Calculus Examples

$f (x) = \sqrt[3]{9 x^{2} + 18}$

Step 1

Find the second derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{9 x^{2} + 18}$ as ${(9 x^{2} + 18)}^{\frac{1}{3}}$ .

$\frac{d}{d x} [{(9 x^{2} + 18)}^{\frac{1}{3}}]$

Step 1.1.2

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 $g (x) = 9 x^{2} + 18$ .

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Step 1.1.2.1

To apply the Chain Rule, set $u$ as $9 x^{2} + 18$ .

$\frac{d}{d u} [u^{\frac{1}{3}}] \frac{d}{d x} [9 x^{2} + 18]$

Step 1.1.2.2

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

$\frac{1}{3} u^{\frac{1}{3} - 1} \frac{d}{d x} [9 x^{2} + 18]$

Step 1.1.2.3

Replace all occurrences of $u$ with $9 x^{2} + 18$ .

$\frac{1}{3} {(9 x^{2} + 18)}^{\frac{1}{3} - 1} \frac{d}{d x} [9 x^{2} + 18]$

Step 1.1.3

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

$\frac{1}{3} {(9 x^{2} + 18)}^{\frac{1}{3} - 1 \cdot \frac{3}{3}} \frac{d}{d x} [9 x^{2} + 18]$

Step 1.1.4

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

$\frac{1}{3} {(9 x^{2} + 18)}^{\frac{1}{3} + \frac{- 1 \cdot 3}{3}} \frac{d}{d x} [9 x^{2} + 18]$

Step 1.1.5

Combine the numerators over the common denominator.

$\frac{1}{3} {(9 x^{2} + 18)}^{\frac{1 - 1 \cdot 3}{3}} \frac{d}{d x} [9 x^{2} + 18]$

Step 1.1.6

Simplify the numerator.

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Step 1.1.6.1

Multiply $- 1$ by $3$ .

$\frac{1}{3} {(9 x^{2} + 18)}^{\frac{1 - 3}{3}} \frac{d}{d x} [9 x^{2} + 18]$

Step 1.1.6.2

Subtract $3$ from $1$ .

$\frac{1}{3} {(9 x^{2} + 18)}^{\frac{- 2}{3}} \frac{d}{d x} [9 x^{2} + 18]$

Step 1.1.7

Combine fractions.

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Step 1.1.7.1

Move the negative in front of the fraction.

$\frac{1}{3} {(9 x^{2} + 18)}^{- \frac{2}{3}} \frac{d}{d x} [9 x^{2} + 18]$

Step 1.1.7.2

Combine $\frac{1}{3}$ and ${(9 x^{2} + 18)}^{- \frac{2}{3}}$ .

$\frac{{(9 x^{2} + 18)}^{- \frac{2}{3}}}{3} \frac{d}{d x} [9 x^{2} + 18]$

Step 1.1.7.3

Move ${(9 x^{2} + 18)}^{- \frac{2}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{3 {(9 x^{2} + 18)}^{\frac{2}{3}}} \frac{d}{d x} [9 x^{2} + 18]$

Step 1.1.8

By the Sum Rule, the derivative of $9 x^{2} + 18$ with respect to $x$ is $\frac{d}{d x} [9 x^{2}] + \frac{d}{d x} [18]$ .

$\frac{1}{3 {(9 x^{2} + 18)}^{\frac{2}{3}}} (\frac{d}{d x} [9 x^{2}] + \frac{d}{d x} [18])$

Step 1.1.9

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

$\frac{1}{3 {(9 x^{2} + 18)}^{\frac{2}{3}}} (9 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [18])$

Step 1.1.10

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

$\frac{1}{3 {(9 x^{2} + 18)}^{\frac{2}{3}}} (9 (2 x) + \frac{d}{d x} [18])$

Step 1.1.11

Multiply $2$ by $9$ .

$\frac{1}{3 {(9 x^{2} + 18)}^{\frac{2}{3}}} (18 x + \frac{d}{d x} [18])$

Step 1.1.12

Since $18$ is constant with respect to $x$ , the derivative of $18$ with respect to $x$ is $0$ .

$\frac{1}{3 {(9 x^{2} + 18)}^{\frac{2}{3}}} (18 x + 0)$

Step 1.1.13

Simplify terms.

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Step 1.1.13.1

Add $18 x$ and $0$ .

$\frac{1}{3 {(9 x^{2} + 18)}^{\frac{2}{3}}} (18 x)$

Step 1.1.13.2

Combine $18$ and $\frac{1}{3 {(9 x^{2} + 18)}^{\frac{2}{3}}}$ .

$\frac{18}{3 {(9 x^{2} + 18)}^{\frac{2}{3}}} x$

Step 1.1.13.3

Combine $\frac{18}{3 {(9 x^{2} + 18)}^{\frac{2}{3}}}$ and $x$ .

$\frac{18 x}{3 {(9 x^{2} + 18)}^{\frac{2}{3}}}$

Step 1.1.13.4

Factor $3$ out of $18 x$ .

$\frac{3 (6 x)}{3 {(9 x^{2} + 18)}^{\frac{2}{3}}}$

Step 1.1.14

Cancel the common factors.

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Step 1.1.14.1

Factor $3$ out of $3 {(9 x^{2} + 18)}^{\frac{2}{3}}$ .

$\frac{3 (6 x)}{3 ({(9 x^{2} + 18)}^{\frac{2}{3}})}$

Step 1.1.14.2

Cancel the common factor.

$\frac{3 (6 x)}{3 {(9 x^{2} + 18)}^{\frac{2}{3}}}$

Step 1.1.14.3

Rewrite the expression.

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

Step 1.2

Find the second derivative.

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Step 1.2.1

Since $6$ is constant with respect to $x$ , the derivative of $\frac{6 x}{{(9 x^{2} + 18)}^{\frac{2}{3}}}$ with respect to $x$ is $6 \frac{d}{d x} [\frac{x}{{(9 x^{2} + 18)}^{\frac{2}{3}}}]$ .

$6 \frac{d}{d x} [\frac{x}{{(9 x^{2} + 18)}^{\frac{2}{3}}}]$

Step 1.2.2

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 $g (x) = {(9 x^{2} + 18)}^{\frac{2}{3}}$ .

$6 \frac{{(9 x^{2} + 18)}^{\frac{2}{3}} \frac{d}{d x} [x] - x \frac{d}{d x} [{(9 x^{2} + 18)}^{\frac{2}{3}}]}{{({(9 x^{2} + 18)}^{\frac{2}{3}})}^{2}}$

Step 1.2.3

Differentiate using the Power Rule.

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Step 1.2.3.1

Multiply the exponents in ${({(9 x^{2} + 18)}^{\frac{2}{3}})}^{2}$ .

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Step 1.2.3.1.1

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

$6 \frac{{(9 x^{2} + 18)}^{\frac{2}{3}} \frac{d}{d x} [x] - x \frac{d}{d x} [{(9 x^{2} + 18)}^{\frac{2}{3}}]}{{(9 x^{2} + 18)}^{\frac{2}{3} \cdot 2}}$

Step 1.2.3.1.2

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

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Step 1.2.3.1.2.1

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

$6 \frac{{(9 x^{2} + 18)}^{\frac{2}{3}} \frac{d}{d x} [x] - x \frac{d}{d x} [{(9 x^{2} + 18)}^{\frac{2}{3}}]}{{(9 x^{2} + 18)}^{\frac{2 \cdot 2}{3}}}$

Step 1.2.3.1.2.2

Multiply $2$ by $2$ .

$6 \frac{{(9 x^{2} + 18)}^{\frac{2}{3}} \frac{d}{d x} [x] - x \frac{d}{d x} [{(9 x^{2} + 18)}^{\frac{2}{3}}]}{{(9 x^{2} + 18)}^{\frac{4}{3}}}$

Step 1.2.3.2

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

$6 \frac{{(9 x^{2} + 18)}^{\frac{2}{3}} \cdot 1 - x \frac{d}{d x} [{(9 x^{2} + 18)}^{\frac{2}{3}}]}{{(9 x^{2} + 18)}^{\frac{4}{3}}}$

Step 1.2.3.3

Multiply ${(9 x^{2} + 18)}^{\frac{2}{3}}$ by $1$ .

$6 \frac{{(9 x^{2} + 18)}^{\frac{2}{3}} - x \frac{d}{d x} [{(9 x^{2} + 18)}^{\frac{2}{3}}]}{{(9 x^{2} + 18)}^{\frac{4}{3}}}$

Step 1.2.4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{2}{3}}$ and $g (x) = 9 x^{2} + 18$ .

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Step 1.2.4.1

To apply the Chain Rule, set $u$ as $9 x^{2} + 18$ .

$6 \frac{{(9 x^{2} + 18)}^{\frac{2}{3}} - x (\frac{d}{d u} [u^{\frac{2}{3}}] \frac{d}{d x} [9 x^{2} + 18])}{{(9 x^{2} + 18)}^{\frac{4}{3}}}$

Step 1.2.4.2

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

$6 \frac{{(9 x^{2} + 18)}^{\frac{2}{3}} - x (\frac{2}{3} u^{\frac{2}{3} - 1} \frac{d}{d x} [9 x^{2} + 18])}{{(9 x^{2} + 18)}^{\frac{4}{3}}}$

Step 1.2.4.3

Replace all occurrences of $u$ with $9 x^{2} + 18$ .

$6 \frac{{(9 x^{2} + 18)}^{\frac{2}{3}} - x (\frac{2}{3} {(9 x^{2} + 18)}^{\frac{2}{3} - 1} \frac{d}{d x} [9 x^{2} + 18])}{{(9 x^{2} + 18)}^{\frac{4}{3}}}$

Step 1.2.5

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

$6 \frac{{(9 x^{2} + 18)}^{\frac{2}{3}} - x (\frac{2}{3} {(9 x^{2} + 18)}^{\frac{2}{3} - 1 \cdot \frac{3}{3}} \frac{d}{d x} [9 x^{2} + 18])}{{(9 x^{2} + 18)}^{\frac{4}{3}}}$

Step 1.2.6

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

$6 \frac{{(9 x^{2} + 18)}^{\frac{2}{3}} - x (\frac{2}{3} {(9 x^{2} + 18)}^{\frac{2}{3} + \frac{- 1 \cdot 3}{3}} \frac{d}{d x} [9 x^{2} + 18])}{{(9 x^{2} + 18)}^{\frac{4}{3}}}$

Step 1.2.7

Combine the numerators over the common denominator.

$6 \frac{{(9 x^{2} + 18)}^{\frac{2}{3}} - x (\frac{2}{3} {(9 x^{2} + 18)}^{\frac{2 - 1 \cdot 3}{3}} \frac{d}{d x} [9 x^{2} + 18])}{{(9 x^{2} + 18)}^{\frac{4}{3}}}$

Step 1.2.8

Simplify the numerator.

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Step 1.2.8.1

Multiply $- 1$ by $3$ .

$6 \frac{{(9 x^{2} + 18)}^{\frac{2}{3}} - x (\frac{2}{3} {(9 x^{2} + 18)}^{\frac{2 - 3}{3}} \frac{d}{d x} [9 x^{2} + 18])}{{(9 x^{2} + 18)}^{\frac{4}{3}}}$

Step 1.2.8.2

Subtract $3$ from $2$ .

$6 \frac{{(9 x^{2} + 18)}^{\frac{2}{3}} - x (\frac{2}{3} {(9 x^{2} + 18)}^{\frac{- 1}{3}} \frac{d}{d x} [9 x^{2} + 18])}{{(9 x^{2} + 18)}^{\frac{4}{3}}}$

Step 1.2.9

Combine fractions.

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Step 1.2.9.1

Move the negative in front of the fraction.

$6 \frac{{(9 x^{2} + 18)}^{\frac{2}{3}} - x (\frac{2}{3} {(9 x^{2} + 18)}^{- \frac{1}{3}} \frac{d}{d x} [9 x^{2} + 18])}{{(9 x^{2} + 18)}^{\frac{4}{3}}}$

Step 1.2.9.2

Combine $\frac{2}{3}$ and ${(9 x^{2} + 18)}^{- \frac{1}{3}}$ .

$6 \frac{{(9 x^{2} + 18)}^{\frac{2}{3}} - x (\frac{2 {(9 x^{2} + 18)}^{- \frac{1}{3}}}{3} \frac{d}{d x} [9 x^{2} + 18])}{{(9 x^{2} + 18)}^{\frac{4}{3}}}$

Step 1.2.9.3

Move ${(9 x^{2} + 18)}^{- \frac{1}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$6 \frac{{(9 x^{2} + 18)}^{\frac{2}{3}} - x (\frac{2}{3 {(9 x^{2} + 18)}^{\frac{1}{3}}} \frac{d}{d x} [9 x^{2} + 18])}{{(9 x^{2} + 18)}^{\frac{4}{3}}}$

Step 1.2.9.4

Combine $\frac{2}{3 {(9 x^{2} + 18)}^{\frac{1}{3}}}$ and $x$ .

$6 \frac{{(9 x^{2} + 18)}^{\frac{2}{3}} - \frac{2 x}{3 {(9 x^{2} + 18)}^{\frac{1}{3}}} \frac{d}{d x} [9 x^{2} + 18]}{{(9 x^{2} + 18)}^{\frac{4}{3}}}$

Step 1.2.10

By the Sum Rule, the derivative of $9 x^{2} + 18$ with respect to $x$ is $\frac{d}{d x} [9 x^{2}] + \frac{d}{d x} [18]$ .

$6 \frac{{(9 x^{2} + 18)}^{\frac{2}{3}} - \frac{2 x}{3 {(9 x^{2} + 18)}^{\frac{1}{3}}} (\frac{d}{d x} [9 x^{2}] + \frac{d}{d x} [18])}{{(9 x^{2} + 18)}^{\frac{4}{3}}}$

Step 1.2.11

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

$6 \frac{{(9 x^{2} + 18)}^{\frac{2}{3}} - \frac{2 x}{3 {(9 x^{2} + 18)}^{\frac{1}{3}}} (9 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [18])}{{(9 x^{2} + 18)}^{\frac{4}{3}}}$

Step 1.2.12

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

$6 \frac{{(9 x^{2} + 18)}^{\frac{2}{3}} - \frac{2 x}{3 {(9 x^{2} + 18)}^{\frac{1}{3}}} (9 (2 x) + \frac{d}{d x} [18])}{{(9 x^{2} + 18)}^{\frac{4}{3}}}$

Step 1.2.13

Multiply $2$ by $9$ .

$6 \frac{{(9 x^{2} + 18)}^{\frac{2}{3}} - \frac{2 x}{3 {(9 x^{2} + 18)}^{\frac{1}{3}}} (18 x + \frac{d}{d x} [18])}{{(9 x^{2} + 18)}^{\frac{4}{3}}}$

Step 1.2.14

Since $18$ is constant with respect to $x$ , the derivative of $18$ with respect to $x$ is $0$ .

$6 \frac{{(9 x^{2} + 18)}^{\frac{2}{3}} - \frac{2 x}{3 {(9 x^{2} + 18)}^{\frac{1}{3}}} (18 x + 0)}{{(9 x^{2} + 18)}^{\frac{4}{3}}}$

Step 1.2.15

Combine fractions.

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Step 1.2.15.1

Add $18 x$ and $0$ .

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

Step 1.2.15.2

Multiply $18$ by $- 1$ .

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

Step 1.2.15.3

Combine $- 18$ and $\frac{2 x}{3 {(9 x^{2} + 18)}^{\frac{1}{3}}}$ .

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

Step 1.2.15.4

Multiply $2$ by $- 18$ .

$6 \frac{{(9 x^{2} + 18)}^{\frac{2}{3}} + \frac{- 36 x}{3 {(9 x^{2} + 18)}^{\frac{1}{3}}} x}{{(9 x^{2} + 18)}^{\frac{4}{3}}}$

Step 1.2.15.5

Combine $\frac{- 36 x}{3 {(9 x^{2} + 18)}^{\frac{1}{3}}}$ and $x$ .

$6 \frac{{(9 x^{2} + 18)}^{\frac{2}{3}} + \frac{- 36 x \cdot x}{3 {(9 x^{2} + 18)}^{\frac{1}{3}}}}{{(9 x^{2} + 18)}^{\frac{4}{3}}}$

Step 1.2.16

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

$6 \frac{{(9 x^{2} + 18)}^{\frac{2}{3}} + \frac{- 36 (x^{1} x)}{3 {(9 x^{2} + 18)}^{\frac{1}{3}}}}{{(9 x^{2} + 18)}^{\frac{4}{3}}}$

Step 1.2.17

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

$6 \frac{{(9 x^{2} + 18)}^{\frac{2}{3}} + \frac{- 36 (x^{1} x^{1})}{3 {(9 x^{2} + 18)}^{\frac{1}{3}}}}{{(9 x^{2} + 18)}^{\frac{4}{3}}}$

Step 1.2.18

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

$6 \frac{{(9 x^{2} + 18)}^{\frac{2}{3}} + \frac{- 36 x^{1 + 1}}{3 {(9 x^{2} + 18)}^{\frac{1}{3}}}}{{(9 x^{2} + 18)}^{\frac{4}{3}}}$

Step 1.2.19

Add $1$ and $1$ .

$6 \frac{{(9 x^{2} + 18)}^{\frac{2}{3}} + \frac{- 36 x^{2}}{3 {(9 x^{2} + 18)}^{\frac{1}{3}}}}{{(9 x^{2} + 18)}^{\frac{4}{3}}}$

Step 1.2.20

Factor $3$ out of $- 36 x^{2}$ .

$6 \frac{{(9 x^{2} + 18)}^{\frac{2}{3}} + \frac{3 (- 12 x^{2})}{3 {(9 x^{2} + 18)}^{\frac{1}{3}}}}{{(9 x^{2} + 18)}^{\frac{4}{3}}}$

Step 1.2.21

Cancel the common factors.

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Step 1.2.21.1

Factor $3$ out of $3 {(9 x^{2} + 18)}^{\frac{1}{3}}$ .

$6 \frac{{(9 x^{2} + 18)}^{\frac{2}{3}} + \frac{3 (- 12 x^{2})}{3 ({(9 x^{2} + 18)}^{\frac{1}{3}})}}{{(9 x^{2} + 18)}^{\frac{4}{3}}}$

Step 1.2.21.2

Cancel the common factor.

$6 \frac{{(9 x^{2} + 18)}^{\frac{2}{3}} + \frac{3 (- 12 x^{2})}{3 {(9 x^{2} + 18)}^{\frac{1}{3}}}}{{(9 x^{2} + 18)}^{\frac{4}{3}}}$

Step 1.2.21.3

Rewrite the expression.

$6 \frac{{(9 x^{2} + 18)}^{\frac{2}{3}} + \frac{- 12 x^{2}}{{(9 x^{2} + 18)}^{\frac{1}{3}}}}{{(9 x^{2} + 18)}^{\frac{4}{3}}}$

Step 1.2.22

Move the negative in front of the fraction.

$6 \frac{{(9 x^{2} + 18)}^{\frac{2}{3}} - \frac{12 x^{2}}{{(9 x^{2} + 18)}^{\frac{1}{3}}}}{{(9 x^{2} + 18)}^{\frac{4}{3}}}$

Step 1.2.23

To write ${(9 x^{2} + 18)}^{\frac{2}{3}}$ as a fraction with a common denominator, multiply by $\frac{{(9 x^{2} + 18)}^{\frac{1}{3}}}{{(9 x^{2} + 18)}^{\frac{1}{3}}}$ .

$6 \frac{\frac{{(9 x^{2} + 18)}^{\frac{2}{3}} {(9 x^{2} + 18)}^{\frac{1}{3}}}{{(9 x^{2} + 18)}^{\frac{1}{3}}} - \frac{12 x^{2}}{{(9 x^{2} + 18)}^{\frac{1}{3}}}}{{(9 x^{2} + 18)}^{\frac{4}{3}}}$

Step 1.2.24

Combine the numerators over the common denominator.

$6 \frac{\frac{{(9 x^{2} + 18)}^{\frac{2}{3}} {(9 x^{2} + 18)}^{\frac{1}{3}} - 12 x^{2}}{{(9 x^{2} + 18)}^{\frac{1}{3}}}}{{(9 x^{2} + 18)}^{\frac{4}{3}}}$

Step 1.2.25

Multiply ${(9 x^{2} + 18)}^{\frac{2}{3}}$ by ${(9 x^{2} + 18)}^{\frac{1}{3}}$ by adding the exponents.

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Step 1.2.25.1

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

$6 \frac{\frac{{(9 x^{2} + 18)}^{\frac{2}{3} + \frac{1}{3}} - 12 x^{2}}{{(9 x^{2} + 18)}^{\frac{1}{3}}}}{{(9 x^{2} + 18)}^{\frac{4}{3}}}$

Step 1.2.25.2

Combine the numerators over the common denominator.

$6 \frac{\frac{{(9 x^{2} + 18)}^{\frac{2 + 1}{3}} - 12 x^{2}}{{(9 x^{2} + 18)}^{\frac{1}{3}}}}{{(9 x^{2} + 18)}^{\frac{4}{3}}}$

Step 1.2.25.3

Add $2$ and $1$ .

$6 \frac{\frac{{(9 x^{2} + 18)}^{\frac{3}{3}} - 12 x^{2}}{{(9 x^{2} + 18)}^{\frac{1}{3}}}}{{(9 x^{2} + 18)}^{\frac{4}{3}}}$

Step 1.2.25.4

Divide $3$ by $3$ .

$6 \frac{\frac{{(9 x^{2} + 18)}^{1} - 12 x^{2}}{{(9 x^{2} + 18)}^{\frac{1}{3}}}}{{(9 x^{2} + 18)}^{\frac{4}{3}}}$

Step 1.2.26

Simplify ${(9 x^{2} + 18)}^{1}$ .

$6 \frac{\frac{9 x^{2} + 18 - 12 x^{2}}{{(9 x^{2} + 18)}^{\frac{1}{3}}}}{{(9 x^{2} + 18)}^{\frac{4}{3}}}$

Step 1.2.27

Subtract $12 x^{2}$ from $9 x^{2}$ .

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

Step 1.2.28

Rewrite $\frac{\frac{- 3 x^{2} + 18}{{(9 x^{2} + 18)}^{\frac{1}{3}}}}{{(9 x^{2} + 18)}^{\frac{4}{3}}}$ as a product.

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

Step 1.2.29

Multiply $\frac{- 3 x^{2} + 18}{{(9 x^{2} + 18)}^{\frac{1}{3}}}$ by $\frac{1}{{(9 x^{2} + 18)}^{\frac{4}{3}}}$ .

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

Step 1.2.30

Multiply ${(9 x^{2} + 18)}^{\frac{1}{3}}$ by ${(9 x^{2} + 18)}^{\frac{4}{3}}$ by adding the exponents.

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Step 1.2.30.1

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

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

Step 1.2.30.2

Combine the numerators over the common denominator.

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

Step 1.2.30.3

Add $1$ and $4$ .

$6 \frac{- 3 x^{2} + 18}{{(9 x^{2} + 18)}^{\frac{5}{3}}}$

Step 1.2.31

Combine $6$ and $\frac{- 3 x^{2} + 18}{{(9 x^{2} + 18)}^{\frac{5}{3}}}$ .

$\frac{6 (- 3 x^{2} + 18)}{{(9 x^{2} + 18)}^{\frac{5}{3}}}$

Step 1.2.32

Simplify.

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Step 1.2.32.1

Apply the distributive property.

$\frac{6 (- 3 x^{2}) + 6 \cdot 18}{{(9 x^{2} + 18)}^{\frac{5}{3}}}$

Step 1.2.32.2

Simplify each term.

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Step 1.2.32.2.1

Multiply $- 3$ by $6$ .

$\frac{- 18 x^{2} + 6 \cdot 18}{{(9 x^{2} + 18)}^{\frac{5}{3}}}$

Step 1.2.32.2.2

Multiply $6$ by $18$ .

$\frac{- 18 x^{2} + 108}{{(9 x^{2} + 18)}^{\frac{5}{3}}}$

Step 1.2.32.3

Factor $18$ out of $- 18 x^{2} + 108$ .

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Step 1.2.32.3.1

Factor $18$ out of $- 18 x^{2}$ .

$\frac{18 (- x^{2}) + 108}{{(9 x^{2} + 18)}^{\frac{5}{3}}}$

Step 1.2.32.3.2

Factor $18$ out of $108$ .

$\frac{18 (- x^{2}) + 18 (6)}{{(9 x^{2} + 18)}^{\frac{5}{3}}}$

Step 1.2.32.3.3

Factor $18$ out of $18 (- x^{2}) + 18 (6)$ .

$\frac{18 (- x^{2} + 6)}{{(9 x^{2} + 18)}^{\frac{5}{3}}}$

Step 1.2.32.4

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

$\frac{18 (- (x^{2}) + 6)}{{(9 x^{2} + 18)}^{\frac{5}{3}}}$

Step 1.2.32.5

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

$\frac{18 (- (x^{2}) - 1 (- 6))}{{(9 x^{2} + 18)}^{\frac{5}{3}}}$

Step 1.2.32.6

Factor $- 1$ out of $- (x^{2}) - 1 (- 6)$ .

$\frac{18 (- (x^{2} - 6))}{{(9 x^{2} + 18)}^{\frac{5}{3}}}$

Step 1.2.32.7

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

$\frac{18 (- 1 (x^{2} - 6))}{{(9 x^{2} + 18)}^{\frac{5}{3}}}$

Step 1.2.32.8

Move the negative in front of the fraction.

$f'' (x) = - \frac{18 (x^{2} - 6)}{{(9 x^{2} + 18)}^{\frac{5}{3}}}$

Step 1.3

The second derivative of $f (x)$ with respect to $x$ is $- \frac{18 (x^{2} - 6)}{{(9 x^{2} + 18)}^{\frac{5}{3}}}$ .

$- \frac{18 (x^{2} - 6)}{{(9 x^{2} + 18)}^{\frac{5}{3}}}$

Step 2

Set the second derivative equal to $0$ then solve the equation $- \frac{18 (x^{2} - 6)}{{(9 x^{2} + 18)}^{\frac{5}{3}}} = 0$ .

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Step 2.1

Set the second derivative equal to $0$ .

$- \frac{18 (x^{2} - 6)}{{(9 x^{2} + 18)}^{\frac{5}{3}}} = 0$

Step 2.2

Set the numerator equal to zero.

$18 (x^{2} - 6) = 0$

Step 2.3

Solve the equation for $x$ .

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Step 2.3.1

Divide each term in $18 (x^{2} - 6) = 0$ by $18$ and simplify.

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Step 2.3.1.1

Divide each term in $18 (x^{2} - 6) = 0$ by $18$ .

$\frac{18 (x^{2} - 6)}{18} = \frac{0}{18}$

Step 2.3.1.2

Simplify the left side.

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Step 2.3.1.2.1

Cancel the common factor of $18$ .

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Step 2.3.1.2.1.1

Cancel the common factor.

$\frac{18 (x^{2} - 6)}{18} = \frac{0}{18}$

Step 2.3.1.2.1.2

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

$x^{2} - 6 = \frac{0}{18}$

Step 2.3.1.3

Simplify the right side.

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Step 2.3.1.3.1

Divide $0$ by $18$ .

$x^{2} - 6 = 0$

Step 2.3.2

Add $6$ to both sides of the equation.

$x^{2} = 6$

Step 2.3.3

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

$x = \pm \sqrt{6}$

Step 2.3.4

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

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Step 2.3.4.1

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

$x = \sqrt{6}$

Step 2.3.4.2

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

$x = - \sqrt{6}$

Step 2.3.4.3

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

$x = \sqrt{6}, - \sqrt{6}$

Step 3

Find the points where the second derivative is $0$ .

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Step 3.1

Substitute $\sqrt{6}$ in $f (x) = \sqrt[3]{9 x^{2} + 18}$ to find the value of $y$ .

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Step 3.1.1

Replace the variable $x$ with $\sqrt{6}$ in the expression.

$f (\sqrt{6}) = \sqrt[3]{9 {(\sqrt{6})}^{2} + 18}$

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 3.1.2.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$f (\sqrt{6}) = \sqrt[3]{9 {(6^{\frac{1}{2}})}^{2} + 18}$

Step 3.1.2.1.2

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

$f (\sqrt{6}) = \sqrt[3]{9 \cdot 6^{\frac{1}{2} \cdot 2} + 18}$

Step 3.1.2.1.3

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

$f (\sqrt{6}) = \sqrt[3]{9 \cdot 6^{\frac{2}{2}} + 18}$

Step 3.1.2.1.4

Cancel the common factor of $2$ .

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Step 3.1.2.1.4.1

Cancel the common factor.

$f (\sqrt{6}) = \sqrt[3]{9 \cdot 6^{\frac{2}{2}} + 18}$

Step 3.1.2.1.4.2

Rewrite the expression.

$f (\sqrt{6}) = \sqrt[3]{9 \cdot 6 + 18}$

Step 3.1.2.1.5

Evaluate the exponent.

$f (\sqrt{6}) = \sqrt[3]{9 \cdot 6 + 18}$

Step 3.1.2.2

Simplify the expression.

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Step 3.1.2.2.1

Multiply $9$ by $6$ .

$f (\sqrt{6}) = \sqrt[3]{54 + 18}$

Step 3.1.2.2.2

Add $54$ and $18$ .

$f (\sqrt{6}) = \sqrt[3]{72}$

Step 3.1.2.3

Rewrite $72$ as $2^{3} \cdot 9$ .

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Step 3.1.2.3.1

Factor $8$ out of $72$ .

$f (\sqrt{6}) = \sqrt[3]{8 (9)}$

Step 3.1.2.3.2

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

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

Step 3.1.2.4

Pull terms out from under the radical.

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

Step 3.1.2.5

The final answer is $2 \sqrt[3]{9}$ .

$2 \sqrt[3]{9}$

Step 3.2

The point found by substituting $\sqrt{6}$ in $f (x) = \sqrt[3]{9 x^{2} + 18}$ is $(\sqrt{6}, 2 \sqrt[3]{9})$ . This point can be an inflection point.

$(\sqrt{6}, 2 \sqrt[3]{9})$

Step 3.3

Substitute $- \sqrt{6}$ in $f (x) = \sqrt[3]{9 x^{2} + 18}$ to find the value of $y$ .

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Step 3.3.1

Replace the variable $x$ with $- \sqrt{6}$ in the expression.

$f (- \sqrt{6}) = \sqrt[3]{9 {(- \sqrt{6})}^{2} + 18}$

Step 3.3.2

Simplify the result.

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Step 3.3.2.1

Simplify the expression.

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Step 3.3.2.1.1

Apply the product rule to $- \sqrt{6}$ .

$f (- \sqrt{6}) = \sqrt[3]{9 ({(- 1)}^{2} {\sqrt{6}}^{2}) + 18}$

Step 3.3.2.1.2

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

$f (- \sqrt{6}) = \sqrt[3]{9 (1 {\sqrt{6}}^{2}) + 18}$

Step 3.3.2.1.3

Multiply ${\sqrt{6}}^{2}$ by $1$ .

$f (- \sqrt{6}) = \sqrt[3]{9 {\sqrt{6}}^{2} + 18}$

Step 3.3.2.2

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 3.3.2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$f (- \sqrt{6}) = \sqrt[3]{9 {(6^{\frac{1}{2}})}^{2} + 18}$

Step 3.3.2.2.2

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

$f (- \sqrt{6}) = \sqrt[3]{9 \cdot 6^{\frac{1}{2} \cdot 2} + 18}$

Step 3.3.2.2.3

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

$f (- \sqrt{6}) = \sqrt[3]{9 \cdot 6^{\frac{2}{2}} + 18}$

Step 3.3.2.2.4

Cancel the common factor of $2$ .

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Step 3.3.2.2.4.1

Cancel the common factor.

$f (- \sqrt{6}) = \sqrt[3]{9 \cdot 6^{\frac{2}{2}} + 18}$

Step 3.3.2.2.4.2

Rewrite the expression.

$f (- \sqrt{6}) = \sqrt[3]{9 \cdot 6 + 18}$

Step 3.3.2.2.5

Evaluate the exponent.

$f (- \sqrt{6}) = \sqrt[3]{9 \cdot 6 + 18}$

Step 3.3.2.3

Simplify the expression.

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Step 3.3.2.3.1

Multiply $9$ by $6$ .

$f (- \sqrt{6}) = \sqrt[3]{54 + 18}$

Step 3.3.2.3.2

Add $54$ and $18$ .

$f (- \sqrt{6}) = \sqrt[3]{72}$

Step 3.3.2.4

Rewrite $72$ as $2^{3} \cdot 9$ .

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Step 3.3.2.4.1

Factor $8$ out of $72$ .

$f (- \sqrt{6}) = \sqrt[3]{8 (9)}$

Step 3.3.2.4.2

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

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

Step 3.3.2.5

Pull terms out from under the radical.

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

Step 3.3.2.6

The final answer is $2 \sqrt[3]{9}$ .

$2 \sqrt[3]{9}$

Step 3.4

The point found by substituting $- \sqrt{6}$ in $f (x) = \sqrt[3]{9 x^{2} + 18}$ is $(- \sqrt{6}, 2 \sqrt[3]{9})$ . This point can be an inflection point.

$(- \sqrt{6}, 2 \sqrt[3]{9})$

Step 3.5

Determine the points that could be inflection points.

$(\sqrt{6}, 2 \sqrt[3]{9}), (- \sqrt{6}, 2 \sqrt[3]{9})$

Step 4

Split $(- \infty, \infty)$ into intervals around the points that could potentially be inflection points.

$(- \infty, - \sqrt{6}) \cup (- \sqrt{6}, \sqrt{6}) \cup (\sqrt{6}, \infty)$

Step 5

Substitute a value from the interval $(- \infty, - \sqrt{6})$ into the second derivative to determine if it is increasing or decreasing.

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Step 5.1

Replace the variable $x$ with $- 2.54948974$ in the expression.

$f'' (- 2.54948974) = - \frac{18 ({(- 2.54948974)}^{2} - 6)}{{(9 {(- 2.54948974)}^{2} + 18)}^{\frac{5}{3}}}$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify the numerator.

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Step 5.2.1.1

Raise $- 2.54948974$ to the power of $2$ .

$f'' (- 2.54948974) = - \frac{18 (6.49989794 - 6)}{{(9 {(- 2.54948974)}^{2} + 18)}^{\frac{5}{3}}}$

Step 5.2.1.2

Subtract $6$ from $6.49989794$ .

$f'' (- 2.54948974) = - \frac{18 \cdot 0.49989794}{{(9 {(- 2.54948974)}^{2} + 18)}^{\frac{5}{3}}}$

Step 5.2.2

Simplify the denominator.

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Step 5.2.2.1

Simplify each term.

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Step 5.2.2.1.1

Raise $- 2.54948974$ to the power of $2$ .

$f'' (- 2.54948974) = - \frac{18 \cdot 0.49989794}{{(9 \cdot 6.49989794 + 18)}^{\frac{5}{3}}}$

Step 5.2.2.1.2

Multiply $9$ by $6.49989794$ .

$f'' (- 2.54948974) = - \frac{18 \cdot 0.49989794}{{(58.49908153 + 18)}^{\frac{5}{3}}}$

Step 5.2.2.2

Add $58.49908153$ and $18$ .

$f'' (- 2.54948974) = - \frac{18 \cdot 0.49989794}{{76.49908153}^{\frac{5}{3}}}$

Step 5.2.2.3

Raise $76.49908153$ to the power of $\frac{5}{3}$ .

$f'' (- 2.54948974) = - \frac{18 \cdot 0.49989794}{1378.56432329}$

Step 5.2.3

Simplify the expression.

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Step 5.2.3.1

Multiply $18$ by $0.49989794$ .

$f'' (- 2.54948974) = - \frac{8.99816307}{1378.56432329}$

Step 5.2.3.2

Divide $8.99816307$ by $1378.56432329$ .

$f'' (- 2.54948974) = - 1 \cdot 0.00652719$

Step 5.2.3.3

Multiply $- 1$ by $0.00652719$ .

$f'' (- 2.54948974) = - 0.00652719$

Step 5.2.4

The final answer is $- 0.00652719$ .

$- 0.00652719$

Step 5.3

At $- 2.54948974$ , the second derivative is $- 0.00652719$ . Since this is negative, the second derivative is decreasing on the interval $(- \infty, - \sqrt{6})$

Decreasing on $(- \infty, - \sqrt{6})$ since $f'' (x) < 0$

Step 6

Substitute a value from the interval $(- \sqrt{6}, \sqrt{6})$ into the second derivative to determine if it is increasing or decreasing.

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Step 6.1

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

$f'' (0) = - \frac{18 ({(0)}^{2} - 6)}{{(9 {(0)}^{2} + 18)}^{\frac{5}{3}}}$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify the numerator.

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Step 6.2.1.1

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

$f'' (0) = - \frac{18 (0 - 6)}{{(9 \cdot 0^{2} + 18)}^{\frac{5}{3}}}$

Step 6.2.1.2

Subtract $6$ from $0$ .

$f'' (0) = - \frac{18 \cdot - 6}{{(9 \cdot 0^{2} + 18)}^{\frac{5}{3}}}$

Step 6.2.2

Simplify the denominator.

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Step 6.2.2.1

Simplify each term.

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Step 6.2.2.1.1

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

$f'' (0) = - \frac{18 \cdot - 6}{{(9 \cdot 0 + 18)}^{\frac{5}{3}}}$

Step 6.2.2.1.2

Multiply $9$ by $0$ .

$f'' (0) = - \frac{18 \cdot - 6}{{(0 + 18)}^{\frac{5}{3}}}$

Step 6.2.2.2

Add $0$ and $18$ .

$f'' (0) = - \frac{18 \cdot - 6}{18^{\frac{5}{3}}}$

Step 6.2.3

Simplify the expression.

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Step 6.2.3.1

Multiply $18$ by $- 6$ .

$f'' (0) = - \frac{- 108}{18^{\frac{5}{3}}}$

Step 6.2.3.2

Move the negative in front of the fraction.

$f'' (0) = \frac{108}{18^{\frac{5}{3}}}$

Step 6.2.4

The final answer is $\frac{108}{18^{\frac{5}{3}}}$ .

$\frac{108}{18^{\frac{5}{3}}}$

Step 6.3

At $0$ , the second derivative is $0.87358046$ . Since this is positive, the second derivative is increasing on the interval $(- \sqrt{6}, \sqrt{6})$ .

Increasing on $(- \sqrt{6}, \sqrt{6})$ since $f'' (x) > 0$

Step 7

Substitute a value from the interval $(\sqrt{6}, \infty)$ into the second derivative to determine if it is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $2.54948974$ in the expression.

$f'' (2.54948974) = - \frac{18 ({(2.54948974)}^{2} - 6)}{{(9 {(2.54948974)}^{2} + 18)}^{\frac{5}{3}}}$

Step 7.2

Simplify the result.

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Step 7.2.1

Simplify the numerator.

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Step 7.2.1.1

Raise $2.54948974$ to the power of $2$ .

$f'' (2.54948974) = - \frac{18 (6.49989794 - 6)}{{(9 \cdot {2.54948974}^{2} + 18)}^{\frac{5}{3}}}$

Step 7.2.1.2

Subtract $6$ from $6.49989794$ .

$f'' (2.54948974) = - \frac{18 \cdot 0.49989794}{{(9 \cdot {2.54948974}^{2} + 18)}^{\frac{5}{3}}}$

Step 7.2.2

Simplify the denominator.

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Step 7.2.2.1

Simplify each term.

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Step 7.2.2.1.1

Raise $2.54948974$ to the power of $2$ .

$f'' (2.54948974) = - \frac{18 \cdot 0.49989794}{{(9 \cdot 6.49989794 + 18)}^{\frac{5}{3}}}$

Step 7.2.2.1.2

Multiply $9$ by $6.49989794$ .

$f'' (2.54948974) = - \frac{18 \cdot 0.49989794}{{(58.49908153 + 18)}^{\frac{5}{3}}}$

Step 7.2.2.2

Add $58.49908153$ and $18$ .

$f'' (2.54948974) = - \frac{18 \cdot 0.49989794}{{76.49908153}^{\frac{5}{3}}}$

Step 7.2.2.3

Raise $76.49908153$ to the power of $\frac{5}{3}$ .

$f'' (2.54948974) = - \frac{18 \cdot 0.49989794}{1378.56432329}$

Step 7.2.3

Simplify the expression.

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Step 7.2.3.1

Multiply $18$ by $0.49989794$ .

$f'' (2.54948974) = - \frac{8.99816307}{1378.56432329}$

Step 7.2.3.2

Divide $8.99816307$ by $1378.56432329$ .

$f'' (2.54948974) = - 1 \cdot 0.00652719$

Step 7.2.3.3

Multiply $- 1$ by $0.00652719$ .

$f'' (2.54948974) = - 0.00652719$

Step 7.2.4

The final answer is $- 0.00652719$ .

$- 0.00652719$

Step 7.3

At $2.54948974$ , the second derivative is $- 0.00652719$ . Since this is negative, the second derivative is decreasing on the interval $(\sqrt{6}, \infty)$

Decreasing on $(\sqrt{6}, \infty)$ since $f'' (x) < 0$

Step 8

An inflection point is a point on a curve at which the concavity changes sign from plus to minus or from minus to plus. The inflection points in this case are $(- \sqrt{6}, 2 \sqrt[3]{9}), (\sqrt{6}, 2 \sqrt[3]{9})$ .

$(- \sqrt{6}, 2 \sqrt[3]{9}), (\sqrt{6}, 2 \sqrt[3]{9})$

Step 9