Calculus Examples

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

Step 1

Write $x^{\frac{1}{3}} (x + 4)$ as a function.

$f (x) = x^{\frac{1}{3}} (x + 4)$

Step 2

Find the first derivative of the function.

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

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

Step 2.2

Differentiate.

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

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

$x^{\frac{1}{3}} (1 + 0) + (x + 4) \frac{d}{d x} [x^{\frac{1}{3}}]$

Step 2.2.4

Simplify the expression.

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Step 2.2.4.1

Add $1$ and $0$ .

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

Step 2.2.4.2

Multiply $x^{\frac{1}{3}}$ by $1$ .

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

Step 2.2.5

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Combine the numerators over the common denominator.

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

Step 2.6

Simplify the numerator.

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Step 2.6.1

Multiply $- 1$ by $3$ .

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

Step 2.6.2

Subtract $3$ from $1$ .

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

Step 2.7

Move the negative in front of the fraction.

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

Step 2.8

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

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

Step 2.9

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

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

Step 2.10

Simplify.

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Step 2.10.1

Apply the distributive property.

$x^{\frac{1}{3}} + x \frac{1}{3 x^{\frac{2}{3}}} + 4 \frac{1}{3 x^{\frac{2}{3}}}$

Step 2.10.2

Combine terms.

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Step 2.10.2.1

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

$x^{\frac{1}{3}} + \frac{x}{3 x^{\frac{2}{3}}} + 4 \frac{1}{3 x^{\frac{2}{3}}}$

Step 2.10.2.2

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

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

Step 2.10.2.3

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

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Step 2.10.2.3.1

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

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Step 2.10.2.3.1.1

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

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

Step 2.10.2.3.1.2

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

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

Step 2.10.2.3.2

Write $1$ as a fraction with a common denominator.

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

Step 2.10.2.3.3

Combine the numerators over the common denominator.

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

Step 2.10.2.3.4

Subtract $2$ from $3$ .

$x^{\frac{1}{3}} + \frac{x^{\frac{1}{3}}}{3} + 4 \frac{1}{3 x^{\frac{2}{3}}}$

Step 2.10.2.4

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

$x^{\frac{1}{3}} + \frac{x^{\frac{1}{3}}}{3} + \frac{4}{3 x^{\frac{2}{3}}}$

Step 2.10.2.5

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

$x^{\frac{1}{3}} \cdot \frac{3}{3} + \frac{x^{\frac{1}{3}}}{3} + \frac{4}{3 x^{\frac{2}{3}}}$

Step 2.10.2.6

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

$\frac{x^{\frac{1}{3}} \cdot 3}{3} + \frac{x^{\frac{1}{3}}}{3} + \frac{4}{3 x^{\frac{2}{3}}}$

Step 2.10.2.7

Combine the numerators over the common denominator.

$\frac{x^{\frac{1}{3}} \cdot 3 + x^{\frac{1}{3}}}{3} + \frac{4}{3 x^{\frac{2}{3}}}$

Step 2.10.2.8

Move $3$ to the left of $x^{\frac{1}{3}}$ .

$\frac{3 \cdot x^{\frac{1}{3}} + x^{\frac{1}{3}}}{3} + \frac{4}{3 x^{\frac{2}{3}}}$

Step 2.10.2.9

Add $3 x^{\frac{1}{3}}$ and $x^{\frac{1}{3}}$ .

$\frac{4 x^{\frac{1}{3}}}{3} + \frac{4}{3 x^{\frac{2}{3}}}$

Step 3

Find the second derivative of the function.

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Step 3.1

By the Sum Rule, the derivative of $\frac{4 x^{\frac{1}{3}}}{3} + \frac{4}{3 x^{\frac{2}{3}}}$ with respect to $x$ is $\frac{d}{d x} [\frac{4 x^{\frac{1}{3}}}{3}] + \frac{d}{d x} [\frac{4}{3 x^{\frac{2}{3}}}]$ .

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

Step 3.2

Evaluate $\frac{d}{d x} [\frac{4 x^{\frac{1}{3}}}{3}]$ .

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.2.4

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

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

Step 3.2.5

Combine the numerators over the common denominator.

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

Step 3.2.6

Simplify the numerator.

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Step 3.2.6.1

Multiply $- 1$ by $3$ .

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

Step 3.2.6.2

Subtract $3$ from $1$ .

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

Step 3.2.7

Move the negative in front of the fraction.

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

Step 3.2.8

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

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

Step 3.2.9

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

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

Step 3.2.10

Multiply $3$ by $3$ .

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

Step 3.2.11

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

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

Step 3.3

Evaluate $\frac{d}{d x} [\frac{4}{3 x^{\frac{2}{3}}}]$ .

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Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

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

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Step 3.3.3.1

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

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

Step 3.3.3.2

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

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

Step 3.3.3.3

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

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

Step 3.3.4

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

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

Step 3.3.5

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

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Step 3.3.5.1

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

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

Step 3.3.5.2

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

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Step 3.3.5.2.1

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

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

Step 3.3.5.2.2

Multiply $2$ by $- 2$ .

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

Step 3.3.5.3

Move the negative in front of the fraction.

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

Step 3.3.6

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

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

Step 3.3.7

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

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

Step 3.3.8

Combine the numerators over the common denominator.

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

Step 3.3.9

Simplify the numerator.

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Step 3.3.9.1

Multiply $- 1$ by $3$ .

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

Step 3.3.9.2

Subtract $3$ from $2$ .

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

Step 3.3.10

Move the negative in front of the fraction.

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

Step 3.3.11

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

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

Step 3.3.12

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

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

Step 3.3.13

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

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Step 3.3.13.1

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

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

Step 3.3.13.2

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

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

Step 3.3.13.3

Combine the numerators over the common denominator.

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

Step 3.3.13.4

Subtract $1$ from $- 4$ .

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

Step 3.3.13.5

Move the negative in front of the fraction.

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

Step 3.3.14

Move $x^{- \frac{5}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 3.3.15

Multiply $\frac{4}{3}$ by $\frac{2}{3 x^{\frac{5}{3}}}$ .

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

Step 3.3.16

Multiply $4$ by $2$ .

$f'' (x) = \frac{4}{9 x^{\frac{2}{3}}} - \frac{8}{3 (3 x^{\frac{5}{3}})}$

Step 3.3.17

Multiply $3$ by $3$ .

$f'' (x) = \frac{4}{9 x^{\frac{2}{3}}} - \frac{8}{9 x^{\frac{5}{3}}}$

Step 4

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

$\frac{4 x^{\frac{1}{3}}}{3} + \frac{4}{3 x^{\frac{2}{3}}} = 0$

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

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

Step 5.1.2

Differentiate.

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Step 5.1.2.1

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

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

Step 5.1.2.2

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

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

Step 5.1.2.3

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

$x^{\frac{1}{3}} (1 + 0) + (x + 4) \frac{d}{d x} [x^{\frac{1}{3}}]$

Step 5.1.2.4

Simplify the expression.

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Step 5.1.2.4.1

Add $1$ and $0$ .

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

Step 5.1.2.4.2

Multiply $x^{\frac{1}{3}}$ by $1$ .

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

Step 5.1.2.5

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

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

Step 5.1.3

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

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

Step 5.1.4

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

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

Step 5.1.5

Combine the numerators over the common denominator.

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

Step 5.1.6

Simplify the numerator.

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Step 5.1.6.1

Multiply $- 1$ by $3$ .

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

Step 5.1.6.2

Subtract $3$ from $1$ .

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

Step 5.1.7

Move the negative in front of the fraction.

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

Step 5.1.8

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

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

Step 5.1.9

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

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

Step 5.1.10

Simplify.

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Step 5.1.10.1

Apply the distributive property.

$x^{\frac{1}{3}} + x \frac{1}{3 x^{\frac{2}{3}}} + 4 \frac{1}{3 x^{\frac{2}{3}}}$

Step 5.1.10.2

Combine terms.

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Step 5.1.10.2.1

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

$x^{\frac{1}{3}} + \frac{x}{3 x^{\frac{2}{3}}} + 4 \frac{1}{3 x^{\frac{2}{3}}}$

Step 5.1.10.2.2

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

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

Step 5.1.10.2.3

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

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Step 5.1.10.2.3.1

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

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Step 5.1.10.2.3.1.1

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

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

Step 5.1.10.2.3.1.2

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

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

Step 5.1.10.2.3.2

Write $1$ as a fraction with a common denominator.

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

Step 5.1.10.2.3.3

Combine the numerators over the common denominator.

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

Step 5.1.10.2.3.4

Subtract $2$ from $3$ .

$x^{\frac{1}{3}} + \frac{x^{\frac{1}{3}}}{3} + 4 \frac{1}{3 x^{\frac{2}{3}}}$

Step 5.1.10.2.4

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

$x^{\frac{1}{3}} + \frac{x^{\frac{1}{3}}}{3} + \frac{4}{3 x^{\frac{2}{3}}}$

Step 5.1.10.2.5

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

$x^{\frac{1}{3}} \cdot \frac{3}{3} + \frac{x^{\frac{1}{3}}}{3} + \frac{4}{3 x^{\frac{2}{3}}}$

Step 5.1.10.2.6

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

$\frac{x^{\frac{1}{3}} \cdot 3}{3} + \frac{x^{\frac{1}{3}}}{3} + \frac{4}{3 x^{\frac{2}{3}}}$

Step 5.1.10.2.7

Combine the numerators over the common denominator.

$\frac{x^{\frac{1}{3}} \cdot 3 + x^{\frac{1}{3}}}{3} + \frac{4}{3 x^{\frac{2}{3}}}$

Step 5.1.10.2.8

Move $3$ to the left of $x^{\frac{1}{3}}$ .

$\frac{3 \cdot x^{\frac{1}{3}} + x^{\frac{1}{3}}}{3} + \frac{4}{3 x^{\frac{2}{3}}}$

Step 5.1.10.2.9

Add $3 x^{\frac{1}{3}}$ and $x^{\frac{1}{3}}$ .

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

Step 5.2

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

$\frac{4 x^{\frac{1}{3}}}{3} + \frac{4}{3 x^{\frac{2}{3}}}$

Step 6

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

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Step 6.1

Set the first derivative equal to $0$ .

$\frac{4 x^{\frac{1}{3}}}{3} + \frac{4}{3 x^{\frac{2}{3}}} = 0$

Step 6.2

Find the LCD of the terms in the equation.

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Step 6.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$3, 3 x^{\frac{2}{3}}, 1$

Step 6.2.2

Since $3, 3 x^{\frac{2}{3}}, 1$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $3, 3, 1$ then find LCM for the variable part $x^{\frac{2}{3}}$ .

Step 6.2.3

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 6.2.4

Since $3$ has no factors besides $1$ and $3$ .

$3$ is a prime number

Step 6.2.5

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 6.2.6

The LCM of $3, 3, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$3$

Step 6.2.7

The LCM of $x^{\frac{2}{3}}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

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

Step 6.2.8

The LCM for $3, 3 x^{\frac{2}{3}}, 1$ is the numeric part $3$ multiplied by the variable part.

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

Step 6.3

Multiply each term in $\frac{4 x^{\frac{1}{3}}}{3} + \frac{4}{3 x^{\frac{2}{3}}} = 0$ by $3 x^{\frac{2}{3}}$ to eliminate the fractions.

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Step 6.3.1

Multiply each term in $\frac{4 x^{\frac{1}{3}}}{3} + \frac{4}{3 x^{\frac{2}{3}}} = 0$ by $3 x^{\frac{2}{3}}$ .

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

Step 6.3.2

Simplify the left side.

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Step 6.3.2.1

Simplify each term.

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Step 6.3.2.1.1

Rewrite using the commutative property of multiplication.

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

Step 6.3.2.1.2

Cancel the common factor of $3$ .

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Step 6.3.2.1.2.1

Cancel the common factor.

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

Step 6.3.2.1.2.2

Rewrite the expression.

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

Step 6.3.2.1.3

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

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Step 6.3.2.1.3.1

Move $x^{\frac{2}{3}}$ .

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

Step 6.3.2.1.3.2

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

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

Step 6.3.2.1.3.3

Combine the numerators over the common denominator.

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

Step 6.3.2.1.3.4

Add $2$ and $1$ .

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

Step 6.3.2.1.3.5

Divide $3$ by $3$ .

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

Step 6.3.2.1.4

Simplify $4 x^{1}$ .

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

Step 6.3.2.1.5

Rewrite using the commutative property of multiplication.

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

Step 6.3.2.1.6

Cancel the common factor of $3$ .

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Step 6.3.2.1.6.1

Cancel the common factor.

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

Step 6.3.2.1.6.2

Rewrite the expression.

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

Step 6.3.2.1.7

Cancel the common factor of $x^{\frac{2}{3}}$ .

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Step 6.3.2.1.7.1

Cancel the common factor.

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

Step 6.3.2.1.7.2

Rewrite the expression.

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

Step 6.3.3

Simplify the right side.

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Step 6.3.3.1

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

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Step 6.3.3.1.1

Multiply $3$ by $0$ .

$4 x + 4 = 0 x^{\frac{2}{3}}$

Step 6.3.3.1.2

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

$4 x + 4 = 0$

Step 6.4

Solve the equation.

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Step 6.4.1

Subtract $4$ from both sides of the equation.

$4 x = - 4$

Step 6.4.2

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

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Step 6.4.2.1

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

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

Step 6.4.2.2

Simplify the left side.

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Step 6.4.2.2.1

Cancel the common factor of $4$ .

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Step 6.4.2.2.1.1

Cancel the common factor.

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

Step 6.4.2.2.1.2

Divide $x$ by $1$ .

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

Step 6.4.2.3

Simplify the right side.

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Step 6.4.2.3.1

Divide $- 4$ by $4$ .

$x = - 1$

Step 7

Find the values where the derivative is undefined.

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Step 7.1

Convert expressions with fractional exponents to radicals.

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Step 7.1.1

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

$\frac{4 \sqrt[3]{x^{1}}}{3} + \frac{4}{3 x^{\frac{2}{3}}}$

Step 7.1.2

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

$\frac{4 \sqrt[3]{x^{1}}}{3} + \frac{4}{3 \sqrt[3]{x^{2}}}$

Step 7.1.3

Anything raised to $1$ is the base itself.

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

Step 7.2

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

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

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.

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

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

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

Step 7.3.2.2

Simplify the left side.

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Step 7.3.2.2.1

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

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Step 7.3.2.2.1.1

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

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

Step 7.3.2.2.1.2

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

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

Step 7.3.2.2.1.3

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

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Step 7.3.2.2.1.3.1

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

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

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.

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

Step 7.3.2.2.1.3.2.2

Rewrite the expression.

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

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 $0$ .

$27 x^{2} = 0$

Step 7.3.3

Solve for $x$ .

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Step 7.3.3.1

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

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Step 7.3.3.1.1

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

$\frac{27 x^{2}}{27} = \frac{0}{27}$

Step 7.3.3.1.2

Simplify the left side.

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Step 7.3.3.1.2.1

Cancel the common factor of $27$ .

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Step 7.3.3.1.2.1.1

Cancel the common factor.

$\frac{27 x^{2}}{27} = \frac{0}{27}$

Step 7.3.3.1.2.1.2

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

$x^{2} = \frac{0}{27}$

Step 7.3.3.1.3

Simplify the right side.

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Step 7.3.3.1.3.1

Divide $0$ by $27$ .

$x^{2} = 0$

Step 7.3.3.2

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

Simplify $\pm \sqrt{0}$ .

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Step 7.3.3.3.1

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

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

Step 7.3.3.3.2

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

$x = \pm 0$

Step 7.3.3.3.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 8

Critical points to evaluate.

$x = - 1, 0$

Step 9

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.

$\frac{4}{9 {(- 1)}^{\frac{2}{3}}} - \frac{8}{9 {(- 1)}^{\frac{5}{3}}}$

Step 10

Evaluate the second derivative.

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Step 10.1

Simplify each term.

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Step 10.1.1

Simplify the denominator.

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Step 10.1.1.1

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

$\frac{4}{9 {({(- 1)}^{3})}^{\frac{2}{3}}} - \frac{8}{9 {(- 1)}^{\frac{5}{3}}}$

Step 10.1.1.2

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

$\frac{4}{9 {(- 1)}^{3 (\frac{2}{3})}} - \frac{8}{9 {(- 1)}^{\frac{5}{3}}}$

Step 10.1.1.3

Cancel the common factor of $3$ .

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Step 10.1.1.3.1

Cancel the common factor.

$\frac{4}{9 {(- 1)}^{3 (\frac{2}{3})}} - \frac{8}{9 {(- 1)}^{\frac{5}{3}}}$

Step 10.1.1.3.2

Rewrite the expression.

$\frac{4}{9 {(- 1)}^{2}} - \frac{8}{9 {(- 1)}^{\frac{5}{3}}}$

Step 10.1.1.4

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

$\frac{4}{9 \cdot 1} - \frac{8}{9 {(- 1)}^{\frac{5}{3}}}$

Step 10.1.2

Multiply $9$ by $1$ .

$\frac{4}{9} - \frac{8}{9 {(- 1)}^{\frac{5}{3}}}$

Step 10.1.3

Simplify the denominator.

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Step 10.1.3.1

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

$\frac{4}{9} - \frac{8}{9 {({(- 1)}^{3})}^{\frac{5}{3}}}$

Step 10.1.3.2

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

$\frac{4}{9} - \frac{8}{9 {(- 1)}^{3 (\frac{5}{3})}}$

Step 10.1.3.3

Cancel the common factor of $3$ .

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Step 10.1.3.3.1

Cancel the common factor.

$\frac{4}{9} - \frac{8}{9 {(- 1)}^{3 (\frac{5}{3})}}$

Step 10.1.3.3.2

Rewrite the expression.

$\frac{4}{9} - \frac{8}{9 {(- 1)}^{5}}$

Step 10.1.3.4

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

$\frac{4}{9} - \frac{8}{9 \cdot - 1}$

Step 10.1.4

Multiply $9$ by $- 1$ .

$\frac{4}{9} - \frac{8}{- 9}$

Step 10.1.5

Move the negative in front of the fraction.

$\frac{4}{9} - - \frac{8}{9}$

Step 10.1.6

Multiply $- - \frac{8}{9}$ .

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Step 10.1.6.1

Multiply $- 1$ by $- 1$ .

$\frac{4}{9} + 1 (\frac{8}{9})$

Step 10.1.6.2

Multiply $\frac{8}{9}$ by $1$ .

$\frac{4}{9} + \frac{8}{9}$

Step 10.2

Simplify terms.

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Step 10.2.1

Combine the numerators over the common denominator.

$\frac{4 + 8}{9}$

Step 10.2.2

Add $4$ and $8$ .

$\frac{12}{9}$

Step 10.2.3

Cancel the common factor of $12$ and $9$ .

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Step 10.2.3.1

Factor $3$ out of $12$ .

$\frac{3 (4)}{9}$

Step 10.2.3.2

Cancel the common factors.

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Step 10.2.3.2.1

Factor $3$ out of $9$ .

$\frac{3 \cdot 4}{3 \cdot 3}$

Step 10.2.3.2.2

Cancel the common factor.

$\frac{3 \cdot 4}{3 \cdot 3}$

Step 10.2.3.2.3

Rewrite the expression.

$\frac{4}{3}$

Step 11

$x = - 1$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = - 1$ is a local minimum

Step 12

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

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Step 12.1

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

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

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

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

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

Step 12.2.1.2

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

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

Step 12.2.2

Cancel the common factor of $3$ .

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Step 12.2.2.1

Cancel the common factor.

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

Step 12.2.2.2

Rewrite the expression.

$f (- 1) = (- 1) ((- 1) + 4)$

Step 12.2.3

Evaluate the exponent.

$f (- 1) = - 1 ((- 1) + 4)$

Step 12.2.4

Add $- 1$ and $4$ .

$f (- 1) = - 1 \cdot 3$

Step 12.2.5

Multiply $- 1$ by $3$ .

$f (- 1) = - 3$

Step 12.2.6

The final answer is $- 3$ .

$y = - 3$

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.

$\frac{4}{9 {(0)}^{\frac{2}{3}}} - \frac{8}{9 {(0)}^{\frac{5}{3}}}$

Step 14

Evaluate the second derivative.

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Step 14.1

Simplify the expression.

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Step 14.1.1

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

$\frac{4}{9 {(0^{3})}^{\frac{2}{3}}} - \frac{8}{9 {(0)}^{\frac{5}{3}}}$

Step 14.1.2

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

$\frac{4}{9 \cdot 0^{3 (\frac{2}{3})}} - \frac{8}{9 {(0)}^{\frac{5}{3}}}$

Step 14.2

Cancel the common factor of $3$ .

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Step 14.2.1

Cancel the common factor.

$\frac{4}{9 \cdot 0^{3 (\frac{2}{3})}} - \frac{8}{9 {(0)}^{\frac{5}{3}}}$

Step 14.2.2

Rewrite the expression.

$\frac{4}{9 \cdot 0^{2}} - \frac{8}{9 {(0)}^{\frac{5}{3}}}$

Step 14.3

Simplify the expression.

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Step 14.3.1

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

$\frac{4}{9 \cdot 0} - \frac{8}{9 {(0)}^{\frac{5}{3}}}$

Step 14.3.2

Multiply $9$ by $0$ .

$\frac{4}{0} - \frac{8}{9 {(0)}^{\frac{5}{3}}}$

Step 14.3.3

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

Undefined

$\frac{4}{0} - \frac{8}{9 {(0)}^{\frac{5}{3}}}$

Step 14.4

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.

$(- \infty, - 1) \cup (- 1, 0) \cup (0, \infty)$

Step 15.2

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

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Step 15.2.1

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

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

Step 15.2.2

The final answer is $\frac{4 {(- 4)}^{\frac{1}{3}}}{3} + \frac{4}{3 {(- 4)}^{\frac{2}{3}}}$ .

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

Step 15.3

Substitute any number, such as $- 0.5$ , from the interval $(- 1, 0)$ in the first derivative $\frac{4 x^{\frac{1}{3}}}{3} + \frac{4}{3 x^{\frac{2}{3}}}$ to check if the result is negative or positive.

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Step 15.3.1

Replace the variable $x$ with $- 0.5$ in the expression.

$f' (- 0.5) = \frac{4 {(- 0.5)}^{\frac{1}{3}}}{3} + \frac{4}{3 {(- 0.5)}^{\frac{2}{3}}}$

Step 15.3.2

The final answer is $\frac{4 {(- 0.5)}^{\frac{1}{3}}}{3} + \frac{4}{3 {(- 0.5)}^{\frac{2}{3}}}$ .

$\frac{4 {(- 0.5)}^{\frac{1}{3}}}{3} + \frac{4}{3 {(- 0.5)}^{\frac{2}{3}}}$

Step 15.4

Substitute any number, such as $2$ , from the interval $(0, \infty)$ in the first derivative $\frac{4 x^{\frac{1}{3}}}{3} + \frac{4}{3 x^{\frac{2}{3}}}$ 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.

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

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

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

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

Step 15.4.2.1.2

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

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

Step 15.4.2.1.3

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 15.4.2.1.4

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

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

Step 15.4.2.1.5

Combine the numerators over the common denominator.

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

Step 15.4.2.1.6

Simplify the numerator.

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Step 15.4.2.1.6.1

Multiply $2$ by $3$ .

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

Step 15.4.2.1.6.2

Add $6$ and $1$ .

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

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

Step 15.4.2.2

The final answer is $\frac{2^{\frac{7}{3}}}{3} + \frac{4}{3 \cdot 2^{\frac{2}{3}}}$ .

$\frac{2^{\frac{7}{3}}}{3} + \frac{4}{3 \cdot 2^{\frac{2}{3}}}$

Step 15.5

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

$x = - 1$ is a local minimum

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

These are the local extrema for $f (x) = x^{\frac{1}{3}} (x + 4)$ .

$x = - 1$ is a local minimum

Step 16