Calculus Examples

$y = \sqrt[3]{4 x^{2} - 12 x}$

Step 1

Write $y = \sqrt[3]{4 x^{2} - 12 x}$ as a function.

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

Step 2

Find the first derivative of the function.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{4 x^{2} - 12 x}$ as ${(4 x^{2} - 12 x)}^{\frac{1}{3}}$ .

$\frac{d}{d x} [{(4 x^{2} - 12 x)}^{\frac{1}{3}}]$

Step 2.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{3}}$ and $g (x) = 4 x^{2} - 12 x$ .

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Step 2.2.1

To apply the Chain Rule, set $u$ as $4 x^{2} - 12 x$ .

$\frac{d}{d u} [u^{\frac{1}{3}}] \frac{d}{d x} [4 x^{2} - 12 x]$

Step 2.2.2

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

$\frac{1}{3} u^{\frac{1}{3} - 1} \frac{d}{d x} [4 x^{2} - 12 x]$

Step 2.2.3

Replace all occurrences of $u$ with $4 x^{2} - 12 x$ .

$\frac{1}{3} {(4 x^{2} - 12 x)}^{\frac{1}{3} - 1} \frac{d}{d x} [4 x^{2} - 12 x]$

Step 2.3

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

$\frac{1}{3} {(4 x^{2} - 12 x)}^{\frac{1}{3} - 1 \cdot \frac{3}{3}} \frac{d}{d x} [4 x^{2} - 12 x]$

Step 2.4

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

$\frac{1}{3} {(4 x^{2} - 12 x)}^{\frac{1}{3} + \frac{- 1 \cdot 3}{3}} \frac{d}{d x} [4 x^{2} - 12 x]$

Step 2.5

Combine the numerators over the common denominator.

$\frac{1}{3} {(4 x^{2} - 12 x)}^{\frac{1 - 1 \cdot 3}{3}} \frac{d}{d x} [4 x^{2} - 12 x]$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Multiply $- 1$ by $3$ .

$\frac{1}{3} {(4 x^{2} - 12 x)}^{\frac{1 - 3}{3}} \frac{d}{d x} [4 x^{2} - 12 x]$

Step 2.6.2

Subtract $3$ from $1$ .

$\frac{1}{3} {(4 x^{2} - 12 x)}^{\frac{- 2}{3}} \frac{d}{d x} [4 x^{2} - 12 x]$

Step 2.7

Combine fractions.

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Step 2.7.1

Move the negative in front of the fraction.

$\frac{1}{3} {(4 x^{2} - 12 x)}^{- \frac{2}{3}} \frac{d}{d x} [4 x^{2} - 12 x]$

Step 2.7.2

Combine $\frac{1}{3}$ and ${(4 x^{2} - 12 x)}^{- \frac{2}{3}}$ .

$\frac{{(4 x^{2} - 12 x)}^{- \frac{2}{3}}}{3} \frac{d}{d x} [4 x^{2} - 12 x]$

Step 2.7.3

Move ${(4 x^{2} - 12 x)}^{- \frac{2}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{3 {(4 x^{2} - 12 x)}^{\frac{2}{3}}} \frac{d}{d x} [4 x^{2} - 12 x]$

Step 2.8

By the Sum Rule, the derivative of $4 x^{2} - 12 x$ with respect to $x$ is $\frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [- 12 x]$ .

$\frac{1}{3 {(4 x^{2} - 12 x)}^{\frac{2}{3}}} (\frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [- 12 x])$

Step 2.9

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

$\frac{1}{3 {(4 x^{2} - 12 x)}^{\frac{2}{3}}} (4 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 12 x])$

Step 2.10

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

$\frac{1}{3 {(4 x^{2} - 12 x)}^{\frac{2}{3}}} (4 (2 x) + \frac{d}{d x} [- 12 x])$

Step 2.11

Multiply $2$ by $4$ .

$\frac{1}{3 {(4 x^{2} - 12 x)}^{\frac{2}{3}}} (8 x + \frac{d}{d x} [- 12 x])$

Step 2.12

Since $- 12$ is constant with respect to $x$ , the derivative of $- 12 x$ with respect to $x$ is $- 12 \frac{d}{d x} [x]$ .

$\frac{1}{3 {(4 x^{2} - 12 x)}^{\frac{2}{3}}} (8 x - 12 \frac{d}{d x} [x])$

Step 2.13

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

$\frac{1}{3 {(4 x^{2} - 12 x)}^{\frac{2}{3}}} (8 x - 12 \cdot 1)$

Step 2.14

Multiply $- 12$ by $1$ .

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

Step 2.15

Simplify.

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Step 2.15.1

Reorder the factors of $\frac{1}{3 {(4 x^{2} - 12 x)}^{\frac{2}{3}}} (8 x - 12)$ .

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

Step 2.15.2

Multiply $8 x - 12$ by $\frac{1}{3 {(4 x^{2} - 12 x)}^{\frac{2}{3}}}$ .

$\frac{8 x - 12}{3 {(4 x^{2} - 12 x)}^{\frac{2}{3}}}$

Step 2.15.3

Factor $4$ out of $8 x - 12$ .

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Step 2.15.3.1

Factor $4$ out of $8 x$ .

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

Step 2.15.3.2

Factor $4$ out of $- 12$ .

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

Step 2.15.3.3

Factor $4$ out of $4 (2 x) + 4 (- 3)$ .

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

Step 3

Find the second derivative of the function.

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Step 3.1

Since $\frac{4}{3}$ is constant with respect to $x$ , the derivative of $\frac{4 (2 x - 3)}{3 {(4 x^{2} - 12 x)}^{\frac{2}{3}}}$ with respect to $x$ is $\frac{4}{3} \frac{d}{d x} [\frac{2 x - 3}{{(4 x^{2} - 12 x)}^{\frac{2}{3}}}]$ .

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

Step 3.2

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = 2 x - 3$ and $g (x) = {(4 x^{2} - 12 x)}^{\frac{2}{3}}$ .

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

Step 3.3

Differentiate.

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Step 3.3.1

Multiply the exponents in ${({(4 x^{2} - 12 x)}^{\frac{2}{3}})}^{2}$ .

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Step 3.3.1.1

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

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

Step 3.3.1.2

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

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Step 3.3.1.2.1

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

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

Step 3.3.1.2.2

Multiply $2$ by $2$ .

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

Step 3.3.2

By the Sum Rule, the derivative of $2 x - 3$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [- 3]$ .

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

Step 3.3.3

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

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

Step 3.3.4

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

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

Step 3.3.5

Multiply $2$ by $1$ .

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

Step 3.3.6

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

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

Step 3.3.7

Simplify the expression.

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Step 3.3.7.1

Add $2$ and $0$ .

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

Step 3.3.7.2

Move $2$ to the left of ${(4 x^{2} - 12 x)}^{\frac{2}{3}}$ .

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

Step 3.4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{2}{3}}$ and $g (x) = 4 x^{2} - 12 x$ .

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Step 3.4.1

To apply the Chain Rule, set $u$ as $4 x^{2} - 12 x$ .

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

Step 3.4.2

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

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

Step 3.4.3

Replace all occurrences of $u$ with $4 x^{2} - 12 x$ .

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

Combine the numerators over the common denominator.

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

Step 3.8

Simplify the numerator.

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Step 3.8.1

Multiply $- 1$ by $3$ .

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

Step 3.8.2

Subtract $3$ from $2$ .

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

Step 3.9

Combine fractions.

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Step 3.9.1

Move the negative in front of the fraction.

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

Step 3.9.2

Combine $\frac{2}{3}$ and ${(4 x^{2} - 12 x)}^{- \frac{1}{3}}$ .

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

Step 3.9.3

Move ${(4 x^{2} - 12 x)}^{- \frac{1}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 3.10

By the Sum Rule, the derivative of $4 x^{2} - 12 x$ with respect to $x$ is $\frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [- 12 x]$ .

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

Step 3.11

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

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

Step 3.12

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

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

Step 3.13

Multiply $2$ by $4$ .

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

Step 3.14

Since $- 12$ is constant with respect to $x$ , the derivative of $- 12 x$ with respect to $x$ is $- 12 \frac{d}{d x} [x]$ .

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

Step 3.15

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

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

Step 3.16

Combine fractions.

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Step 3.16.1

Multiply $- 12$ by $1$ .

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

Step 3.16.2

Multiply $\frac{4}{3}$ by $\frac{2 {(4 x^{2} - 12 x)}^{\frac{2}{3}} - (2 x - 3) (\frac{2}{3 {(4 x^{2} - 12 x)}^{\frac{1}{3}}} (8 x - 12))}{{(4 x^{2} - 12 x)}^{\frac{4}{3}}}$ .

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

Step 3.17

Simplify.

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Step 3.17.1

Apply the distributive property.

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

Step 3.17.2

Apply the distributive property.

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

Step 3.17.3

Simplify the numerator.

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Step 3.17.3.1

Multiply $2$ by $4$ .

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

Step 3.17.3.2

Simplify each term.

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Step 3.17.3.2.1

Multiply $2$ by $- 1$ .

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

Step 3.17.3.2.2

Multiply $- 1$ by $- 3$ .

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

Step 3.17.3.3

Multiply $\frac{2}{3 {(4 x^{2} - 12 x)}^{\frac{1}{3}}}$ by $8 x - 12$ .

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

Step 3.17.3.4

Simplify the numerator.

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Step 3.17.3.4.1

Factor $4$ out of $8 x - 12$ .

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Step 3.17.3.4.1.1

Factor $4$ out of $8 x$ .

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

Step 3.17.3.4.1.2

Factor $4$ out of $- 12$ .

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

Step 3.17.3.4.1.3

Factor $4$ out of $4 (2 x) + 4 (- 3)$ .

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

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

Step 3.17.3.4.2

Multiply $2$ by $4$ .

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

Step 3.17.3.5

Multiply $- 2 x + 3$ by $\frac{8 (2 x - 3)}{3 {(4 x^{2} - 12 x)}^{\frac{1}{3}}}$ .

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

Step 3.17.3.6

Simplify the numerator.

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Step 3.17.3.6.1

Factor $- 1$ out of $- 2 x$ .

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

Step 3.17.3.6.2

Rewrite $3$ as $- 1 (- 3)$ .

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

Step 3.17.3.6.3

Factor $- 1$ out of $- (2 x) - 1 (- 3)$ .

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

Step 3.17.3.6.4

Rewrite $- (2 x - 3)$ as $- 1 (2 x - 3)$ .

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

Step 3.17.3.6.5

Raise $2 x - 3$ to the power of $1$ .

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

Step 3.17.3.6.6

Raise $2 x - 3$ to the power of $1$ .

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

Step 3.17.3.6.7

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

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

Step 3.17.3.6.8

Add $1$ and $1$ .

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

Step 3.17.3.6.9

Multiply $- 1$ by $8$ .

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

Step 3.17.3.7

Move the negative in front of the fraction.

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

Step 3.17.3.8

Multiply $4 (- \frac{8 {(2 x - 3)}^{2}}{3 {(4 x^{2} - 12 x)}^{\frac{1}{3}}})$ .

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Step 3.17.3.8.1

Multiply $- 1$ by $4$ .

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

Step 3.17.3.8.2

Combine $- 4$ and $\frac{8 {(2 x - 3)}^{2}}{3 {(4 x^{2} - 12 x)}^{\frac{1}{3}}}$ .

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

Step 3.17.3.8.3

Multiply $8$ by $- 4$ .

$f'' (x) = \frac{8 {(4 x^{2} - 12 x)}^{\frac{2}{3}} + \frac{- 32 {(2 x - 3)}^{2}}{3 {(4 x^{2} - 12 x)}^{\frac{1}{3}}}}{3 {(4 x^{2} - 12 x)}^{\frac{4}{3}}}$

Step 3.17.3.9

Move the negative in front of the fraction.

$f'' (x) = \frac{8 {(4 x^{2} - 12 x)}^{\frac{2}{3}} - \frac{32 {(2 x - 3)}^{2}}{3 {(4 x^{2} - 12 x)}^{\frac{1}{3}}}}{3 {(4 x^{2} - 12 x)}^{\frac{4}{3}}}$

Step 3.17.3.10

To write $8 {(4 x^{2} - 12 x)}^{\frac{2}{3}}$ as a fraction with a common denominator, multiply by $\frac{3 {(4 x^{2} - 12 x)}^{\frac{1}{3}}}{3 {(4 x^{2} - 12 x)}^{\frac{1}{3}}}$ .

$f'' (x) = \frac{8 {(4 x^{2} - 12 x)}^{\frac{2}{3}} \cdot \frac{3 {(4 x^{2} - 12 x)}^{\frac{1}{3}}}{3 {(4 x^{2} - 12 x)}^{\frac{1}{3}}} - \frac{32 {(2 x - 3)}^{2}}{3 {(4 x^{2} - 12 x)}^{\frac{1}{3}}}}{3 {(4 x^{2} - 12 x)}^{\frac{4}{3}}}$

Step 3.17.3.11

Combine $8 {(4 x^{2} - 12 x)}^{\frac{2}{3}}$ and $\frac{3 {(4 x^{2} - 12 x)}^{\frac{1}{3}}}{3 {(4 x^{2} - 12 x)}^{\frac{1}{3}}}$ .

$f'' (x) = \frac{\frac{8 {(4 x^{2} - 12 x)}^{\frac{2}{3}} (3 {(4 x^{2} - 12 x)}^{\frac{1}{3}})}{3 {(4 x^{2} - 12 x)}^{\frac{1}{3}}} - \frac{32 {(2 x - 3)}^{2}}{3 {(4 x^{2} - 12 x)}^{\frac{1}{3}}}}{3 {(4 x^{2} - 12 x)}^{\frac{4}{3}}}$

Step 3.17.3.12

Combine the numerators over the common denominator.

$f'' (x) = \frac{\frac{8 {(4 x^{2} - 12 x)}^{\frac{2}{3}} (3 {(4 x^{2} - 12 x)}^{\frac{1}{3}}) - 32 {(2 x - 3)}^{2}}{3 {(4 x^{2} - 12 x)}^{\frac{1}{3}}}}{3 {(4 x^{2} - 12 x)}^{\frac{4}{3}}}$

Step 3.17.3.13

Reorder terms.

$f'' (x) = \frac{\frac{- 32 {(2 x - 3)}^{2} + 3 \cdot (8 {(4 x^{2} - 12 x)}^{\frac{2}{3}} {(4 x^{2} - 12 x)}^{\frac{1}{3}})}{3 {(4 x^{2} - 12 x)}^{\frac{1}{3}}}}{3 {(4 x^{2} - 12 x)}^{\frac{4}{3}}}$

Step 3.17.3.14

Rewrite $\frac{- 32 {(2 x - 3)}^{2} + 3 \cdot 8 {(4 x^{2} - 12 x)}^{\frac{2}{3}} {(4 x^{2} - 12 x)}^{\frac{1}{3}}}{3 {(4 x^{2} - 12 x)}^{\frac{1}{3}}}$ in a factored form.

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Step 3.17.3.14.1

Rewrite ${(2 x - 3)}^{2}$ as $(2 x - 3) (2 x - 3)$ .

$f'' (x) = \frac{\frac{- 32 ((2 x - 3) (2 x - 3)) + 3 \cdot (8 {(4 x^{2} - 12 x)}^{\frac{2}{3}} {(4 x^{2} - 12 x)}^{\frac{1}{3}})}{3 {(4 x^{2} - 12 x)}^{\frac{1}{3}}}}{3 {(4 x^{2} - 12 x)}^{\frac{4}{3}}}$

Step 3.17.3.14.2

Expand $(2 x - 3) (2 x - 3)$ using the FOIL Method.

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Step 3.17.3.14.2.1

Apply the distributive property.

$f'' (x) = \frac{\frac{- 32 (2 x (2 x - 3) - 3 (2 x - 3)) + 3 \cdot (8 {(4 x^{2} - 12 x)}^{\frac{2}{3}} {(4 x^{2} - 12 x)}^{\frac{1}{3}})}{3 {(4 x^{2} - 12 x)}^{\frac{1}{3}}}}{3 {(4 x^{2} - 12 x)}^{\frac{4}{3}}}$

Step 3.17.3.14.2.2

Apply the distributive property.

$f'' (x) = \frac{\frac{- 32 (2 x (2 x) + 2 x \cdot - 3 - 3 (2 x - 3)) + 3 \cdot (8 {(4 x^{2} - 12 x)}^{\frac{2}{3}} {(4 x^{2} - 12 x)}^{\frac{1}{3}})}{3 {(4 x^{2} - 12 x)}^{\frac{1}{3}}}}{3 {(4 x^{2} - 12 x)}^{\frac{4}{3}}}$

Step 3.17.3.14.2.3

Apply the distributive property.

$f'' (x) = \frac{\frac{- 32 (2 x (2 x) + 2 x \cdot - 3 - 3 (2 x) - 3 \cdot - 3) + 3 \cdot (8 {(4 x^{2} - 12 x)}^{\frac{2}{3}} {(4 x^{2} - 12 x)}^{\frac{1}{3}})}{3 {(4 x^{2} - 12 x)}^{\frac{1}{3}}}}{3 {(4 x^{2} - 12 x)}^{\frac{4}{3}}}$

Step 3.17.3.14.3

Simplify and combine like terms.

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Step 3.17.3.14.3.1

Simplify each term.

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Step 3.17.3.14.3.1.1

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{\frac{- 32 (2 \cdot (2 x \cdot x) + 2 x \cdot - 3 - 3 (2 x) - 3 \cdot - 3) + 3 \cdot (8 {(4 x^{2} - 12 x)}^{\frac{2}{3}} {(4 x^{2} - 12 x)}^{\frac{1}{3}})}{3 {(4 x^{2} - 12 x)}^{\frac{1}{3}}}}{3 {(4 x^{2} - 12 x)}^{\frac{4}{3}}}$

Step 3.17.3.14.3.1.2

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

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Step 3.17.3.14.3.1.2.1

Move $x$ .

$f'' (x) = \frac{\frac{- 32 (2 \cdot (2 (x \cdot x)) + 2 x \cdot - 3 - 3 (2 x) - 3 \cdot - 3) + 3 \cdot (8 {(4 x^{2} - 12 x)}^{\frac{2}{3}} {(4 x^{2} - 12 x)}^{\frac{1}{3}})}{3 {(4 x^{2} - 12 x)}^{\frac{1}{3}}}}{3 {(4 x^{2} - 12 x)}^{\frac{4}{3}}}$

Step 3.17.3.14.3.1.2.2

Multiply $x$ by $x$ .

$f'' (x) = \frac{\frac{- 32 (2 \cdot (2 x^{2}) + 2 x \cdot - 3 - 3 (2 x) - 3 \cdot - 3) + 3 \cdot (8 {(4 x^{2} - 12 x)}^{\frac{2}{3}} {(4 x^{2} - 12 x)}^{\frac{1}{3}})}{3 {(4 x^{2} - 12 x)}^{\frac{1}{3}}}}{3 {(4 x^{2} - 12 x)}^{\frac{4}{3}}}$

Step 3.17.3.14.3.1.3

Multiply $2$ by $2$ .

$f'' (x) = \frac{\frac{- 32 (4 x^{2} + 2 x \cdot - 3 - 3 (2 x) - 3 \cdot - 3) + 3 \cdot (8 {(4 x^{2} - 12 x)}^{\frac{2}{3}} {(4 x^{2} - 12 x)}^{\frac{1}{3}})}{3 {(4 x^{2} - 12 x)}^{\frac{1}{3}}}}{3 {(4 x^{2} - 12 x)}^{\frac{4}{3}}}$

Step 3.17.3.14.3.1.4

Multiply $- 3$ by $2$ .

$f'' (x) = \frac{\frac{- 32 (4 x^{2} - 6 x - 3 (2 x) - 3 \cdot - 3) + 3 \cdot (8 {(4 x^{2} - 12 x)}^{\frac{2}{3}} {(4 x^{2} - 12 x)}^{\frac{1}{3}})}{3 {(4 x^{2} - 12 x)}^{\frac{1}{3}}}}{3 {(4 x^{2} - 12 x)}^{\frac{4}{3}}}$

Step 3.17.3.14.3.1.5

Multiply $2$ by $- 3$ .

$f'' (x) = \frac{\frac{- 32 (4 x^{2} - 6 x - 6 x - 3 \cdot - 3) + 3 \cdot (8 {(4 x^{2} - 12 x)}^{\frac{2}{3}} {(4 x^{2} - 12 x)}^{\frac{1}{3}})}{3 {(4 x^{2} - 12 x)}^{\frac{1}{3}}}}{3 {(4 x^{2} - 12 x)}^{\frac{4}{3}}}$

Step 3.17.3.14.3.1.6

Multiply $- 3$ by $- 3$ .

$f'' (x) = \frac{\frac{- 32 (4 x^{2} - 6 x - 6 x + 9) + 3 \cdot (8 {(4 x^{2} - 12 x)}^{\frac{2}{3}} {(4 x^{2} - 12 x)}^{\frac{1}{3}})}{3 {(4 x^{2} - 12 x)}^{\frac{1}{3}}}}{3 {(4 x^{2} - 12 x)}^{\frac{4}{3}}}$

Step 3.17.3.14.3.2

Subtract $6 x$ from $- 6 x$ .

$f'' (x) = \frac{\frac{- 32 (4 x^{2} - 12 x + 9) + 3 \cdot (8 {(4 x^{2} - 12 x)}^{\frac{2}{3}} {(4 x^{2} - 12 x)}^{\frac{1}{3}})}{3 {(4 x^{2} - 12 x)}^{\frac{1}{3}}}}{3 {(4 x^{2} - 12 x)}^{\frac{4}{3}}}$

Step 3.17.3.14.4

Apply the distributive property.

$f'' (x) = \frac{\frac{- 32 (4 x^{2}) - 32 (- 12 x) - 32 \cdot 9 + 3 \cdot (8 {(4 x^{2} - 12 x)}^{\frac{2}{3}} {(4 x^{2} - 12 x)}^{\frac{1}{3}})}{3 {(4 x^{2} - 12 x)}^{\frac{1}{3}}}}{3 {(4 x^{2} - 12 x)}^{\frac{4}{3}}}$

Step 3.17.3.14.5

Simplify.

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Step 3.17.3.14.5.1

Multiply $4$ by $- 32$ .

$f'' (x) = \frac{\frac{- 128 x^{2} - 32 (- 12 x) - 32 \cdot 9 + 3 \cdot (8 {(4 x^{2} - 12 x)}^{\frac{2}{3}} {(4 x^{2} - 12 x)}^{\frac{1}{3}})}{3 {(4 x^{2} - 12 x)}^{\frac{1}{3}}}}{3 {(4 x^{2} - 12 x)}^{\frac{4}{3}}}$

Step 3.17.3.14.5.2

Multiply $- 12$ by $- 32$ .

$f'' (x) = \frac{\frac{- 128 x^{2} + 384 x - 32 \cdot 9 + 3 \cdot (8 {(4 x^{2} - 12 x)}^{\frac{2}{3}} {(4 x^{2} - 12 x)}^{\frac{1}{3}})}{3 {(4 x^{2} - 12 x)}^{\frac{1}{3}}}}{3 {(4 x^{2} - 12 x)}^{\frac{4}{3}}}$

Step 3.17.3.14.5.3

Multiply $- 32$ by $9$ .

$f'' (x) = \frac{\frac{- 128 x^{2} + 384 x - 288 + 3 \cdot (8 {(4 x^{2} - 12 x)}^{\frac{2}{3}} {(4 x^{2} - 12 x)}^{\frac{1}{3}})}{3 {(4 x^{2} - 12 x)}^{\frac{1}{3}}}}{3 {(4 x^{2} - 12 x)}^{\frac{4}{3}}}$

Step 3.17.3.14.6

Multiply ${(4 x^{2} - 12 x)}^{\frac{2}{3}}$ by ${(4 x^{2} - 12 x)}^{\frac{1}{3}}$ by adding the exponents.

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Step 3.17.3.14.6.1

Move ${(4 x^{2} - 12 x)}^{\frac{1}{3}}$ .

$f'' (x) = \frac{\frac{- 128 x^{2} + 384 x - 288 + 3 \cdot (8 ({(4 x^{2} - 12 x)}^{\frac{1}{3}} {(4 x^{2} - 12 x)}^{\frac{2}{3}}))}{3 {(4 x^{2} - 12 x)}^{\frac{1}{3}}}}{3 {(4 x^{2} - 12 x)}^{\frac{4}{3}}}$

Step 3.17.3.14.6.2

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

$f'' (x) = \frac{\frac{- 128 x^{2} + 384 x - 288 + 3 \cdot (8 {(4 x^{2} - 12 x)}^{\frac{1}{3} + \frac{2}{3}})}{3 {(4 x^{2} - 12 x)}^{\frac{1}{3}}}}{3 {(4 x^{2} - 12 x)}^{\frac{4}{3}}}$

Step 3.17.3.14.6.3

Combine the numerators over the common denominator.

$f'' (x) = \frac{\frac{- 128 x^{2} + 384 x - 288 + 3 \cdot (8 {(4 x^{2} - 12 x)}^{\frac{1 + 2}{3}})}{3 {(4 x^{2} - 12 x)}^{\frac{1}{3}}}}{3 {(4 x^{2} - 12 x)}^{\frac{4}{3}}}$

Step 3.17.3.14.6.4

Add $1$ and $2$ .

$f'' (x) = \frac{\frac{- 128 x^{2} + 384 x - 288 + 3 \cdot (8 {(4 x^{2} - 12 x)}^{\frac{3}{3}})}{3 {(4 x^{2} - 12 x)}^{\frac{1}{3}}}}{3 {(4 x^{2} - 12 x)}^{\frac{4}{3}}}$

Step 3.17.3.14.6.5

Divide $3$ by $3$ .

$f'' (x) = \frac{\frac{- 128 x^{2} + 384 x - 288 + 3 \cdot (8 (4 x^{2} - 12 x))}{3 {(4 x^{2} - 12 x)}^{\frac{1}{3}}}}{3 {(4 x^{2} - 12 x)}^{\frac{4}{3}}}$

Step 3.17.3.14.7

Simplify $3 \cdot 8 {(4 x^{2} - 12 x)}^{1}$ .

$f'' (x) = \frac{\frac{- 128 x^{2} + 384 x - 288 + 3 \cdot (8 (4 x^{2} - 12 x))}{3 {(4 x^{2} - 12 x)}^{\frac{1}{3}}}}{3 {(4 x^{2} - 12 x)}^{\frac{4}{3}}}$

Step 3.17.3.14.8

Multiply $3$ by $8$ .

$f'' (x) = \frac{\frac{- 128 x^{2} + 384 x - 288 + 24 (4 x^{2} - 12 x)}{3 {(4 x^{2} - 12 x)}^{\frac{1}{3}}}}{3 {(4 x^{2} - 12 x)}^{\frac{4}{3}}}$

Step 3.17.3.14.9

Apply the distributive property.

$f'' (x) = \frac{\frac{- 128 x^{2} + 384 x - 288 + 24 (4 x^{2}) + 24 (- 12 x)}{3 {(4 x^{2} - 12 x)}^{\frac{1}{3}}}}{3 {(4 x^{2} - 12 x)}^{\frac{4}{3}}}$

Step 3.17.3.14.10

Multiply $4$ by $24$ .

$f'' (x) = \frac{\frac{- 128 x^{2} + 384 x - 288 + 96 x^{2} + 24 (- 12 x)}{3 {(4 x^{2} - 12 x)}^{\frac{1}{3}}}}{3 {(4 x^{2} - 12 x)}^{\frac{4}{3}}}$

Step 3.17.3.14.11

Multiply $- 12$ by $24$ .

$f'' (x) = \frac{\frac{- 128 x^{2} + 384 x - 288 + 96 x^{2} - 288 x}{3 {(4 x^{2} - 12 x)}^{\frac{1}{3}}}}{3 {(4 x^{2} - 12 x)}^{\frac{4}{3}}}$

Step 3.17.3.14.12

Add $- 128 x^{2}$ and $96 x^{2}$ .

$f'' (x) = \frac{\frac{- 32 x^{2} + 384 x - 288 - 288 x}{3 {(4 x^{2} - 12 x)}^{\frac{1}{3}}}}{3 {(4 x^{2} - 12 x)}^{\frac{4}{3}}}$

Step 3.17.3.14.13

Subtract $288 x$ from $384 x$ .

$f'' (x) = \frac{\frac{- 32 x^{2} + 96 x - 288}{3 {(4 x^{2} - 12 x)}^{\frac{1}{3}}}}{3 {(4 x^{2} - 12 x)}^{\frac{4}{3}}}$

Step 3.17.3.14.14

Factor $32$ out of $- 32 x^{2} + 96 x - 288$ .

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Step 3.17.3.14.14.1

Factor $32$ out of $- 32 x^{2}$ .

$f'' (x) = \frac{\frac{32 (- x^{2}) + 96 x - 288}{3 {(4 x^{2} - 12 x)}^{\frac{1}{3}}}}{3 {(4 x^{2} - 12 x)}^{\frac{4}{3}}}$

Step 3.17.3.14.14.2

Factor $32$ out of $96 x$ .

$f'' (x) = \frac{\frac{32 (- x^{2}) + 32 (3 x) - 288}{3 {(4 x^{2} - 12 x)}^{\frac{1}{3}}}}{3 {(4 x^{2} - 12 x)}^{\frac{4}{3}}}$

Step 3.17.3.14.14.3

Factor $32$ out of $- 288$ .

$f'' (x) = \frac{\frac{32 (- x^{2}) + 32 (3 x) + 32 (- 9)}{3 {(4 x^{2} - 12 x)}^{\frac{1}{3}}}}{3 {(4 x^{2} - 12 x)}^{\frac{4}{3}}}$

Step 3.17.3.14.14.4

Factor $32$ out of $32 (- x^{2}) + 32 (3 x)$ .

$f'' (x) = \frac{\frac{32 (- x^{2} + 3 x) + 32 (- 9)}{3 {(4 x^{2} - 12 x)}^{\frac{1}{3}}}}{3 {(4 x^{2} - 12 x)}^{\frac{4}{3}}}$

Step 3.17.3.14.14.5

Factor $32$ out of $32 (- x^{2} + 3 x) + 32 (- 9)$ .

$f'' (x) = \frac{\frac{32 (- x^{2} + 3 x - 9)}{3 {(4 x^{2} - 12 x)}^{\frac{1}{3}}}}{3 {(4 x^{2} - 12 x)}^{\frac{4}{3}}}$

Step 3.17.4

Combine terms.

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Step 3.17.4.1

Rewrite $\frac{\frac{32 (- x^{2} + 3 x - 9)}{3 {(4 x^{2} - 12 x)}^{\frac{1}{3}}}}{3 {(4 x^{2} - 12 x)}^{\frac{4}{3}}}$ as a product.

$f'' (x) = \frac{32 (- x^{2} + 3 x - 9)}{3 {(4 x^{2} - 12 x)}^{\frac{1}{3}}} \cdot \frac{1}{3 {(4 x^{2} - 12 x)}^{\frac{4}{3}}}$

Step 3.17.4.2

Multiply $\frac{32 (- x^{2} + 3 x - 9)}{3 {(4 x^{2} - 12 x)}^{\frac{1}{3}}}$ by $\frac{1}{3 {(4 x^{2} - 12 x)}^{\frac{4}{3}}}$ .

$f'' (x) = \frac{32 (- x^{2} + 3 x - 9)}{3 {(4 x^{2} - 12 x)}^{\frac{1}{3}} (3 {(4 x^{2} - 12 x)}^{\frac{4}{3}})}$

Step 3.17.4.3

Multiply $3$ by $3$ .

$f'' (x) = \frac{32 (- x^{2} + 3 x - 9)}{9 {(4 x^{2} - 12 x)}^{\frac{1}{3}} {(4 x^{2} - 12 x)}^{\frac{4}{3}}}$

Step 3.17.4.4

Multiply ${(4 x^{2} - 12 x)}^{\frac{1}{3}}$ by ${(4 x^{2} - 12 x)}^{\frac{4}{3}}$ by adding the exponents.

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Step 3.17.4.4.1

Move ${(4 x^{2} - 12 x)}^{\frac{4}{3}}$ .

$f'' (x) = \frac{32 (- x^{2} + 3 x - 9)}{9 ({(4 x^{2} - 12 x)}^{\frac{4}{3}} {(4 x^{2} - 12 x)}^{\frac{1}{3}})}$

Step 3.17.4.4.2

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

$f'' (x) = \frac{32 (- x^{2} + 3 x - 9)}{9 {(4 x^{2} - 12 x)}^{\frac{4}{3} + \frac{1}{3}}}$

Step 3.17.4.4.3

Combine the numerators over the common denominator.

$f'' (x) = \frac{32 (- x^{2} + 3 x - 9)}{9 {(4 x^{2} - 12 x)}^{\frac{4 + 1}{3}}}$

Step 3.17.4.4.4

Add $4$ and $1$ .

$f'' (x) = \frac{32 (- x^{2} + 3 x - 9)}{9 {(4 x^{2} - 12 x)}^{\frac{5}{3}}}$

Step 3.17.5

Factor $- 1$ out of $- x^{2}$ .

$f'' (x) = \frac{32 (- (x^{2}) + 3 x - 9)}{9 {(4 x^{2} - 12 x)}^{\frac{5}{3}}}$

Step 3.17.6

Factor $- 1$ out of $3 x$ .

$f'' (x) = \frac{32 (- (x^{2}) - (- 3 x) - 9)}{9 {(4 x^{2} - 12 x)}^{\frac{5}{3}}}$

Step 3.17.7

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

$f'' (x) = \frac{32 (- (x^{2} - 3 x) - 9)}{9 {(4 x^{2} - 12 x)}^{\frac{5}{3}}}$

Step 3.17.8

Rewrite $- 9$ as $- 1 (9)$ .

$f'' (x) = \frac{32 (- (x^{2} - 3 x) - 1 \cdot 9)}{9 {(4 x^{2} - 12 x)}^{\frac{5}{3}}}$

Step 3.17.9

Factor $- 1$ out of $- (x^{2} - 3 x) - 1 (9)$ .

$f'' (x) = \frac{32 (- (x^{2} - 3 x + 9))}{9 {(4 x^{2} - 12 x)}^{\frac{5}{3}}}$

Step 3.17.10

Rewrite $- (x^{2} - 3 x + 9)$ as $- 1 (x^{2} - 3 x + 9)$ .

$f'' (x) = \frac{32 (- 1 (x^{2} - 3 x + 9))}{9 {(4 x^{2} - 12 x)}^{\frac{5}{3}}}$

Step 3.17.11

Move the negative in front of the fraction.

$f'' (x) = - \frac{32 (x^{2} - 3 x + 9)}{9 {(4 x^{2} - 12 x)}^{\frac{5}{3}}}$

Step 4

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

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

Step 5

Find the first derivative.

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Step 5.1

Find the first derivative.

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Step 5.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{4 x^{2} - 12 x}$ as ${(4 x^{2} - 12 x)}^{\frac{1}{3}}$ .

$\frac{d}{d x} [{(4 x^{2} - 12 x)}^{\frac{1}{3}}]$

Step 5.1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{3}}$ and $g (x) = 4 x^{2} - 12 x$ .

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Step 5.1.2.1

To apply the Chain Rule, set $u$ as $4 x^{2} - 12 x$ .

$\frac{d}{d u} [u^{\frac{1}{3}}] \frac{d}{d x} [4 x^{2} - 12 x]$

Step 5.1.2.2

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

$\frac{1}{3} u^{\frac{1}{3} - 1} \frac{d}{d x} [4 x^{2} - 12 x]$

Step 5.1.2.3

Replace all occurrences of $u$ with $4 x^{2} - 12 x$ .

$\frac{1}{3} {(4 x^{2} - 12 x)}^{\frac{1}{3} - 1} \frac{d}{d x} [4 x^{2} - 12 x]$

Step 5.1.3

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

$\frac{1}{3} {(4 x^{2} - 12 x)}^{\frac{1}{3} - 1 \cdot \frac{3}{3}} \frac{d}{d x} [4 x^{2} - 12 x]$

Step 5.1.4

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

$\frac{1}{3} {(4 x^{2} - 12 x)}^{\frac{1}{3} + \frac{- 1 \cdot 3}{3}} \frac{d}{d x} [4 x^{2} - 12 x]$

Step 5.1.5

Combine the numerators over the common denominator.

$\frac{1}{3} {(4 x^{2} - 12 x)}^{\frac{1 - 1 \cdot 3}{3}} \frac{d}{d x} [4 x^{2} - 12 x]$

Step 5.1.6

Simplify the numerator.

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Step 5.1.6.1

Multiply $- 1$ by $3$ .

$\frac{1}{3} {(4 x^{2} - 12 x)}^{\frac{1 - 3}{3}} \frac{d}{d x} [4 x^{2} - 12 x]$

Step 5.1.6.2

Subtract $3$ from $1$ .

$\frac{1}{3} {(4 x^{2} - 12 x)}^{\frac{- 2}{3}} \frac{d}{d x} [4 x^{2} - 12 x]$

Step 5.1.7

Combine fractions.

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Step 5.1.7.1

Move the negative in front of the fraction.

$\frac{1}{3} {(4 x^{2} - 12 x)}^{- \frac{2}{3}} \frac{d}{d x} [4 x^{2} - 12 x]$

Step 5.1.7.2

Combine $\frac{1}{3}$ and ${(4 x^{2} - 12 x)}^{- \frac{2}{3}}$ .

$\frac{{(4 x^{2} - 12 x)}^{- \frac{2}{3}}}{3} \frac{d}{d x} [4 x^{2} - 12 x]$

Step 5.1.7.3

Move ${(4 x^{2} - 12 x)}^{- \frac{2}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{3 {(4 x^{2} - 12 x)}^{\frac{2}{3}}} \frac{d}{d x} [4 x^{2} - 12 x]$

Step 5.1.8

By the Sum Rule, the derivative of $4 x^{2} - 12 x$ with respect to $x$ is $\frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [- 12 x]$ .

$\frac{1}{3 {(4 x^{2} - 12 x)}^{\frac{2}{3}}} (\frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [- 12 x])$

Step 5.1.9

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

$\frac{1}{3 {(4 x^{2} - 12 x)}^{\frac{2}{3}}} (4 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 12 x])$

Step 5.1.10

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

$\frac{1}{3 {(4 x^{2} - 12 x)}^{\frac{2}{3}}} (4 (2 x) + \frac{d}{d x} [- 12 x])$

Step 5.1.11

Multiply $2$ by $4$ .

$\frac{1}{3 {(4 x^{2} - 12 x)}^{\frac{2}{3}}} (8 x + \frac{d}{d x} [- 12 x])$

Step 5.1.12

Since $- 12$ is constant with respect to $x$ , the derivative of $- 12 x$ with respect to $x$ is $- 12 \frac{d}{d x} [x]$ .

$\frac{1}{3 {(4 x^{2} - 12 x)}^{\frac{2}{3}}} (8 x - 12 \frac{d}{d x} [x])$

Step 5.1.13

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

$\frac{1}{3 {(4 x^{2} - 12 x)}^{\frac{2}{3}}} (8 x - 12 \cdot 1)$

Step 5.1.14

Multiply $- 12$ by $1$ .

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

Step 5.1.15

Simplify.

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Step 5.1.15.1

Reorder the factors of $\frac{1}{3 {(4 x^{2} - 12 x)}^{\frac{2}{3}}} (8 x - 12)$ .

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

Step 5.1.15.2

Multiply $8 x - 12$ by $\frac{1}{3 {(4 x^{2} - 12 x)}^{\frac{2}{3}}}$ .

$\frac{8 x - 12}{3 {(4 x^{2} - 12 x)}^{\frac{2}{3}}}$

Step 5.1.15.3

Factor $4$ out of $8 x - 12$ .

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Step 5.1.15.3.1

Factor $4$ out of $8 x$ .

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

Step 5.1.15.3.2

Factor $4$ out of $- 12$ .

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

Step 5.1.15.3.3

Factor $4$ out of $4 (2 x) + 4 (- 3)$ .

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

Step 5.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{4 (2 x - 3)}{3 {(4 x^{2} - 12 x)}^{\frac{2}{3}}}$ .

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

Step 6

Set the first derivative equal to $0$ then solve the equation $\frac{4 (2 x - 3)}{3 {(4 x^{2} - 12 x)}^{\frac{2}{3}}} = 0$ .

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Step 6.1

Set the first derivative equal to $0$ .

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

Step 6.2

Set the numerator equal to zero.

$4 (2 x - 3) = 0$

Step 6.3

Solve the equation for $x$ .

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Step 6.3.1

Divide each term in $4 (2 x - 3) = 0$ by $4$ and simplify.

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Step 6.3.1.1

Divide each term in $4 (2 x - 3) = 0$ by $4$ .

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

Step 6.3.1.2

Simplify the left side.

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Step 6.3.1.2.1

Cancel the common factor of $4$ .

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Step 6.3.1.2.1.1

Cancel the common factor.

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

Step 6.3.1.2.1.2

Divide $2 x - 3$ by $1$ .

$2 x - 3 = \frac{0}{4}$

Step 6.3.1.3

Simplify the right side.

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Step 6.3.1.3.1

Divide $0$ by $4$ .

$2 x - 3 = 0$

Step 6.3.2

Add $3$ to both sides of the equation.

$2 x = 3$

Step 6.3.3

Divide each term in $2 x = 3$ by $2$ and simplify.

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Step 6.3.3.1

Divide each term in $2 x = 3$ by $2$ .

$\frac{2 x}{2} = \frac{3}{2}$

Step 6.3.3.2

Simplify the left side.

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Step 6.3.3.2.1

Cancel the common factor of $2$ .

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Step 6.3.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{3}{2}$

Step 6.3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{3}{2}$

Step 7

Find the values where the derivative is undefined.

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Step 7.1

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

$\frac{4 (2 x - 3)}{3 \sqrt[3]{{(4 x^{2} - 12 x)}^{2}}}$

Step 7.2

Set the denominator in $\frac{4 (2 x - 3)}{3 \sqrt[3]{{(4 x^{2} - 12 x)}^{2}}}$ equal to $0$ to find where the expression is undefined.

$3 \sqrt[3]{{(4 x^{2} - 12 x)}^{2}} = 0$

Step 7.3

Solve for $x$ .

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Step 7.3.1

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

${(3 \sqrt[3]{{(4 x^{2} - 12 x)}^{2}})}^{3} = 0^{3}$

Step 7.3.2

Simplify each side of the equation.

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Step 7.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{{(4 x^{2} - 12 x)}^{2}}$ as ${(4 x^{2} - 12 x)}^{\frac{2}{3}}$ .

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

Step 7.3.2.2

Simplify the left side.

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Step 7.3.2.2.1

Simplify ${(3 {(4 x^{2} - 12 x)}^{\frac{2}{3}})}^{3}$ .

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Step 7.3.2.2.1.1

Apply the product rule to $3 {(4 x^{2} - 12 x)}^{\frac{2}{3}}$ .

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

Step 7.3.2.2.1.2

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

$27 {({(4 x^{2} - 12 x)}^{\frac{2}{3}})}^{3} = 0^{3}$

Step 7.3.2.2.1.3

Multiply the exponents in ${({(4 x^{2} - 12 x)}^{\frac{2}{3}})}^{3}$ .

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Step 7.3.2.2.1.3.1

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

$27 {(4 x^{2} - 12 x)}^{\frac{2}{3} \cdot 3} = 0^{3}$

Step 7.3.2.2.1.3.2

Cancel the common factor of $3$ .

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Step 7.3.2.2.1.3.2.1

Cancel the common factor.

$27 {(4 x^{2} - 12 x)}^{\frac{2}{3} \cdot 3} = 0^{3}$

Step 7.3.2.2.1.3.2.2

Rewrite the expression.

$27 {(4 x^{2} - 12 x)}^{2} = 0^{3}$

Step 7.3.2.3

Simplify the right side.

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Step 7.3.2.3.1

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

$27 {(4 x^{2} - 12 x)}^{2} = 0$

Step 7.3.3

Solve for $x$ .

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Step 7.3.3.1

Factor the left side of the equation.

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Step 7.3.3.1.1

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

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Step 7.3.3.1.1.1

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

$27 {(4 x (x) - 12 x)}^{2} = 0$

Step 7.3.3.1.1.2

Factor $4 x$ out of $- 12 x$ .

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

Step 7.3.3.1.1.3

Factor $4 x$ out of $4 x (x) + 4 x (- 3)$ .

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

Step 7.3.3.1.2

Factor.

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Step 7.3.3.1.2.1

Apply the product rule to $4 x (x - 3)$ .

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

Step 7.3.3.1.2.2

Remove unnecessary parentheses.

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

Step 7.3.3.1.3

Factor.

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Step 7.3.3.1.3.1

Apply the product rule to $4 x$ .

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

Step 7.3.3.1.3.2

Remove unnecessary parentheses.

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

Step 7.3.3.2

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

$x^{2} = 0$

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

Step 7.3.3.3

Set $x^{2}$ equal to $0$ and solve for $x$ .

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Step 7.3.3.3.1

Set $x^{2}$ equal to $0$ .

$x^{2} = 0$

Step 7.3.3.3.2

Solve $x^{2} = 0$ for $x$ .

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Step 7.3.3.3.2.1

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

$x = \pm \sqrt{0}$

Step 7.3.3.3.2.2

Simplify $\pm \sqrt{0}$ .

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Step 7.3.3.3.2.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 7.3.3.3.2.2.2

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

$x = \pm 0$

Step 7.3.3.3.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 7.3.3.4

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

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Step 7.3.3.4.1

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

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

Step 7.3.3.4.2

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

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Step 7.3.3.4.2.1

Set the $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 7.3.3.4.2.2

Add $3$ to both sides of the equation.

$x = 3$

Step 7.3.3.5

The final solution is all the values that make $27 \cdot (4^{2} x^{2} {(x - 3)}^{2}) = 0$ true.

$x = 0, 3$

Step 7.4

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 $0$ .

$x = 0, x = 3$

Step 8

Critical points to evaluate.

$x = \frac{3}{2}, 0, 3$

Step 9

Evaluate the second derivative at $x = \frac{3}{2}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- \frac{32 ({(\frac{3}{2})}^{2} - 3 (\frac{3}{2}) + 9)}{9 {(4 {(\frac{3}{2})}^{2} - 12 (\frac{3}{2}))}^{\frac{5}{3}}}$

Step 10

Evaluate the second derivative.

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Step 10.1

Simplify the numerator.

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Step 10.1.1

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

$- \frac{32 (\frac{3^{2}}{2^{2}} - 3 (\frac{3}{2}) + 9)}{9 {(4 {(\frac{3}{2})}^{2} - 12 (\frac{3}{2}))}^{\frac{5}{3}}}$

Step 10.1.2

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

$- \frac{32 (\frac{9}{2^{2}} - 3 (\frac{3}{2}) + 9)}{9 {(4 {(\frac{3}{2})}^{2} - 12 (\frac{3}{2}))}^{\frac{5}{3}}}$

Step 10.1.3

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

$- \frac{32 (\frac{9}{4} - 3 (\frac{3}{2}) + 9)}{9 {(4 {(\frac{3}{2})}^{2} - 12 (\frac{3}{2}))}^{\frac{5}{3}}}$

Step 10.1.4

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

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Step 10.1.4.1

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

$- \frac{32 (\frac{9}{4} + \frac{- 3 \cdot 3}{2} + 9)}{9 {(4 {(\frac{3}{2})}^{2} - 12 (\frac{3}{2}))}^{\frac{5}{3}}}$

Step 10.1.4.2

Multiply $- 3$ by $3$ .

$- \frac{32 (\frac{9}{4} + \frac{- 9}{2} + 9)}{9 {(4 {(\frac{3}{2})}^{2} - 12 (\frac{3}{2}))}^{\frac{5}{3}}}$

Step 10.1.5

Move the negative in front of the fraction.

$- \frac{32 (\frac{9}{4} - \frac{9}{2} + 9)}{9 {(4 {(\frac{3}{2})}^{2} - 12 (\frac{3}{2}))}^{\frac{5}{3}}}$

Step 10.1.6

To write $- \frac{9}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{32 (\frac{9}{4} - \frac{9}{2} \cdot \frac{2}{2} + 9)}{9 {(4 {(\frac{3}{2})}^{2} - 12 (\frac{3}{2}))}^{\frac{5}{3}}}$

Step 10.1.7

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 10.1.7.1

Multiply $\frac{9}{2}$ by $\frac{2}{2}$ .

$- \frac{32 (\frac{9}{4} - \frac{9 \cdot 2}{2 \cdot 2} + 9)}{9 {(4 {(\frac{3}{2})}^{2} - 12 (\frac{3}{2}))}^{\frac{5}{3}}}$

Step 10.1.7.2

Multiply $2$ by $2$ .

$- \frac{32 (\frac{9}{4} - \frac{9 \cdot 2}{4} + 9)}{9 {(4 {(\frac{3}{2})}^{2} - 12 (\frac{3}{2}))}^{\frac{5}{3}}}$

Step 10.1.8

Combine the numerators over the common denominator.

$- \frac{32 (\frac{9 - 9 \cdot 2}{4} + 9)}{9 {(4 {(\frac{3}{2})}^{2} - 12 (\frac{3}{2}))}^{\frac{5}{3}}}$

Step 10.1.9

Simplify the numerator.

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Step 10.1.9.1

Multiply $- 9$ by $2$ .

$- \frac{32 (\frac{9 - 18}{4} + 9)}{9 {(4 {(\frac{3}{2})}^{2} - 12 (\frac{3}{2}))}^{\frac{5}{3}}}$

Step 10.1.9.2

Subtract $18$ from $9$ .

$- \frac{32 (\frac{- 9}{4} + 9)}{9 {(4 {(\frac{3}{2})}^{2} - 12 (\frac{3}{2}))}^{\frac{5}{3}}}$

Step 10.1.10

To write $9$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$- \frac{32 (\frac{- 9}{4} + 9 \cdot \frac{4}{4})}{9 {(4 {(\frac{3}{2})}^{2} - 12 (\frac{3}{2}))}^{\frac{5}{3}}}$

Step 10.1.11

Combine $9$ and $\frac{4}{4}$ .

$- \frac{32 (\frac{- 9}{4} + \frac{9 \cdot 4}{4})}{9 {(4 {(\frac{3}{2})}^{2} - 12 (\frac{3}{2}))}^{\frac{5}{3}}}$

Step 10.1.12

Combine the numerators over the common denominator.

$- \frac{32 \frac{- 9 + 9 \cdot 4}{4}}{9 {(4 {(\frac{3}{2})}^{2} - 12 (\frac{3}{2}))}^{\frac{5}{3}}}$

Step 10.1.13

Simplify the numerator.

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Step 10.1.13.1

Multiply $9$ by $4$ .

$- \frac{32 \frac{- 9 + 36}{4}}{9 {(4 {(\frac{3}{2})}^{2} - 12 (\frac{3}{2}))}^{\frac{5}{3}}}$

Step 10.1.13.2

Add $- 9$ and $36$ .

$- \frac{32 (\frac{27}{4})}{9 {(4 {(\frac{3}{2})}^{2} - 12 (\frac{3}{2}))}^{\frac{5}{3}}}$

Step 10.2

Simplify the denominator.

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Step 10.2.1

Simplify each term.

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Step 10.2.1.1

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

$- \frac{32 (\frac{27}{4})}{9 {(4 \frac{3^{2}}{2^{2}} - 12 (\frac{3}{2}))}^{\frac{5}{3}}}$

Step 10.2.1.2

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

$- \frac{32 (\frac{27}{4})}{9 {(4 \frac{9}{2^{2}} - 12 (\frac{3}{2}))}^{\frac{5}{3}}}$

Step 10.2.1.3

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

$- \frac{32 (\frac{27}{4})}{9 {(4 (\frac{9}{4}) - 12 (\frac{3}{2}))}^{\frac{5}{3}}}$

Step 10.2.1.4

Cancel the common factor of $4$ .

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Step 10.2.1.4.1

Cancel the common factor.

$- \frac{32 (\frac{27}{4})}{9 {(4 (\frac{9}{4}) - 12 (\frac{3}{2}))}^{\frac{5}{3}}}$

Step 10.2.1.4.2

Rewrite the expression.

$- \frac{32 (\frac{27}{4})}{9 {(9 - 12 (\frac{3}{2}))}^{\frac{5}{3}}}$

Step 10.2.1.5

Cancel the common factor of $2$ .

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Step 10.2.1.5.1

Factor $2$ out of $- 12$ .

$- \frac{32 (\frac{27}{4})}{9 {(9 + 2 (- 6) \frac{3}{2})}^{\frac{5}{3}}}$

Step 10.2.1.5.2

Cancel the common factor.

$- \frac{32 (\frac{27}{4})}{9 {(9 + 2 \cdot - 6 \frac{3}{2})}^{\frac{5}{3}}}$

Step 10.2.1.5.3

Rewrite the expression.

$- \frac{32 (\frac{27}{4})}{9 {(9 - 6 \cdot 3)}^{\frac{5}{3}}}$

Step 10.2.1.6

Multiply $- 6$ by $3$ .

$- \frac{32 (\frac{27}{4})}{9 {(9 - 18)}^{\frac{5}{3}}}$

Step 10.2.2

Subtract $18$ from $9$ .

$- \frac{32 (\frac{27}{4})}{9 {(- 9)}^{\frac{5}{3}}}$

Step 10.3

Simplify terms.

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Step 10.3.1

Combine $32$ and $\frac{27}{4}$ .

$- \frac{\frac{32 \cdot 27}{4}}{9 {(- 9)}^{\frac{5}{3}}}$

Step 10.3.2

Simplify the expression.

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Step 10.3.2.1

Multiply $32$ by $27$ .

$- \frac{\frac{864}{4}}{9 {(- 9)}^{\frac{5}{3}}}$

Step 10.3.2.2

Divide $864$ by $4$ .

$- \frac{216}{9 {(- 9)}^{\frac{5}{3}}}$

Step 10.3.3

Factor $9$ out of $216$ .

$- \frac{9 \cdot 24}{9 {(- 9)}^{\frac{5}{3}}}$

Step 10.4

Cancel the common factors.

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Step 10.4.1

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

$- \frac{9 \cdot 24}{9 ({(- 9)}^{\frac{5}{3}})}$

Step 10.4.2

Cancel the common factor.

$- \frac{9 \cdot 24}{9 {(- 9)}^{\frac{5}{3}}}$

Step 10.4.3

Rewrite the expression.

$- \frac{24}{{(- 9)}^{\frac{5}{3}}}$

Step 11

$x = \frac{3}{2}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = \frac{3}{2}$ is a local minimum

Step 12

Find the y-value when $x = \frac{3}{2}$ .

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Step 12.1

Replace the variable $x$ with $\frac{3}{2}$ in the expression.

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

Step 12.2

Simplify the result.

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Step 12.2.1

Simplify the expression.

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Step 12.2.1.1

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

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

Step 12.2.1.2

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

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

Step 12.2.1.3

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

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

Step 12.2.2

Cancel the common factor of $4$ .

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Step 12.2.2.1

Cancel the common factor.

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

Step 12.2.2.2

Rewrite the expression.

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

Step 12.2.3

Cancel the common factor of $2$ .

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Step 12.2.3.1

Factor $2$ out of $- 12$ .

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

Step 12.2.3.2

Cancel the common factor.

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

Step 12.2.3.3

Rewrite the expression.

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

Step 12.2.4

Simplify the expression.

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Step 12.2.4.1

Multiply $- 6$ by $3$ .

$f (\frac{3}{2}) = \sqrt[3]{9 - 18}$

Step 12.2.4.2

Subtract $18$ from $9$ .

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

Step 12.2.5

Rewrite $- 9$ as ${(- 1)}^{3} \cdot 9$ .

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Step 12.2.5.1

Rewrite $- 9$ as $- 1 (9)$ .

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

Step 12.2.5.2

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

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

Step 12.2.6

Pull terms out from under the radical.

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

Step 12.2.7

Rewrite $- 1 \sqrt[3]{9}$ as $- \sqrt[3]{9}$ .

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

Step 12.2.8

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

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

Step 13

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.

$- \frac{32 ({(0)}^{2} - 3 \cdot 0 + 9)}{9 {(4 {(0)}^{2} - 12 \cdot 0)}^{\frac{5}{3}}}$

Step 14

Evaluate the second derivative.

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Step 14.1

Simplify each term.

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Step 14.1.1

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

$- \frac{32 ({(0)}^{2} - 3 \cdot 0 + 9)}{9 {(4 \cdot 0 - 12 \cdot 0)}^{\frac{5}{3}}}$

Step 14.1.2

Multiply $4$ by $0$ .

$- \frac{32 ({(0)}^{2} - 3 \cdot 0 + 9)}{9 {(0 - 12 \cdot 0)}^{\frac{5}{3}}}$

Step 14.1.3

Multiply $- 12$ by $0$ .

$- \frac{32 ({(0)}^{2} - 3 \cdot 0 + 9)}{9 {(0 + 0)}^{\frac{5}{3}}}$

Step 14.2

Reduce the expression by cancelling the common factors.

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Step 14.2.1

Add $0$ and $0$ .

$- \frac{32 ({(0)}^{2} - 3 \cdot 0 + 9)}{9 \cdot 0^{\frac{5}{3}}}$

Step 14.2.2

Simplify the expression.

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Step 14.2.2.1

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

$- \frac{32 ({(0)}^{2} - 3 \cdot 0 + 9)}{9 \cdot {(0^{3})}^{\frac{5}{3}}}$

Step 14.2.2.2

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

$- \frac{32 ({(0)}^{2} - 3 \cdot 0 + 9)}{9 \cdot 0^{3 (\frac{5}{3})}}$

Step 14.2.3

Cancel the common factor of $3$ .

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Step 14.2.3.1

Cancel the common factor.

$- \frac{32 ({(0)}^{2} - 3 \cdot 0 + 9)}{9 \cdot 0^{3 (\frac{5}{3})}}$

Step 14.2.3.2

Rewrite the expression.

$- \frac{32 ({(0)}^{2} - 3 \cdot 0 + 9)}{9 \cdot 0^{5}}$

Step 14.2.4

Simplify the expression.

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Step 14.2.4.1

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

$- \frac{32 ({(0)}^{2} - 3 \cdot 0 + 9)}{9 \cdot 0}$

Step 14.2.4.2

Multiply $9$ by $0$ .

$- \frac{32 ({(0)}^{2} - 3 \cdot 0 + 9)}{0}$

Step 14.2.4.3

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

Undefined

$- \frac{32 ({(0)}^{2} - 3 \cdot 0 + 9)}{0}$

Step 14.2.5

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

Undefined

$- \frac{32 ({(0)}^{2} - 3 \cdot 0 + 9)}{0}$

Step 14.3

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

Undefined

Step 15

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

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Step 15.1

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

$(- \infty, 0) \cup (0, \frac{3}{2}) \cup (\frac{3}{2}, 3) \cup (3, \infty)$

Step 15.2

Substitute any number, such as $- 2$ , from the interval $(- \infty, 0)$ in the first derivative $\frac{4 (2 x - 3)}{3 {(4 x^{2} - 12 x)}^{\frac{2}{3}}}$ to check if the result is negative or positive.

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Step 15.2.1

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

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

Step 15.2.2

Simplify the result.

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Step 15.2.2.1

Simplify the numerator.

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Step 15.2.2.1.1

Multiply $2$ by $- 2$ .

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

Step 15.2.2.1.2

Subtract $3$ from $- 4$ .

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

Step 15.2.2.2

Simplify the denominator.

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Step 15.2.2.2.1

Simplify each term.

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Step 15.2.2.2.1.1

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

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

Step 15.2.2.2.1.2

Multiply $4$ by $4$ .

$f' (- 2) = \frac{4 \cdot - 7}{3 {(16 - 12 \cdot - 2)}^{\frac{2}{3}}}$

Step 15.2.2.2.1.3

Multiply $- 12$ by $- 2$ .

$f' (- 2) = \frac{4 \cdot - 7}{3 {(16 + 24)}^{\frac{2}{3}}}$

Step 15.2.2.2.2

Add $16$ and $24$ .

$f' (- 2) = \frac{4 \cdot - 7}{3 \cdot 40^{\frac{2}{3}}}$

Step 15.2.2.3

Simplify the expression.

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Step 15.2.2.3.1

Multiply $4$ by $- 7$ .

$f' (- 2) = \frac{- 28}{3 \cdot 40^{\frac{2}{3}}}$

Step 15.2.2.3.2

Move the negative in front of the fraction.

$f' (- 2) = - \frac{28}{3 \cdot 40^{\frac{2}{3}}}$

Step 15.2.2.4

The final answer is $- \frac{28}{3 \cdot 40^{\frac{2}{3}}}$ .

$- \frac{28}{3 \cdot 40^{\frac{2}{3}}}$

Step 15.3

Substitute any number, such as $1$ , from the interval $(0, \frac{3}{2})$ in the first derivative $\frac{4 (2 x - 3)}{3 {(4 x^{2} - 12 x)}^{\frac{2}{3}}}$ to check if the result is negative or positive.

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Step 15.3.1

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

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

Step 15.3.2

Simplify the result.

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Step 15.3.2.1

Simplify the numerator.

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Step 15.3.2.1.1

Multiply $2$ by $1$ .

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

Step 15.3.2.1.2

Subtract $3$ from $2$ .

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

Step 15.3.2.2

Simplify the denominator.

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Step 15.3.2.2.1

Simplify each term.

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Step 15.3.2.2.1.1

One to any power is one.

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

Step 15.3.2.2.1.2

Multiply $4$ by $1$ .

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

Step 15.3.2.2.1.3

Multiply $- 12$ by $1$ .

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

Step 15.3.2.2.2

Subtract $12$ from $4$ .

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

Step 15.3.2.2.3

Rewrite $- 8$ as ${(- 2)}^{3}$ .

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

Step 15.3.2.2.4

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

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

Step 15.3.2.2.5

Cancel the common factor of $3$ .

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Step 15.3.2.2.5.1

Cancel the common factor.

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

Step 15.3.2.2.5.2

Rewrite the expression.

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

Step 15.3.2.2.6

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

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

Step 15.3.2.3

Reduce the expression by cancelling the common factors.

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Step 15.3.2.3.1

Multiply $4$ by $- 1$ .

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

Step 15.3.2.3.2

Multiply $3$ by $4$ .

$f' (1) = \frac{- 4}{12}$

Step 15.3.2.3.3

Cancel the common factor of $- 4$ and $12$ .

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Step 15.3.2.3.3.1

Factor $4$ out of $- 4$ .

$f' (1) = \frac{4 (- 1)}{12}$

Step 15.3.2.3.3.2

Cancel the common factors.

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Step 15.3.2.3.3.2.1

Factor $4$ out of $12$ .

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

Step 15.3.2.3.3.2.2

Cancel the common factor.

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

Step 15.3.2.3.3.2.3

Rewrite the expression.

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

Step 15.3.2.3.4

Move the negative in front of the fraction.

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

Step 15.3.2.4

The final answer is $- \frac{1}{3}$ .

$- \frac{1}{3}$

Step 15.4

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

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Step 15.4.1

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

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

Step 15.4.2

Simplify the result.

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Step 15.4.2.1

Simplify the numerator.

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Step 15.4.2.1.1

Multiply $2$ by $2$ .

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

Step 15.4.2.1.2

Subtract $3$ from $4$ .

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

Step 15.4.2.2

Simplify the denominator.

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Step 15.4.2.2.1

Simplify each term.

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Step 15.4.2.2.1.1

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

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

Step 15.4.2.2.1.2

Multiply $4$ by $4$ .

$f' (2) = \frac{4 \cdot 1}{3 {(16 - 12 \cdot 2)}^{\frac{2}{3}}}$

Step 15.4.2.2.1.3

Multiply $- 12$ by $2$ .

$f' (2) = \frac{4 \cdot 1}{3 {(16 - 24)}^{\frac{2}{3}}}$

Step 15.4.2.2.2

Subtract $24$ from $16$ .

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

Step 15.4.2.2.3

Rewrite $- 8$ as ${(- 2)}^{3}$ .

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

Step 15.4.2.2.4

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

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

Step 15.4.2.2.5

Cancel the common factor of $3$ .

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Step 15.4.2.2.5.1

Cancel the common factor.

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

Step 15.4.2.2.5.2

Rewrite the expression.

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

Step 15.4.2.2.6

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

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

Step 15.4.2.3

Reduce the expression by cancelling the common factors.

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Step 15.4.2.3.1

Multiply $4$ by $1$ .

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

Step 15.4.2.3.2

Multiply $3$ by $4$ .

$f' (2) = \frac{4}{12}$

Step 15.4.2.3.3

Cancel the common factor of $4$ and $12$ .

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Step 15.4.2.3.3.1

Factor $4$ out of $4$ .

$f' (2) = \frac{4 (1)}{12}$

Step 15.4.2.3.3.2

Cancel the common factors.

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Step 15.4.2.3.3.2.1

Factor $4$ out of $12$ .

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

Step 15.4.2.3.3.2.2

Cancel the common factor.

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

Step 15.4.2.3.3.2.3

Rewrite the expression.

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

Step 15.4.2.4

The final answer is $\frac{1}{3}$ .

$\frac{1}{3}$

Step 15.5

Substitute any number, such as $6$ , from the interval $(3, \infty)$ in the first derivative $\frac{4 (2 x - 3)}{3 {(4 x^{2} - 12 x)}^{\frac{2}{3}}}$ to check if the result is negative or positive.

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Step 15.5.1

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

$f' (6) = \frac{4 (2 (6) - 3)}{3 {(4 {(6)}^{2} - 12 \cdot 6)}^{\frac{2}{3}}}$

Step 15.5.2

Simplify the result.

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Step 15.5.2.1

Factor $3$ out of $4 (2 (6) - 3)$ .

$f' (6) = \frac{3 (4 (2 \cdot (2) - 1))}{3 {(4 {(6)}^{2} - 12 \cdot 6)}^{\frac{2}{3}}}$

Step 15.5.2.2

Cancel the common factors.

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Step 15.5.2.2.1

Factor $3$ out of $3 {(4 {(6)}^{2} - 12 \cdot 6)}^{\frac{2}{3}}$ .

$f' (6) = \frac{3 (4 (2 \cdot (2) - 1))}{3 ({(4 {(6)}^{2} - 12 \cdot 6)}^{\frac{2}{3}})}$

Step 15.5.2.2.2

Cancel the common factor.

$f' (6) = \frac{3 (4 (2 \cdot (2) - 1))}{3 {(4 {(6)}^{2} - 12 \cdot 6)}^{\frac{2}{3}}}$

Step 15.5.2.2.3

Rewrite the expression.

$f' (6) = \frac{4 (2 \cdot (2) - 1)}{{(4 {(6)}^{2} - 12 \cdot 6)}^{\frac{2}{3}}}$

Step 15.5.2.3

Simplify the numerator.

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Step 15.5.2.3.1

Multiply $2$ by $2$ .

$f' (6) = \frac{4 (4 - 1)}{{(4 \cdot 6^{2} - 12 \cdot 6)}^{\frac{2}{3}}}$

Step 15.5.2.3.2

Subtract $1$ from $4$ .

$f' (6) = \frac{4 \cdot 3}{{(4 \cdot 6^{2} - 12 \cdot 6)}^{\frac{2}{3}}}$

Step 15.5.2.4

Simplify the denominator.

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Step 15.5.2.4.1

Simplify each term.

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Step 15.5.2.4.1.1

Raise $6$ to the power of $2$ .

$f' (6) = \frac{4 \cdot 3}{{(4 \cdot 36 - 12 \cdot 6)}^{\frac{2}{3}}}$

Step 15.5.2.4.1.2

Multiply $4$ by $36$ .

$f' (6) = \frac{4 \cdot 3}{{(144 - 12 \cdot 6)}^{\frac{2}{3}}}$

Step 15.5.2.4.1.3

Multiply $- 12$ by $6$ .

$f' (6) = \frac{4 \cdot 3}{{(144 - 72)}^{\frac{2}{3}}}$

Step 15.5.2.4.2

Subtract $72$ from $144$ .

$f' (6) = \frac{4 \cdot 3}{72^{\frac{2}{3}}}$

Step 15.5.2.5

Multiply $4$ by $3$ .

$f' (6) = \frac{12}{72^{\frac{2}{3}}}$

Step 15.5.2.6

The final answer is $\frac{12}{72^{\frac{2}{3}}}$ .

$\frac{12}{72^{\frac{2}{3}}}$

Step 15.6

Since the first derivative did not change signs around $x = 0$ , this is not a local maximum or minimum.

Not a local maximum or minimum

Step 15.7

Since the first derivative changed signs from negative to positive around $x = \frac{3}{2}$ , then $x = \frac{3}{2}$ is a local minimum.

$x = \frac{3}{2}$ is a local minimum

Step 15.8

Since the first derivative did not change signs around $x = 3$ , this is not a local maximum or minimum.

Not a local maximum or minimum

Step 15.9

These are the local extrema for $f (x) = \sqrt[3]{4 x^{2} - 12 x}$ .

$x = \frac{3}{2}$ is a local minimum

Step 16