Calculus Examples

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

Step 1

Find the $x$ values where the second derivative is equal to $0$ .

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Step 1.1

Find the second derivative.

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Step 1.1.1

Find the first derivative.

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Step 1.1.1.1

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 $g (x) = x + 4$ .

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

Step 1.1.1.2

Differentiate.

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Step 1.1.1.2.1

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

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

Step 1.1.1.2.2

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

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

Step 1.1.1.2.3

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

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

Step 1.1.1.2.4

Simplify the expression.

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Step 1.1.1.2.4.1

Add $1$ and $0$ .

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

Step 1.1.1.2.4.2

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

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

Step 1.1.1.2.5

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

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

Step 1.1.1.3

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

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

Step 1.1.1.4

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

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

Step 1.1.1.5

Combine the numerators over the common denominator.

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

Step 1.1.1.6

Simplify the numerator.

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Step 1.1.1.6.1

Multiply $- 1$ by $3$ .

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

Step 1.1.1.6.2

Subtract $3$ from $1$ .

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

Step 1.1.1.7

Move the negative in front of the fraction.

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

Step 1.1.1.8

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

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

Step 1.1.1.9

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

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

Step 1.1.1.10

Simplify.

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Step 1.1.1.10.1

Apply the distributive property.

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

Step 1.1.1.10.2

Combine terms.

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Step 1.1.1.10.2.1

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

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

Step 1.1.1.10.2.2

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

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

Step 1.1.1.10.2.3

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

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Step 1.1.1.10.2.3.1

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

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Step 1.1.1.10.2.3.1.1

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

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

Step 1.1.1.10.2.3.1.2

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

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

Step 1.1.1.10.2.3.2

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

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

Step 1.1.1.10.2.3.3

Combine the numerators over the common denominator.

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

Step 1.1.1.10.2.3.4

Subtract $2$ from $3$ .

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

Step 1.1.1.10.2.4

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

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

Step 1.1.1.10.2.5

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

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

Step 1.1.1.10.2.6

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

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

Step 1.1.1.10.2.7

Combine the numerators over the common denominator.

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

Step 1.1.1.10.2.8

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

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

Step 1.1.1.10.2.9

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 1.1.2

Find the second derivative.

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Step 1.1.2.1

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 1.1.2.2

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

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Step 1.1.2.2.1

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 1.1.2.2.2

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 1.1.2.2.3

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 1.1.2.2.4

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 1.1.2.2.5

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 1.1.2.2.6

Simplify the numerator.

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Step 1.1.2.2.6.1

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 1.1.2.2.6.2

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 1.1.2.2.7

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 1.1.2.2.8

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 1.1.2.2.9

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 1.1.2.2.10

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 1.1.2.2.11

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 1.1.2.3

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

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Step 1.1.2.3.1

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 1.1.2.3.2

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 1.1.2.3.3

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 1.1.2.3.3.1

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 1.1.2.3.3.2

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 1.1.2.3.3.3

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 1.1.2.3.4

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 1.1.2.3.5

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

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Step 1.1.2.3.5.1

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 1.1.2.3.5.2

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

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Step 1.1.2.3.5.2.1

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 1.1.2.3.5.2.2

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 1.1.2.3.5.3

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 1.1.2.3.6

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 1.1.2.3.7

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 1.1.2.3.8

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 1.1.2.3.9

Simplify the numerator.

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Step 1.1.2.3.9.1

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 1.1.2.3.9.2

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 1.1.2.3.10

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 1.1.2.3.11

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 1.1.2.3.12

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 1.1.2.3.13

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

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Step 1.1.2.3.13.1

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 1.1.2.3.13.2

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 1.1.2.3.13.3

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 1.1.2.3.13.4

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 1.1.2.3.13.5

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 1.1.2.3.14

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 1.1.2.3.15

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 1.1.2.3.16

Multiply $4$ by $2$ .

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

Step 1.1.2.3.17

Multiply $3$ by $3$ .

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

Step 1.1.3

The second derivative of $f (x)$ with respect to $x$ is $\frac{4}{9 x^{\frac{2}{3}}} - \frac{8}{9 x^{\frac{5}{3}}}$ .

$\frac{4}{9 x^{\frac{2}{3}}} - \frac{8}{9 x^{\frac{5}{3}}}$

Step 1.2

Set the second derivative equal to $0$ then solve the equation $\frac{4}{9 x^{\frac{2}{3}}} - \frac{8}{9 x^{\frac{5}{3}}} = 0$ .

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Step 1.2.1

Set the second derivative equal to $0$ .

$\frac{4}{9 x^{\frac{2}{3}}} - \frac{8}{9 x^{\frac{5}{3}}} = 0$

Step 1.2.2

Find the LCD of the terms in the equation.

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Step 1.2.2.1

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

$9 x^{\frac{2}{3}}, 9 x^{\frac{5}{3}}, 1$

Step 1.2.2.2

Since $9 x^{\frac{2}{3}}, 9 x^{\frac{5}{3}}, 1$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $9, 9, 1$ then find LCM for the variable part $x^{\frac{2}{3}}, x^{\frac{5}{3}}$ .

Step 1.2.2.3

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 1.2.2.4

$9$ has factors of $3$ and $3$ .

$3 \cdot 3$

Step 1.2.2.5

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

Not prime

Step 1.2.2.6

The LCM of $9, 9, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$3 \cdot 3$

Step 1.2.2.7

Multiply $3$ by $3$ .

$9$

Step 1.2.2.8

The LCM of $x^{\frac{2}{3}}, x^{\frac{5}{3}}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$x^{\frac{5}{3}}$

Step 1.2.2.9

The LCM for $9 x^{\frac{2}{3}}, 9 x^{\frac{5}{3}}, 1$ is the numeric part $9$ multiplied by the variable part.

$9 x^{\frac{5}{3}}$

Step 1.2.3

Multiply each term in $\frac{4}{9 x^{\frac{2}{3}}} - \frac{8}{9 x^{\frac{5}{3}}} = 0$ by $9 x^{\frac{5}{3}}$ to eliminate the fractions.

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Step 1.2.3.1

Multiply each term in $\frac{4}{9 x^{\frac{2}{3}}} - \frac{8}{9 x^{\frac{5}{3}}} = 0$ by $9 x^{\frac{5}{3}}$ .

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

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Simplify each term.

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Step 1.2.3.2.1.1

Rewrite using the commutative property of multiplication.

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

Step 1.2.3.2.1.2

Cancel the common factor of $9$ .

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Step 1.2.3.2.1.2.1

Cancel the common factor.

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

Step 1.2.3.2.1.2.2

Rewrite the expression.

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

Step 1.2.3.2.1.3

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

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Step 1.2.3.2.1.3.1

Factor $x^{\frac{2}{3}}$ out of $x^{\frac{5}{3}}$ .

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

Step 1.2.3.2.1.3.2

Cancel the common factor.

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

Step 1.2.3.2.1.3.3

Rewrite the expression.

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

Step 1.2.3.2.1.4

Divide $3$ by $3$ .

$4 x^{1} - \frac{8}{9 x^{\frac{5}{3}}} (9 x^{\frac{5}{3}}) = 0 (9 x^{\frac{5}{3}})$

Step 1.2.3.2.1.5

Simplify.

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

Step 1.2.3.2.1.6

Cancel the common factor of $9 x^{\frac{5}{3}}$ .

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Step 1.2.3.2.1.6.1

Move the leading negative in $- \frac{8}{9 x^{\frac{5}{3}}}$ into the numerator.

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

Step 1.2.3.2.1.6.2

Cancel the common factor.

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

Step 1.2.3.2.1.6.3

Rewrite the expression.

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

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Multiply $0 (9 x^{\frac{5}{3}})$ .

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Step 1.2.3.3.1.1

Multiply $9$ by $0$ .

$4 x - 8 = 0 x^{\frac{5}{3}}$

Step 1.2.3.3.1.2

Multiply $0$ by $x^{\frac{5}{3}}$ .

$4 x - 8 = 0$

Step 1.2.4

Solve the equation.

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Step 1.2.4.1

Add $8$ to both sides of the equation.

$4 x = 8$

Step 1.2.4.2

Divide each term in $4 x = 8$ by $4$ and simplify.

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Step 1.2.4.2.1

Divide each term in $4 x = 8$ by $4$ .

$\frac{4 x}{4} = \frac{8}{4}$

Step 1.2.4.2.2

Simplify the left side.

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Step 1.2.4.2.2.1

Cancel the common factor of $4$ .

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Step 1.2.4.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{8}{4}$

Step 1.2.4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{8}{4}$

Step 1.2.4.2.3

Simplify the right side.

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Step 1.2.4.2.3.1

Divide $8$ by $4$ .

$x = 2$

Step 2

Find the domain of $f (x) = x^{\frac{1}{3}} (x + 4)$ .

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Step 2.1

Convert expressions with fractional exponents to radicals.

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Step 2.1.1

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

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

Step 2.1.2

Anything raised to $1$ is the base itself.

$\sqrt[3]{x} (x + 4)$

Step 2.2

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 3

Create intervals around the $x$ -values where the second derivative is zero or undefined.

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

Step 4

Substitute any number from the interval $(- \infty, 2)$ into the second derivative and evaluate to determine the concavity.

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Step 4.1

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

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

Step 4.2

Simplify the expression.

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Step 4.2.1

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

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

Step 4.2.2

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

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

Step 4.3

Cancel the common factor of $3$ .

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Step 4.3.1

Cancel the common factor.

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

Step 4.3.2

Rewrite the expression.

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

Step 4.4

Simplify the expression.

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Step 4.4.1

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

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

Step 4.4.2

Multiply $9$ by $0$ .

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

Step 4.4.3

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

$f'' (0) = Undefined$

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

Step 4.5

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

$f'' (0) = Undefined$

Step 4.6

The graph is concave up on the interval $(- \infty, 2)$ because $f'' (0)$ is positive.

Concave up on $(- \infty, 2)$ since $f'' (x)$ is positive

Step 5

Substitute any number from the interval $(2, \infty)$ into the second derivative and evaluate to determine the concavity.

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Step 5.1

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

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

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

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

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

Step 5.2.1.2

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

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Step 5.2.1.2.1

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

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Step 5.2.1.2.1.1

Raise $4$ to the power of $1$ .

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

Step 5.2.1.2.1.2

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

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

Step 5.2.1.2.2

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

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

Step 5.2.1.2.3

Combine the numerators over the common denominator.

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

Step 5.2.1.2.4

Subtract $2$ from $3$ .

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

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

Step 5.2.2

The final answer is $\frac{4^{\frac{1}{3}}}{9} - \frac{8}{9 \cdot 4^{\frac{5}{3}}}$ .

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

Step 5.3

The graph is concave up on the interval $(2, \infty)$ because $f'' (4)$ is positive.

Concave up on $(2, \infty)$ since $f'' (x)$ is positive

Step 6

The graph is concave down when the second derivative is negative and concave up when the second derivative is positive.

Concave up on $(- \infty, 2)$ since $f'' (x)$ is positive

Concave up on $(2, \infty)$ since $f'' (x)$ is positive

Step 7