(- 2)}^{2} - 12)}{9 \cdot 0}$

Step 13.3.2

Multiply $9$ by $0$ .

$\frac{4 ({(- 2)}^{2} - 12)}{0}$

Step 13.3.3

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

Undefined

$\frac{4 ({(- 2)}^{2} - 12)}{0}$

Step 13.4

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

Undefined

Step 14

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

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Step 14.1

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

$(- \infty, - 2) \cup (- 2, 0) \cup (0, 2) \cup (2, \infty)$

Step 14.2

Substitute any number, such as $- 4$ , from the interval $(- \infty, - 2)$ in the first derivative $\frac{4 x}{3 {(x^{2} - 4)}^{\frac{1}{3}}}$ to check if the result is negative or positive.

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Step 14.2.1

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

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

Step 14.2.2

Simplify the result.

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Step 14.2.2.1

Multiply $4$ by $- 4$ .

$f' (- 4) = \frac{- 16}{3 {({(- 4)}^{2} - 4)}^{\frac{1}{3}}}$

Step 14.2.2.2

Simplify the denominator.

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Step 14.2.2.2.1

Raise $- 4$ to the power of $2$ .

$f' (- 4) = \frac{- 16}{3 {(16 - 4)}^{\frac{1}{3}}}$

Step 14.2.2.2.2

Subtract $4$ from $16$ .

$f' (- 4) = \frac{- 16}{3 \cdot 12^{\frac{1}{3}}}$

Step 14.2.2.3

Move the negative in front of the fraction.

$f' (- 4) = - \frac{16}{3 \cdot 12^{\frac{1}{3}}}$

Step 14.2.2.4

The final answer is $- \frac{16}{3 \cdot 12^{\frac{1}{3}}}$ .

$- \frac{16}{3 \cdot 12^{\frac{1}{3}}}$

Step 14.3

Substitute any number, such as $- 1$ , from the interval $(- 2, 0)$ in the first derivative $\frac{4 x}{3 {(x^{2} - 4)}^{\frac{1}{3}}}$ to check if the result is negative or positive.

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Step 14.3.1

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

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

Step 14.3.2

Simplify the result.

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Step 14.3.2.1

Multiply $4$ by $- 1$ .

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

Step 14.3.2.2

Simplify the denominator.

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Step 14.3.2.2.1

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

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

Step 14.3.2.2.2

Subtract $4$ from $1$ .

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

Step 14.3.2.3

Move the negative in front of the fraction.

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

Step 14.3.2.4

The final answer is $- \frac{4}{3 {(- 3)}^{\frac{1}{3}}}$ .

$- \frac{4}{3 {(- 3)}^{\frac{1}{3}}}$

Step 14.4

Substitute any number, such as $1$ , from the interval $(0, 2)$ in the first derivative $\frac{4 x}{3 {(x^{2} - 4)}^{\frac{1}{3}}}$ to check if the result is negative or positive.

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Step 14.4.1

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

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

Step 14.4.2

Simplify the result.

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Step 14.4.2.1

Multiply $4$ by $1$ .

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

Step 14.4.2.2

Simplify the denominator.

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Step 14.4.2.2.1

One to any power is one.

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

Step 14.4.2.2.2

Subtract $4$ from $1$ .

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

Step 14.4.2.3

The final answer is $\frac{4}{3 {(- 3)}^{\frac{1}{3}}}$ .

$\frac{4}{3 {(- 3)}^{\frac{1}{3}}}$

Step 14.5

Substitute any number, such as $4$ , from the interval $(2, \infty)$ in the first derivative $\frac{4 x}{3 {(x^{2} - 4)}^{\frac{1}{3}}}$ to check if the result is negative or positive.

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Step 14.5.1

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

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

Step 14.5.2

Simplify the result.

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Step 14.5.2.1

Multiply $4$ by $4$ .

$f' (4) = \frac{16}{3 {(4^{2} - 4)}^{\frac{1}{3}}}$

Step 14.5.2.2

Simplify the denominator.

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Step 14.5.2.2.1

Raise $4$ to the power of $2$ .

$f' (4) = \frac{16}{3 {(16 - 4)}^{\frac{1}{3}}}$

Step 14.5.2.2.2

Subtract $4$ from $16$ .

$f' (4) = \frac{16}{3 \cdot 12^{\frac{1}{3}}}$

Step 14.5.2.3

The final answer is $\frac{16}{3 \cdot 12^{\frac{1}{3}}}$ .

$\frac{16}{3 \cdot 12^{\frac{1}{3}}}$

Step 14.6

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

$x = - 2$ is a local minimum

Step 14.7

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 14.8

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

$x = 2$ is a local minimum

Step 14.9

These are the local extrema for $f (x) = {(x^{2} - 4)}^{\frac{2}{3}}$ .

$x = - 2$ is a local minimum

$x = 0$ is a local maximum

$x = 2$ is a local minimum

$x = - 2$ is a local minimum

$x = 0$ is a local maximum

$x = 2$ is a local minimum

Step 15