Calculus Examples

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

Step 1

Find the first derivative of the function.

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Step 1.1

Differentiate.

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Step 1.1.1

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

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

Step 1.1.2

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

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

Step 1.2

Evaluate $\frac{d}{d x} [\sqrt[3]{2 - x^{3}}]$ .

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Step 1.2.1

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

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

Step 1.2.2

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 $g (x) = 2 - x^{3}$ .

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Step 1.2.2.1

To apply the Chain Rule, set $u$ as $2 - x^{3}$ .

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

Step 1.2.2.2

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

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

Step 1.2.2.3

Replace all occurrences of $u$ with $2 - x^{3}$ .

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

Step 1.2.3

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

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

Step 1.2.4

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

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

Step 1.2.5

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

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

Step 1.2.6

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

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

Step 1.2.7

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

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

Step 1.2.8

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

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

Step 1.2.9

Combine the numerators over the common denominator.

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

Step 1.2.10

Simplify the numerator.

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Step 1.2.10.1

Multiply $- 1$ by $3$ .

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

Step 1.2.10.2

Subtract $3$ from $1$ .

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

Step 1.2.11

Move the negative in front of the fraction.

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

Step 1.2.12

Multiply $3$ by $- 1$ .

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

Step 1.2.13

Subtract $3 x^{2}$ from $0$ .

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

Step 1.2.14

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

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

Step 1.2.15

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

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

Step 1.2.16

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

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

Step 1.2.17

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

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

Step 1.2.18

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

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

Step 1.2.19

Cancel the common factors.

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Step 1.2.19.1

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

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

Step 1.2.19.2

Cancel the common factor.

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

Step 1.2.19.3

Rewrite the expression.

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

Step 1.2.20

Move the negative in front of the fraction.

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

Step 1.3

Reorder terms.

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

Step 2

Find the second derivative of the function.

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Step 2.1

By the Sum Rule, the derivative of $- \frac{x^{2}}{{(2 - x^{3})}^{\frac{2}{3}}} + 1$ with respect to $x$ is $\frac{d}{d x} [- \frac{x^{2}}{{(2 - x^{3})}^{\frac{2}{3}}}] + \frac{d}{d x} [1]$ .

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

Step 2.2

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

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Step 2.2.1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = - 1$ and $g (x) = \frac{x^{2}}{{(2 - x^{3})}^{\frac{2}{3}}}$ .

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

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 $g (x) = 2 - x^{3}$ .

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Step 2.2.4.1

To apply the Chain Rule, set $u$ as $2 - x^{3}$ .

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

Step 2.2.4.2

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

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

Step 2.2.4.3

Replace all occurrences of $u$ with $2 - x^{3}$ .

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

Step 2.2.5

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

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

Step 2.2.6

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

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

Step 2.2.7

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

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

Step 2.2.8

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

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

Step 2.2.9

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

Step 2.2.10

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

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

Step 2.2.11

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

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

Step 2.2.12

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

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

Step 2.2.13

Combine the numerators over the common denominator.

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

Step 2.2.14

Simplify the numerator.

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Step 2.2.14.1

Multiply $- 1$ by $3$ .

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

Step 2.2.14.2

Subtract $3$ from $2$ .

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

Step 2.2.15

Move the negative in front of the fraction.

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

Step 2.2.16

Multiply $3$ by $- 1$ .

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

Step 2.2.17

Subtract $3 x^{2}$ from $0$ .

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

Step 2.2.18

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

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

Step 2.2.19

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

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

Step 2.2.20

Multiply $2$ by $- 3$ .

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

Step 2.2.21

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

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

Step 2.2.22

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

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

Step 2.2.23

Factor $3$ out of $- 6 x^{2}$ .

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

Step 2.2.24

Cancel the common factors.

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Step 2.2.24.1

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

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

Step 2.2.24.2

Cancel the common factor.

Step 2.2.24.3

Rewrite the expression.

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

Step 2.2.25

Move the negative in front of the fraction.

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

Step 2.2.26

Multiply $- 1$ by $- 1$ .

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

Step 2.2.27

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

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

Step 2.2.28

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

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

Step 2.2.29

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 2.2.29.1

Move $x^{2}$ .

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

Step 2.2.29.2

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

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

Step 2.2.29.3

Add $2$ and $2$ .

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

Step 2.2.30

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

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

Step 2.2.31

Reorder $2$ and $- x^{3}$ .

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

Step 2.2.32

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

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

Step 2.2.33

Combine the numerators over the common denominator.

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

Step 2.2.34

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

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Step 2.2.34.1

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

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

Step 2.2.34.2

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

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

Step 2.2.34.3

Combine the numerators over the common denominator.

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

Step 2.2.34.4

Add $1$ and $2$ .

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

Step 2.2.34.5

Divide $3$ by $3$ .

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

Step 2.2.35

Simplify $2 x {(- x^{3} + 2)}^{1}$ .

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

Step 2.2.36

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

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Step 2.2.36.1

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

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

Step 2.2.36.2

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

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Step 2.2.36.2.1

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

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

Step 2.2.36.2.2

Multiply $2$ by $2$ .

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

Step 2.2.37

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

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

Step 2.2.38

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

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

Step 2.2.39

Reorder terms.

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

Step 2.2.40

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

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Step 2.2.40.1

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

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

Step 2.2.40.2

Combine the numerators over the common denominator.

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

Step 2.2.40.3

Add $1$ and $4$ .

$f'' (x) = - \frac{2 x (- x^{3} + 2) + 2 x^{4}}{{(- x^{3} + 2)}^{\frac{5}{3}}} + \frac{x^{2}}{{(2 - x^{3})}^{\frac{2}{3}}} \cdot 0 + \frac{d}{d x} (1)$

Step 2.2.41

Multiply $\frac{x^{2}}{{(2 - x^{3})}^{\frac{2}{3}}}$ by $0$ .

$f'' (x) = - \frac{2 x (- x^{3} + 2) + 2 x^{4}}{{(- x^{3} + 2)}^{\frac{5}{3}}} + 0 + \frac{d}{d x} (1)$

Step 2.2.42

Add $- \frac{2 x (- x^{3} + 2) + 2 x^{4}}{{(- x^{3} + 2)}^{\frac{5}{3}}}$ and $0$ .

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

Step 2.3

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$f'' (x) = - \frac{2 x (- x^{3} + 2) + 2 x^{4}}{{(- x^{3} + 2)}^{\frac{5}{3}}} + 0$

Step 2.4

Simplify.

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Step 2.4.1

Apply the distributive property.

$f'' (x) = - \frac{2 x (- x^{3}) + 2 x \cdot 2 + 2 x^{4}}{{(- x^{3} + 2)}^{\frac{5}{3}}} + 0$

Step 2.4.2

Combine terms.

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Step 2.4.2.1

Multiply $- 1$ by $2$ .

$f'' (x) = - \frac{- 2 x \cdot x^{3} + 2 x \cdot 2 + 2 x^{4}}{{(- x^{3} + 2)}^{\frac{5}{3}}} + 0$

Step 2.4.2.2

Multiply $x$ by $x^{3}$ by adding the exponents.

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Step 2.4.2.2.1

Move $x^{3}$ .

$f'' (x) = - \frac{- 2 (x^{3} x) + 2 x \cdot 2 + 2 x^{4}}{{(- x^{3} + 2)}^{\frac{5}{3}}} + 0$

Step 2.4.2.2.2

Multiply $x^{3}$ by $x$ .

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Step 2.4.2.2.2.1

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

$f'' (x) = - \frac{- 2 (x^{3} x) + 2 x \cdot 2 + 2 x^{4}}{{(- x^{3} + 2)}^{\frac{5}{3}}} + 0$

Step 2.4.2.2.2.2

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

$f'' (x) = - \frac{- 2 x^{3 + 1} + 2 x \cdot 2 + 2 x^{4}}{{(- x^{3} + 2)}^{\frac{5}{3}}} + 0$

Step 2.4.2.2.3

Add $3$ and $1$ .

$f'' (x) = - \frac{- 2 x^{4} + 2 x \cdot 2 + 2 x^{4}}{{(- x^{3} + 2)}^{\frac{5}{3}}} + 0$

Step 2.4.2.3

Multiply $2$ by $2$ .

$f'' (x) = - \frac{- 2 x^{4} + 4 x + 2 x^{4}}{{(- x^{3} + 2)}^{\frac{5}{3}}} + 0$

Step 2.4.2.4

Add $- 2 x^{4}$ and $2 x^{4}$ .

$f'' (x) = - \frac{4 x + 0}{{(- x^{3} + 2)}^{\frac{5}{3}}} + 0$

Step 2.4.2.5

Add $4 x$ and $0$ .

$f'' (x) = - \frac{4 x}{{(- x^{3} + 2)}^{\frac{5}{3}}} + 0$

Step 2.4.2.6

Add $- \frac{4 x}{{(- x^{3} + 2)}^{\frac{5}{3}}}$ and $0$ .

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

Step 3

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

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

Step 4

Find the first derivative.

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Step 4.1

Find the first derivative.

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Step 4.1.1

Differentiate.

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Step 4.1.1.1

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

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

Step 4.1.1.2

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

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

Step 4.1.2

Evaluate $\frac{d}{d x} [\sqrt[3]{2 - x^{3}}]$ .

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Step 4.1.2.1

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

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

Step 4.1.2.2

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 $g (x) = 2 - x^{3}$ .

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Step 4.1.2.2.1

To apply the Chain Rule, set $u$ as $2 - x^{3}$ .

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

Step 4.1.2.2.2

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

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

Step 4.1.2.2.3

Replace all occurrences of $u$ with $2 - x^{3}$ .

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

Step 4.1.2.3

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

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

Step 4.1.2.4

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

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

Step 4.1.2.5

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

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

Step 4.1.2.6

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

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

Step 4.1.2.7

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

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

Step 4.1.2.8

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

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

Step 4.1.2.9

Combine the numerators over the common denominator.

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

Step 4.1.2.10

Simplify the numerator.

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Step 4.1.2.10.1

Multiply $- 1$ by $3$ .

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

Step 4.1.2.10.2

Subtract $3$ from $1$ .

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

Step 4.1.2.11

Move the negative in front of the fraction.

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

Step 4.1.2.12

Multiply $3$ by $- 1$ .

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

Step 4.1.2.13

Subtract $3 x^{2}$ from $0$ .

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

Step 4.1.2.14

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

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

Step 4.1.2.15

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

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

Step 4.1.2.16

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

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

Step 4.1.2.17

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

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

Step 4.1.2.18

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

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

Step 4.1.2.19

Cancel the common factors.

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Step 4.1.2.19.1

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

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

Step 4.1.2.19.2

Cancel the common factor.

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

Step 4.1.2.19.3

Rewrite the expression.

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

Step 4.1.2.20

Move the negative in front of the fraction.

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

Step 4.1.3

Reorder terms.

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

Step 4.2

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

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

Step 5

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

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Step 5.1

Set the first derivative equal to $0$ .

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

Step 5.2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x = 1$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

Solve for $x$ .

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Step 6.3.1

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

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

Step 6.3.2

Simplify each side of the equation.

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Step 6.3.2.1

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

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

Step 6.3.2.2

Simplify the left side.

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Step 6.3.2.2.1

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

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Step 6.3.2.2.1.1

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

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

Step 6.3.2.2.1.2

Cancel the common factor of $3$ .

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Step 6.3.2.2.1.2.1

Cancel the common factor.

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

Step 6.3.2.2.1.2.2

Rewrite the expression.

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

Step 6.3.2.3

Simplify the right side.

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Step 6.3.2.3.1

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

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

Step 6.3.3

Solve for $x$ .

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Step 6.3.3.1

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

$2 - x^{3} = 0$

Step 6.3.3.2

Solve for $x$ .

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Step 6.3.3.2.1

Subtract $2$ from both sides of the equation.

$- x^{3} = - 2$

Step 6.3.3.2.2

Divide each term in $- x^{3} = - 2$ by $- 1$ and simplify.

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Step 6.3.3.2.2.1

Divide each term in $- x^{3} = - 2$ by $- 1$ .

$\frac{- x^{3}}{- 1} = \frac{- 2}{- 1}$

Step 6.3.3.2.2.2

Simplify the left side.

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Step 6.3.3.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x^{3}}{1} = \frac{- 2}{- 1}$

Step 6.3.3.2.2.2.2

Divide $x^{3}$ by $1$ .

$x^{3} = \frac{- 2}{- 1}$

Step 6.3.3.2.2.3

Simplify the right side.

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Step 6.3.3.2.2.3.1

Divide $- 2$ by $- 1$ .

$x^{3} = 2$

Step 6.3.3.2.3

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

$x = \sqrt[3]{2}$

Step 7

Critical points to evaluate.

$x = 1, \sqrt[3]{2}$

Step 8

Evaluate the second derivative at $x = 1$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

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

Step 9

Evaluate the second derivative.

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Step 9.1

Multiply $4$ by $1$ .

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

Step 9.2

Simplify the denominator.

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Step 9.2.1

Simplify each term.

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Step 9.2.1.1

One to any power is one.

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

Step 9.2.1.2

Multiply $- 1$ by $1$ .

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

Step 9.2.2

Add $- 1$ and $2$ .

$- \frac{4}{1^{\frac{5}{3}}}$

Step 9.2.3

One to any power is one.

$- \frac{4}{1}$

Step 9.3

Simplify the expression.

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Step 9.3.1

Divide $4$ by $1$ .

$- 1 \cdot 4$

Step 9.3.2

Multiply $- 1$ by $4$ .

$- 4$

Step 10

$x = 1$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = 1$ is a local maximum

Step 11

Find the y-value when $x = 1$ .

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Step 11.1

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

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

Step 11.2

Simplify the result.

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Step 11.2.1

Simplify each term.

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Step 11.2.1.1

One to any power is one.

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

Step 11.2.1.2

Multiply $- 1$ by $1$ .

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

Step 11.2.1.3

Subtract $1$ from $2$ .

$f (1) = 1 + \sqrt[3]{1}$

Step 11.2.1.4

Any root of $1$ is $1$ .

$f (1) = 1 + 1$

Step 11.2.2

Add $1$ and $1$ .

$f (1) = 2$

Step 11.2.3

The final answer is $2$ .

$y = 2$

Step 12

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

$- \frac{4 (\sqrt[3]{2})}{{(- {(\sqrt[3]{2})}^{3} + 2)}^{\frac{5}{3}}}$

Step 13

Evaluate the second derivative.

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Step 13.1

Simplify each term.

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Step 13.1.1

Remove parentheses.

$- \frac{4 (\sqrt[3]{2})}{{(- {\sqrt[3]{2}}^{3} + 2)}^{\frac{5}{3}}}$

Step 13.1.2

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

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Step 13.1.2.1

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

$- \frac{4 (\sqrt[3]{2})}{{(- {(2^{\frac{1}{3}})}^{3} + 2)}^{\frac{5}{3}}}$

Step 13.1.2.2

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

$- \frac{4 (\sqrt[3]{2})}{{(- 2^{\frac{1}{3} \cdot 3} + 2)}^{\frac{5}{3}}}$

Step 13.1.2.3

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

$- \frac{4 (\sqrt[3]{2})}{{(- 2^{\frac{3}{3}} + 2)}^{\frac{5}{3}}}$

Step 13.1.2.4

Cancel the common factor of $3$ .

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Step 13.1.2.4.1

Cancel the common factor.

$- \frac{4 (\sqrt[3]{2})}{{(- 2^{\frac{3}{3}} + 2)}^{\frac{5}{3}}}$

Step 13.1.2.4.2

Rewrite the expression.

$- \frac{4 (\sqrt[3]{2})}{{(- 2^{1} + 2)}^{\frac{5}{3}}}$

Step 13.1.2.5

Evaluate the exponent.

$- \frac{4 (\sqrt[3]{2})}{{(- 1 \cdot 2 + 2)}^{\frac{5}{3}}}$

Step 13.1.3

Multiply $- 1$ by $2$ .

$- \frac{4 (\sqrt[3]{2})}{{(- 2 + 2)}^{\frac{5}{3}}}$

Step 13.2

Reduce the expression by cancelling the common factors.

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Step 13.2.1

Add $- 2$ and $2$ .

$- \frac{4 (\sqrt[3]{2})}{0^{\frac{5}{3}}}$

Step 13.2.2

Simplify the expression.

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Step 13.2.2.1

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

$- \frac{4 (\sqrt[3]{2})}{{(0^{3})}^{\frac{5}{3}}}$

Step 13.2.2.2

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

$- \frac{4 (\sqrt[3]{2})}{0^{3 (\frac{5}{3})}}$

Step 13.2.3

Cancel the common factor of $3$ .

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Step 13.2.3.1

Cancel the common factor.

$- \frac{4 (\sqrt[3]{2})}{0^{3 (\frac{5}{3})}}$

Step 13.2.3.2

Rewrite the expression.

$- \frac{4 (\sqrt[3]{2})}{0^{5}}$

Step 13.2.4

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

$- \frac{4 (\sqrt[3]{2})}{0}$

Step 13.2.5

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

Undefined

$- \frac{4 (\sqrt[3]{2})}{0}$

Step 13.3

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

Undefined

Step 14

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

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Step 14.1

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

$(- \infty, 1) \cup (1, \sqrt[3]{2}) \cup (\sqrt[3]{2}, \infty)$

Step 14.2

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

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Step 14.2.1

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

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

Step 14.2.2

Simplify the result.

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Step 14.2.2.1

Simplify each term.

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Step 14.2.2.1.1

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

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

Step 14.2.2.1.2

Simplify the denominator.

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Step 14.2.2.1.2.1

Simplify each term.

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Step 14.2.2.1.2.1.1

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

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

Step 14.2.2.1.2.1.2

Multiply $- 1$ by $0$ .

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

Step 14.2.2.1.2.2

Add $2$ and $0$ .

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

Step 14.2.2.1.3

Divide $0$ by $2^{\frac{2}{3}}$ .

$f' (0) = - 0 + 1$

Step 14.2.2.1.4

Multiply $- 1$ by $0$ .

$f' (0) = 0 + 1$

Step 14.2.2.2

Add $0$ and $1$ .

$f' (0) = 1$

Step 14.2.2.3

The final answer is $1$ .

$1$

Step 14.3

Substitute any number, such as $1.13$ , from the interval $(1, \sqrt[3]{2})$ in the first derivative $- \frac{x^{2}}{{(2 - x^{3})}^{\frac{2}{3}}} + 1$ to check if the result is negative or positive.

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Step 14.3.1

Replace the variable $x$ with $1.13$ in the expression.

$f' (1.13) = - \frac{{(1.13)}^{2}}{{(2 - {(1.13)}^{3})}^{\frac{2}{3}}} + 1$

Step 14.3.2

Simplify the result.

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Step 14.3.2.1

Simplify each term.

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Step 14.3.2.1.1

Raise $1.13$ to the power of $2$ .

$f' (1.13) = - \frac{1.2769}{{(2 - {1.13}^{3})}^{\frac{2}{3}}} + 1$

Step 14.3.2.1.2

Simplify the denominator.

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Step 14.3.2.1.2.1

Simplify each term.

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Step 14.3.2.1.2.1.1

Raise $1.13$ to the power of $3$ .

$f' (1.13) = - \frac{1.2769}{{(2 - 1 \cdot 1.442897)}^{\frac{2}{3}}} + 1$

Step 14.3.2.1.2.1.2

Multiply $- 1$ by $1.442897$ .

$f' (1.13) = - \frac{1.2769}{{(2 - 1.442897)}^{\frac{2}{3}}} + 1$

Step 14.3.2.1.2.2

Subtract $1.442897$ from $2$ .

$f' (1.13) = - \frac{1.2769}{{0.557103}^{\frac{2}{3}}} + 1$

Step 14.3.2.1.2.3

Raise $0.557103$ to the power of $\frac{2}{3}$ .

$f' (1.13) = - \frac{1.2769}{0.67705455} + 1$

Step 14.3.2.1.3

Divide $1.2769$ by $0.67705455$ .

$f' (1.13) = - 1 \cdot 1.88596323 + 1$

Step 14.3.2.1.4

Multiply $- 1$ by $1.88596323$ .

$f' (1.13) = - 1.88596323 + 1$

Step 14.3.2.2

Add $- 1.88596323$ and $1$ .

$f' (1.13) = - 0.88596323$

Step 14.3.2.3

The final answer is $- 0.88596323$ .

$- 0.88596323$

Step 14.4

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

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Step 14.4.1

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

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

Step 14.4.2

Simplify the result.

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Step 14.4.2.1

Simplify each term.

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Step 14.4.2.1.1

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

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

Step 14.4.2.1.2

Simplify the denominator.

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Step 14.4.2.1.2.1

Simplify each term.

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Step 14.4.2.1.2.1.1

Raise $4$ to the power of $3$ .

$f' (4) = - \frac{16}{{(2 - 1 \cdot 64)}^{\frac{2}{3}}} + 1$

Step 14.4.2.1.2.1.2

Multiply $- 1$ by $64$ .

$f' (4) = - \frac{16}{{(2 - 64)}^{\frac{2}{3}}} + 1$

Step 14.4.2.1.2.2

Subtract $64$ from $2$ .

$f' (4) = - \frac{16}{{(- 62)}^{\frac{2}{3}}} + 1$

Step 14.4.2.2

The final answer is $- \frac{16}{{(- 62)}^{\frac{2}{3}}} + 1$ .

$- \frac{16}{{(- 62)}^{\frac{2}{3}}} + 1$

Step 14.5

Since the first derivative changed signs from positive to negative around $x = 1$ , then $x = 1$ is a local maximum.

$x = 1$ is a local maximum

Step 14.6

Since the first derivative did not change signs around $x = \sqrt[3]{2}$ , this is not a local maximum or minimum.

Not a local maximum or minimum

Step 14.7

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

$x = 1$ is a local maximum

Step 15