Calculus Examples

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

Step 1

Find the first derivative of the function.

Tap for more steps...

Step 1.1

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

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

Step 1.2

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

Tap for more steps...

Step 1.2.1

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{4}{3}$ .

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

Combine the numerators over the common denominator.

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

Step 1.2.5

Simplify the numerator.

Tap for more steps...

Step 1.2.5.1

Multiply $- 1$ by $3$ .

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

Step 1.2.5.2

Subtract $3$ from $4$ .

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

Step 1.3

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

Tap for more steps...

Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

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

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

Step 1.3.4

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

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

Step 1.3.5

Combine the numerators over the common denominator.

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

Step 1.3.6

Simplify the numerator.

Tap for more steps...

Step 1.3.6.1

Multiply $- 1$ by $3$ .

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

Step 1.3.6.2

Subtract $3$ from $2$ .

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

Step 1.3.7

Move the negative in front of the fraction.

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

Step 1.3.8

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

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

Step 1.3.9

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

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

Step 1.4

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

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

Step 2

Find the second derivative of the function.

Tap for more steps...

Step 2.1

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

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

Step 2.2

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

Tap for more steps...

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

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

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

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

Step 2.2.5

Combine the numerators over the common denominator.

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

Step 2.2.6

Simplify the numerator.

Tap for more steps...

Step 2.2.6.1

Multiply $- 1$ by $3$ .

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

Step 2.2.6.2

Subtract $3$ from $1$ .

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

Step 2.2.7

Move the negative in front of the fraction.

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

Step 2.2.8

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

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

Step 2.2.9

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

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

Step 2.2.10

Multiply $3$ by $3$ .

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

Step 2.2.11

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

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

Step 2.3

Evaluate $\frac{d}{d x} [- \frac{2}{3 x^{\frac{1}{3}}}]$ .

Tap for more steps...

Step 2.3.1

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

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

Step 2.3.2

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

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

Step 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 $g (x) = x^{\frac{1}{3}}$ .

Tap for more steps...

Step 2.3.3.1

To apply the Chain Rule, set $u$ as $x^{\frac{1}{3}}$ .

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

Step 2.3.3.2

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

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

Step 2.3.3.3

Replace all occurrences of $u$ with $x^{\frac{1}{3}}$ .

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

Step 2.3.4

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

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

Step 2.3.5

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

Tap for more steps...

Step 2.3.5.1

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

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

Step 2.3.5.2

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

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

Step 2.3.5.3

Move the negative in front of the fraction.

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

Step 2.3.6

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

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

Step 2.3.7

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

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

Step 2.3.8

Combine the numerators over the common denominator.

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

Step 2.3.9

Simplify the numerator.

Tap for more steps...

Step 2.3.9.1

Multiply $- 1$ by $3$ .

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

Step 2.3.9.2

Subtract $3$ from $1$ .

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

Step 2.3.10

Move the negative in front of the fraction.

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

Step 2.3.11

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

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

Step 2.3.12

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

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

Step 2.3.13

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

Tap for more steps...

Step 2.3.13.1

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

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

Step 2.3.13.2

Combine the numerators over the common denominator.

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

Step 2.3.13.3

Subtract $2$ from $- 2$ .

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

Step 2.3.13.4

Move the negative in front of the fraction.

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

Step 2.3.14

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

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

Step 2.3.15

Multiply $- 1$ by $- 1$ .

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

Step 2.3.16

Multiply $\frac{2}{3}$ by $1$ .

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

Step 2.3.17

Multiply $\frac{2}{3}$ by $\frac{1}{3 x^{\frac{4}{3}}}$ .

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

Step 2.3.18

Multiply $3$ by $3$ .

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

Step 3

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

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

Step 4

Find the first derivative.

Tap for more steps...

Step 4.1

Find the first derivative.

Tap for more steps...

Step 4.1.1

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

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

Step 4.1.2

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

Tap for more steps...

Step 4.1.2.1

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{4}{3}$ .

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

Step 4.1.2.2

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

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

Step 4.1.2.3

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

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

Step 4.1.2.4

Combine the numerators over the common denominator.

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

Step 4.1.2.5

Simplify the numerator.

Tap for more steps...

Step 4.1.2.5.1

Multiply $- 1$ by $3$ .

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

Step 4.1.2.5.2

Subtract $3$ from $4$ .

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

Step 4.1.3

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

Tap for more steps...

Step 4.1.3.1

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

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

Step 4.1.3.2

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

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

Step 4.1.3.3

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

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

Step 4.1.3.4

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

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

Step 4.1.3.5

Combine the numerators over the common denominator.

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

Step 4.1.3.6

Simplify the numerator.

Tap for more steps...

Step 4.1.3.6.1

Multiply $- 1$ by $3$ .

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

Step 4.1.3.6.2

Subtract $3$ from $2$ .

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

Step 4.1.3.7

Move the negative in front of the fraction.

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

Step 4.1.3.8

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

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

Step 4.1.3.9

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

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

Step 4.1.4

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

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

Step 4.2

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

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

Step 5

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

Tap for more steps...

Step 5.1

Set the first derivative equal to $0$ .

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

Step 5.2

Find the LCD of the terms in the equation.

Tap for more steps...

Step 5.2.1

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

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

Step 5.2.2

Since $3, 3 x^{\frac{1}{3}}, 1$ 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 $x^{\frac{1}{3}}$ .

Step 5.2.3

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 5.2.4

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

$3$ is a prime number

Step 5.2.5

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

Not prime

Step 5.2.6

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 5.2.7

The LCM of $x^{\frac{1}{3}}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

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

Step 5.2.8

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

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

Step 5.3

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

Tap for more steps...

Step 5.3.1

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

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

Step 5.3.2

Simplify the left side.

Tap for more steps...

Step 5.3.2.1

Simplify each term.

Tap for more steps...

Step 5.3.2.1.1

Rewrite using the commutative property of multiplication.

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

Step 5.3.2.1.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 5.3.2.1.2.1

Cancel the common factor.

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

Step 5.3.2.1.2.2

Rewrite the expression.

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

Step 5.3.2.1.3

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

Tap for more steps...

Step 5.3.2.1.3.1

Move $x^{\frac{1}{3}}$ .

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

Step 5.3.2.1.3.2

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

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

Step 5.3.2.1.3.3

Combine the numerators over the common denominator.

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

Step 5.3.2.1.3.4

Add $1$ and $1$ .

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

Step 5.3.2.1.4

Cancel the common factor of $3 x^{\frac{1}{3}}$ .

Tap for more steps...

Step 5.3.2.1.4.1

Move the leading negative in $- \frac{2}{3 x^{\frac{1}{3}}}$ into the numerator.

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

Step 5.3.2.1.4.2

Cancel the common factor.

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

Step 5.3.2.1.4.3

Rewrite the expression.

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

Step 5.3.3

Simplify the right side.

Tap for more steps...

Step 5.3.3.1

Multiply $0 (3 x^{\frac{1}{3}})$ .

Tap for more steps...

Step 5.3.3.1.1

Multiply $3$ by $0$ .

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

Step 5.3.3.1.2

Multiply $0$ by $x^{\frac{1}{3}}$ .

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

Step 5.4

Solve the equation.

Tap for more steps...

Step 5.4.1

Add $2$ to both sides of the equation.

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

Step 5.4.2

Raise each side of the equation to the power of $\frac{3}{2}$ to eliminate the fractional exponent on the left side.

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

Step 5.4.3

Simplify the left side.

Tap for more steps...

Step 5.4.3.1

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

Tap for more steps...

Step 5.4.3.1.1

Simplify the expression.

Tap for more steps...

Step 5.4.3.1.1.1

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

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

Step 5.4.3.1.1.2

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

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

Step 5.4.3.1.1.3

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

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

Step 5.4.3.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 5.4.3.1.2.1

Cancel the common factor.

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

Step 5.4.3.1.2.2

Rewrite the expression.

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

Step 5.4.3.1.3

Simplify the expression.

Tap for more steps...

Step 5.4.3.1.3.1

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

$8 {(x^{\frac{2}{3}})}^{\frac{3}{2}} = \pm 2^{\frac{3}{2}}$

Step 5.4.3.1.3.2

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

Tap for more steps...

Step 5.4.3.1.3.2.1

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

$8 x^{\frac{2}{3} \cdot \frac{3}{2}} = \pm 2^{\frac{3}{2}}$

Step 5.4.3.1.3.2.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 5.4.3.1.3.2.2.1

Cancel the common factor.

$8 x^{\frac{2}{3} \cdot \frac{3}{2}} = \pm 2^{\frac{3}{2}}$

Step 5.4.3.1.3.2.2.2

Rewrite the expression.

$8 x^{\frac{1}{3} \cdot 3} = \pm 2^{\frac{3}{2}}$

Step 5.4.3.1.3.2.3

Cancel the common factor of $3$ .

Tap for more steps...

Step 5.4.3.1.3.2.3.1

Cancel the common factor.

$8 x^{\frac{1}{3} \cdot 3} = \pm 2^{\frac{3}{2}}$

Step 5.4.3.1.3.2.3.2

Rewrite the expression.

$8 x^{1} = \pm 2^{\frac{3}{2}}$

Step 5.4.3.1.4

Simplify.

$8 x = \pm 2^{\frac{3}{2}}$

Step 5.4.4

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

Tap for more steps...

Step 5.4.4.1

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

$8 x = 2^{\frac{3}{2}}$

Step 5.4.4.2

Divide each term in $8 x = 2^{\frac{3}{2}}$ by $8$ and simplify.

Tap for more steps...

Step 5.4.4.2.1

Divide each term in $8 x = 2^{\frac{3}{2}}$ by $8$ .

$\frac{8 x}{8} = \frac{2^{\frac{3}{2}}}{8}$

Step 5.4.4.2.2

Simplify the left side.

Tap for more steps...

Step 5.4.4.2.2.1

Cancel the common factor of $8$ .

Tap for more steps...

Step 5.4.4.2.2.1.1

Cancel the common factor.

$\frac{8 x}{8} = \frac{2^{\frac{3}{2}}}{8}$

Step 5.4.4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{2^{\frac{3}{2}}}{8}$

Step 5.4.4.3

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

$8 x = - 2^{\frac{3}{2}}$

Step 5.4.4.4

Divide each term in $8 x = - 2^{\frac{3}{2}}$ by $8$ and simplify.

Tap for more steps...

Step 5.4.4.4.1

Divide each term in $8 x = - 2^{\frac{3}{2}}$ by $8$ .

$\frac{8 x}{8} = \frac{- 2^{\frac{3}{2}}}{8}$

Step 5.4.4.4.2

Simplify the left side.

Tap for more steps...

Step 5.4.4.4.2.1

Cancel the common factor of $8$ .

Tap for more steps...

Step 5.4.4.4.2.1.1

Cancel the common factor.

$\frac{8 x}{8} = \frac{- 2^{\frac{3}{2}}}{8}$

Step 5.4.4.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 2^{\frac{3}{2}}}{8}$

Step 5.4.4.4.3

Simplify the right side.

Tap for more steps...

Step 5.4.4.4.3.1

Move the negative in front of the fraction.

$x = - \frac{2^{\frac{3}{2}}}{8}$

Step 5.4.4.5

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

$x = \frac{2^{\frac{3}{2}}}{8}, - \frac{2^{\frac{3}{2}}}{8}$

Step 6

Find the values where the derivative is undefined.

Tap for more steps...

Step 6.1

Convert expressions with fractional exponents to radicals.

Tap for more steps...

Step 6.1.1

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

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

Step 6.1.2

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

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

Step 6.1.3

Anything raised to $1$ is the base itself.

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

Step 6.1.4

Anything raised to $1$ is the base itself.

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

Step 6.2

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

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

Step 6.3

Solve for $x$ .

Tap for more steps...

Step 6.3.1

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

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

Step 6.3.2

Simplify each side of the equation.

Tap for more steps...

Step 6.3.2.1

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

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

Step 6.3.2.2

Simplify the left side.

Tap for more steps...

Step 6.3.2.2.1

Simplify ${(3 x^{\frac{1}{3}})}^{3}$ .

Tap for more steps...

Step 6.3.2.2.1.1

Apply the product rule to $3 x^{\frac{1}{3}}$ .

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

Step 6.3.2.2.1.2

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

$27 {(x^{\frac{1}{3}})}^{3} = 0^{3}$

Step 6.3.2.2.1.3

Multiply the exponents in ${(x^{\frac{1}{3}})}^{3}$ .

Tap for more steps...

Step 6.3.2.2.1.3.1

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

$27 x^{\frac{1}{3} \cdot 3} = 0^{3}$

Step 6.3.2.2.1.3.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 6.3.2.2.1.3.2.1

Cancel the common factor.

$27 x^{\frac{1}{3} \cdot 3} = 0^{3}$

Step 6.3.2.2.1.3.2.2

Rewrite the expression.

$27 x^{1} = 0^{3}$

Step 6.3.2.2.1.4

Simplify.

$27 x = 0^{3}$

Step 6.3.2.3

Simplify the right side.

Tap for more steps...

Step 6.3.2.3.1

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

$27 x = 0$

Step 6.3.3

Divide each term in $27 x = 0$ by $27$ and simplify.

Tap for more steps...

Step 6.3.3.1

Divide each term in $27 x = 0$ by $27$ .

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

Step 6.3.3.2

Simplify the left side.

Tap for more steps...

Step 6.3.3.2.1

Cancel the common factor of $27$ .

Tap for more steps...

Step 6.3.3.2.1.1

Cancel the common factor.

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

Step 6.3.3.2.1.2

Divide $x$ by $1$ .

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

Step 6.3.3.3

Simplify the right side.

Tap for more steps...

Step 6.3.3.3.1

Divide $0$ by $27$ .

$x = 0$

Step 7

Critical points to evaluate.

$x = \frac{2^{\frac{3}{2}}}{8}, - \frac{2^{\frac{3}{2}}}{8}, 0$

Step 8

Evaluate the second derivative at $x = \frac{2^{\frac{3}{2}}}{8}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{4}{9 {(\frac{2^{\frac{3}{2}}}{8})}^{\frac{2}{3}}} + \frac{2}{9 {(\frac{2^{\frac{3}{2}}}{8})}^{\frac{4}{3}}}$

Step 9

Evaluate the second derivative.

Tap for more steps...

Step 9.1

Simplify each term.

Tap for more steps...

Step 9.1.1

Simplify the denominator.

Tap for more steps...

Step 9.1.1.1

Apply the product rule to $\frac{2^{\frac{3}{2}}}{8}$ .

$\frac{4}{9 \frac{{(2^{\frac{3}{2}})}^{\frac{2}{3}}}{8^{\frac{2}{3}}}} + \frac{2}{9 {(\frac{2^{\frac{3}{2}}}{8})}^{\frac{4}{3}}}$

Step 9.1.1.2

Simplify the numerator.

Tap for more steps...

Step 9.1.1.2.1

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

Tap for more steps...

Step 9.1.1.2.1.1

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

$\frac{4}{9 \frac{2^{\frac{3}{2} \cdot \frac{2}{3}}}{8^{\frac{2}{3}}}} + \frac{2}{9 {(\frac{2^{\frac{3}{2}}}{8})}^{\frac{4}{3}}}$

Step 9.1.1.2.1.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 9.1.1.2.1.2.1

Cancel the common factor.

$\frac{4}{9 \frac{2^{\frac{3}{2} \cdot \frac{2}{3}}}{8^{\frac{2}{3}}}} + \frac{2}{9 {(\frac{2^{\frac{3}{2}}}{8})}^{\frac{4}{3}}}$

Step 9.1.1.2.1.2.2

Rewrite the expression.

$\frac{4}{9 \frac{2^{\frac{1}{2} \cdot 2}}{8^{\frac{2}{3}}}} + \frac{2}{9 {(\frac{2^{\frac{3}{2}}}{8})}^{\frac{4}{3}}}$

Step 9.1.1.2.1.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 9.1.1.2.1.3.1

Cancel the common factor.

$\frac{4}{9 \frac{2^{\frac{1}{2} \cdot 2}}{8^{\frac{2}{3}}}} + \frac{2}{9 {(\frac{2^{\frac{3}{2}}}{8})}^{\frac{4}{3}}}$

Step 9.1.1.2.1.3.2

Rewrite the expression.

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

Step 9.1.1.2.2

Evaluate the exponent.

$\frac{4}{9 \frac{2}{8^{\frac{2}{3}}}} + \frac{2}{9 {(\frac{2^{\frac{3}{2}}}{8})}^{\frac{4}{3}}}$

Step 9.1.1.3

Simplify the denominator.

Tap for more steps...

Step 9.1.1.3.1

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

$\frac{4}{9 \frac{2}{{(2^{3})}^{\frac{2}{3}}}} + \frac{2}{9 {(\frac{2^{\frac{3}{2}}}{8})}^{\frac{4}{3}}}$

Step 9.1.1.3.2

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

$\frac{4}{9 \frac{2}{2^{3 (\frac{2}{3})}}} + \frac{2}{9 {(\frac{2^{\frac{3}{2}}}{8})}^{\frac{4}{3}}}$

Step 9.1.1.3.3

Cancel the common factor of $3$ .

Tap for more steps...

Step 9.1.1.3.3.1

Cancel the common factor.

$\frac{4}{9 \frac{2}{2^{3 (\frac{2}{3})}}} + \frac{2}{9 {(\frac{2^{\frac{3}{2}}}{8})}^{\frac{4}{3}}}$

Step 9.1.1.3.3.2

Rewrite the expression.

$\frac{4}{9 \frac{2}{2^{2}}} + \frac{2}{9 {(\frac{2^{\frac{3}{2}}}{8})}^{\frac{4}{3}}}$

Step 9.1.1.3.4

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

$\frac{4}{9 (\frac{2}{4})} + \frac{2}{9 {(\frac{2^{\frac{3}{2}}}{8})}^{\frac{4}{3}}}$

Step 9.1.1.4

Cancel the common factor of $2$ and $4$ .

Tap for more steps...

Step 9.1.1.4.1

Factor $2$ out of $2$ .

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

Step 9.1.1.4.2

Cancel the common factors.

Tap for more steps...

Step 9.1.1.4.2.1

Factor $2$ out of $4$ .

$\frac{4}{9 \frac{2 \cdot 1}{2 \cdot 2}} + \frac{2}{9 {(\frac{2^{\frac{3}{2}}}{8})}^{\frac{4}{3}}}$

Step 9.1.1.4.2.2

Cancel the common factor.

$\frac{4}{9 \frac{2 \cdot 1}{2 \cdot 2}} + \frac{2}{9 {(\frac{2^{\frac{3}{2}}}{8})}^{\frac{4}{3}}}$

Step 9.1.1.4.2.3

Rewrite the expression.

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

Step 9.1.2

Combine $9$ and $\frac{1}{2}$ .

$\frac{4}{\frac{9}{2}} + \frac{2}{9 {(\frac{2^{\frac{3}{2}}}{8})}^{\frac{4}{3}}}$

Step 9.1.3

Multiply the numerator by the reciprocal of the denominator.

$4 (\frac{2}{9}) + \frac{2}{9 {(\frac{2^{\frac{3}{2}}}{8})}^{\frac{4}{3}}}$

Step 9.1.4

Multiply $4 (\frac{2}{9})$ .

Tap for more steps...

Step 9.1.4.1

Combine $4$ and $\frac{2}{9}$ .

$\frac{4 \cdot 2}{9} + \frac{2}{9 {(\frac{2^{\frac{3}{2}}}{8})}^{\frac{4}{3}}}$

Step 9.1.4.2

Multiply $4$ by $2$ .

$\frac{8}{9} + \frac{2}{9 {(\frac{2^{\frac{3}{2}}}{8})}^{\frac{4}{3}}}$

Step 9.1.5

Simplify the denominator.

Tap for more steps...

Step 9.1.5.1

Apply the product rule to $\frac{2^{\frac{3}{2}}}{8}$ .

$\frac{8}{9} + \frac{2}{9 \frac{{(2^{\frac{3}{2}})}^{\frac{4}{3}}}{8^{\frac{4}{3}}}}$

Step 9.1.5.2

Simplify the numerator.

Tap for more steps...

Step 9.1.5.2.1

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

Tap for more steps...

Step 9.1.5.2.1.1

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

$\frac{8}{9} + \frac{2}{9 \frac{2^{\frac{3}{2} \cdot \frac{4}{3}}}{8^{\frac{4}{3}}}}$

Step 9.1.5.2.1.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 9.1.5.2.1.2.1

Cancel the common factor.

$\frac{8}{9} + \frac{2}{9 \frac{2^{\frac{3}{2} \cdot \frac{4}{3}}}{8^{\frac{4}{3}}}}$

Step 9.1.5.2.1.2.2

Rewrite the expression.

$\frac{8}{9} + \frac{2}{9 \frac{2^{\frac{1}{2} \cdot 4}}{8^{\frac{4}{3}}}}$

Step 9.1.5.2.1.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 9.1.5.2.1.3.1

Factor $2$ out of $4$ .

$\frac{8}{9} + \frac{2}{9 \frac{2^{\frac{1}{2} \cdot (2 (2))}}{8^{\frac{4}{3}}}}$

Step 9.1.5.2.1.3.2

Cancel the common factor.

$\frac{8}{9} + \frac{2}{9 \frac{2^{\frac{1}{2} \cdot (2 \cdot 2)}}{8^{\frac{4}{3}}}}$

Step 9.1.5.2.1.3.3

Rewrite the expression.

$\frac{8}{9} + \frac{2}{9 \frac{2^{2}}{8^{\frac{4}{3}}}}$

Step 9.1.5.2.2

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

$\frac{8}{9} + \frac{2}{9 \frac{4}{8^{\frac{4}{3}}}}$

Step 9.1.5.3

Simplify the denominator.

Tap for more steps...

Step 9.1.5.3.1

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

$\frac{8}{9} + \frac{2}{9 \frac{4}{{(2^{3})}^{\frac{4}{3}}}}$

Step 9.1.5.3.2

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

$\frac{8}{9} + \frac{2}{9 \frac{4}{2^{3 (\frac{4}{3})}}}$

Step 9.1.5.3.3

Cancel the common factor of $3$ .

Tap for more steps...

Step 9.1.5.3.3.1

Cancel the common factor.

$\frac{8}{9} + \frac{2}{9 \frac{4}{2^{3 (\frac{4}{3})}}}$

Step 9.1.5.3.3.2

Rewrite the expression.

$\frac{8}{9} + \frac{2}{9 \frac{4}{2^{4}}}$

Step 9.1.5.3.4

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

$\frac{8}{9} + \frac{2}{9 (\frac{4}{16})}$

Step 9.1.5.4

Cancel the common factor of $4$ and $16$ .

Tap for more steps...

Step 9.1.5.4.1

Factor $4$ out of $4$ .

$\frac{8}{9} + \frac{2}{9 \frac{4 (1)}{16}}$

Step 9.1.5.4.2

Cancel the common factors.

Tap for more steps...

Step 9.1.5.4.2.1

Factor $4$ out of $16$ .

$\frac{8}{9} + \frac{2}{9 \frac{4 \cdot 1}{4 \cdot 4}}$

Step 9.1.5.4.2.2

Cancel the common factor.

$\frac{8}{9} + \frac{2}{9 \frac{4 \cdot 1}{4 \cdot 4}}$

Step 9.1.5.4.2.3

Rewrite the expression.

$\frac{8}{9} + \frac{2}{9 (\frac{1}{4})}$

Step 9.1.6

Combine $9$ and $\frac{1}{4}$ .

$\frac{8}{9} + \frac{2}{\frac{9}{4}}$

Step 9.1.7

Multiply the numerator by the reciprocal of the denominator.

$\frac{8}{9} + 2 (\frac{4}{9})$

Step 9.1.8

Multiply $2 (\frac{4}{9})$ .

Tap for more steps...

Step 9.1.8.1

Combine $2$ and $\frac{4}{9}$ .

$\frac{8}{9} + \frac{2 \cdot 4}{9}$

Step 9.1.8.2

Multiply $2$ by $4$ .

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

Step 9.2

Combine fractions.

Tap for more steps...

Step 9.2.1

Combine the numerators over the common denominator.

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

Step 9.2.2

Add $8$ and $8$ .

$\frac{16}{9}$

Step 10

$x = \frac{2^{\frac{3}{2}}}{8}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = \frac{2^{\frac{3}{2}}}{8}$ is a local minimum

Step 11

Find the y-value when $x = \frac{2^{\frac{3}{2}}}{8}$ .

Tap for more steps...

Step 11.1

Replace the variable $x$ with $\frac{2^{\frac{3}{2}}}{8}$ in the expression.

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

Step 11.2

Simplify the result.

Tap for more steps...

Step 11.2.1

Simplify each term.

Tap for more steps...

Step 11.2.1.1

Apply the product rule to $\frac{2^{\frac{3}{2}}}{8}$ .

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

Step 11.2.1.2

Simplify the numerator.

Tap for more steps...

Step 11.2.1.2.1

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

Tap for more steps...

Step 11.2.1.2.1.1

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

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

Step 11.2.1.2.1.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 11.2.1.2.1.2.1

Cancel the common factor.

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

Step 11.2.1.2.1.2.2

Rewrite the expression.

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

Step 11.2.1.2.1.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 11.2.1.2.1.3.1

Factor $2$ out of $4$ .

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

Step 11.2.1.2.1.3.2

Cancel the common factor.

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

Step 11.2.1.2.1.3.3

Rewrite the expression.

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

Step 11.2.1.2.2

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

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

Step 11.2.1.3

Simplify the denominator.

Tap for more steps...

Step 11.2.1.3.1

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

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

Step 11.2.1.3.2

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

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

Step 11.2.1.3.3

Cancel the common factor of $3$ .

Tap for more steps...

Step 11.2.1.3.3.1

Cancel the common factor.

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

Step 11.2.1.3.3.2

Rewrite the expression.

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

Step 11.2.1.3.4

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

$f (\frac{2^{\frac{3}{2}}}{8}) = \frac{4}{16} - {(\frac{2^{\frac{3}{2}}}{8})}^{\frac{2}{3}}$

Step 11.2.1.4

Cancel the common factor of $4$ and $16$ .

Tap for more steps...

Step 11.2.1.4.1

Factor $4$ out of $4$ .

$f (\frac{2^{\frac{3}{2}}}{8}) = \frac{4 (1)}{16} - {(\frac{2^{\frac{3}{2}}}{8})}^{\frac{2}{3}}$

Step 11.2.1.4.2

Cancel the common factors.

Tap for more steps...

Step 11.2.1.4.2.1

Factor $4$ out of $16$ .

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

Step 11.2.1.4.2.2

Cancel the common factor.

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

Step 11.2.1.4.2.3

Rewrite the expression.

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

Step 11.2.1.5

Apply the product rule to $\frac{2^{\frac{3}{2}}}{8}$ .

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

Step 11.2.1.6

Simplify the numerator.

Tap for more steps...

Step 11.2.1.6.1

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

Tap for more steps...

Step 11.2.1.6.1.1

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

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

Step 11.2.1.6.1.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 11.2.1.6.1.2.1

Cancel the common factor.

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

Step 11.2.1.6.1.2.2

Rewrite the expression.

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

Step 11.2.1.6.1.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 11.2.1.6.1.3.1

Cancel the common factor.

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

Step 11.2.1.6.1.3.2

Rewrite the expression.

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

Step 11.2.1.6.2

Evaluate the exponent.

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

Step 11.2.1.7

Simplify the denominator.

Tap for more steps...

Step 11.2.1.7.1

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

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

Step 11.2.1.7.2

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

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

Step 11.2.1.7.3

Cancel the common factor of $3$ .

Tap for more steps...

Step 11.2.1.7.3.1

Cancel the common factor.

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

Step 11.2.1.7.3.2

Rewrite the expression.

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

Step 11.2.1.7.4

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

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

Step 11.2.1.8

Cancel the common factor of $2$ and $4$ .

Tap for more steps...

Step 11.2.1.8.1

Factor $2$ out of $2$ .

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

Step 11.2.1.8.2

Cancel the common factors.

Tap for more steps...

Step 11.2.1.8.2.1

Factor $2$ out of $4$ .

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

Step 11.2.1.8.2.2

Cancel the common factor.

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

Step 11.2.1.8.2.3

Rewrite the expression.

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

Step 11.2.2

To write $- \frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 11.2.3

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

Tap for more steps...

Step 11.2.3.1

Multiply $\frac{1}{2}$ by $\frac{2}{2}$ .

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

Step 11.2.3.2

Multiply $2$ by $2$ .

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

Step 11.2.4

Combine the numerators over the common denominator.

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

Step 11.2.5

Subtract $2$ from $1$ .

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

Step 11.2.6

Move the negative in front of the fraction.

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

Step 11.2.7

The final answer is $- \frac{1}{4}$ .

$y = - \frac{1}{4}$

Step 12

Evaluate the second derivative at $x = - \frac{2^{\frac{3}{2}}}{8}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

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

Step 13

Evaluate the second derivative.

Tap for more steps...

Step 13.1

Simplify each term.

Tap for more steps...

Step 13.1.1

Simplify the denominator.

Tap for more steps...

Step 13.1.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 13.1.1.1.1

Apply the product rule to $- \frac{2^{\frac{3}{2}}}{8}$ .

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

Step 13.1.1.1.2

Apply the product rule to $\frac{2^{\frac{3}{2}}}{8}$ .

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

Step 13.1.1.2

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

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

Step 13.1.1.3

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

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

Step 13.1.1.4

Cancel the common factor of $3$ .

Tap for more steps...

Step 13.1.1.4.1

Cancel the common factor.

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

Step 13.1.1.4.2

Rewrite the expression.

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

Step 13.1.1.5

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

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

Step 13.1.1.6

Multiply $\frac{{(2^{\frac{3}{2}})}^{\frac{2}{3}}}{8^{\frac{2}{3}}}$ by $1$ .

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

Step 13.1.1.7

Simplify the numerator.

Tap for more steps...

Step 13.1.1.7.1

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

Tap for more steps...

Step 13.1.1.7.1.1

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

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

Step 13.1.1.7.1.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 13.1.1.7.1.2.1

Cancel the common factor.

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

Step 13.1.1.7.1.2.2

Rewrite the expression.

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

Step 13.1.1.7.1.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 13.1.1.7.1.3.1

Cancel the common factor.

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

Step 13.1.1.7.1.3.2

Rewrite the expression.

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

Step 13.1.1.7.2

Evaluate the exponent.

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

Step 13.1.1.8

Simplify the denominator.

Tap for more steps...

Step 13.1.1.8.1

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

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

Step 13.1.1.8.2

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

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

Step 13.1.1.8.3

Cancel the common factor of $3$ .

Tap for more steps...

Step 13.1.1.8.3.1

Cancel the common factor.

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

Step 13.1.1.8.3.2

Rewrite the expression.

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

Step 13.1.1.8.4

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

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

Step 13.1.1.9

Cancel the common factor of $2$ and $4$ .

Tap for more steps...

Step 13.1.1.9.1

Factor $2$ out of $2$ .

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

Step 13.1.1.9.2

Cancel the common factors.

Tap for more steps...

Step 13.1.1.9.2.1

Factor $2$ out of $4$ .

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

Step 13.1.1.9.2.2

Cancel the common factor.

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

Step 13.1.1.9.2.3

Rewrite the expression.

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

Step 13.1.2

Combine $9$ and $\frac{1}{2}$ .

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

Step 13.1.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 13.1.4

Multiply $4 (\frac{2}{9})$ .

Tap for more steps...

Step 13.1.4.1

Combine $4$ and $\frac{2}{9}$ .

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

Step 13.1.4.2

Multiply $4$ by $2$ .

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

Step 13.1.5

Simplify the denominator.

Tap for more steps...

Step 13.1.5.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 13.1.5.1.1

Apply the product rule to $- \frac{2^{\frac{3}{2}}}{8}$ .

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

Step 13.1.5.1.2

Apply the product rule to $\frac{2^{\frac{3}{2}}}{8}$ .

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

Step 13.1.5.2

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

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

Step 13.1.5.3

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

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

Step 13.1.5.4

Cancel the common factor of $3$ .

Tap for more steps...

Step 13.1.5.4.1

Cancel the common factor.

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

Step 13.1.5.4.2

Rewrite the expression.

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

Step 13.1.5.5

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

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

Step 13.1.5.6

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

$\frac{8}{9} + \frac{2}{9 \frac{{(2^{\frac{3}{2}})}^{\frac{4}{3}}}{8^{\frac{4}{3}}}}$

Step 13.1.5.7

Simplify the numerator.

Tap for more steps...

Step 13.1.5.7.1

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

Tap for more steps...

Step 13.1.5.7.1.1

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

$\frac{8}{9} + \frac{2}{9 \frac{2^{\frac{3}{2} \cdot \frac{4}{3}}}{8^{\frac{4}{3}}}}$

Step 13.1.5.7.1.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 13.1.5.7.1.2.1

Cancel the common factor.

$\frac{8}{9} + \frac{2}{9 \frac{2^{\frac{3}{2} \cdot \frac{4}{3}}}{8^{\frac{4}{3}}}}$

Step 13.1.5.7.1.2.2

Rewrite the expression.

$\frac{8}{9} + \frac{2}{9 \frac{2^{\frac{1}{2} \cdot 4}}{8^{\frac{4}{3}}}}$

Step 13.1.5.7.1.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 13.1.5.7.1.3.1

Factor $2$ out of $4$ .

$\frac{8}{9} + \frac{2}{9 \frac{2^{\frac{1}{2} \cdot (2 (2))}}{8^{\frac{4}{3}}}}$

Step 13.1.5.7.1.3.2

Cancel the common factor.

$\frac{8}{9} + \frac{2}{9 \frac{2^{\frac{1}{2} \cdot (2 \cdot 2)}}{8^{\frac{4}{3}}}}$

Step 13.1.5.7.1.3.3

Rewrite the expression.

$\frac{8}{9} + \frac{2}{9 \frac{2^{2}}{8^{\frac{4}{3}}}}$

Step 13.1.5.7.2

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

$\frac{8}{9} + \frac{2}{9 \frac{4}{8^{\frac{4}{3}}}}$

Step 13.1.5.8

Simplify the denominator.

Tap for more steps...

Step 13.1.5.8.1

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

$\frac{8}{9} + \frac{2}{9 \frac{4}{{(2^{3})}^{\frac{4}{3}}}}$

Step 13.1.5.8.2

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

$\frac{8}{9} + \frac{2}{9 \frac{4}{2^{3 (\frac{4}{3})}}}$

Step 13.1.5.8.3

Cancel the common factor of $3$ .

Tap for more steps...

Step 13.1.5.8.3.1

Cancel the common factor.

$\frac{8}{9} + \frac{2}{9 \frac{4}{2^{3 (\frac{4}{3})}}}$

Step 13.1.5.8.3.2

Rewrite the expression.

$\frac{8}{9} + \frac{2}{9 \frac{4}{2^{4}}}$

Step 13.1.5.8.4

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

$\frac{8}{9} + \frac{2}{9 (\frac{4}{16})}$

Step 13.1.5.9

Cancel the common factor of $4$ and $16$ .

Tap for more steps...

Step 13.1.5.9.1

Factor $4$ out of $4$ .

$\frac{8}{9} + \frac{2}{9 \frac{4 (1)}{16}}$

Step 13.1.5.9.2

Cancel the common factors.

Tap for more steps...

Step 13.1.5.9.2.1

Factor $4$ out of $16$ .

$\frac{8}{9} + \frac{2}{9 \frac{4 \cdot 1}{4 \cdot 4}}$

Step 13.1.5.9.2.2

Cancel the common factor.

$\frac{8}{9} + \frac{2}{9 \frac{4 \cdot 1}{4 \cdot 4}}$

Step 13.1.5.9.2.3

Rewrite the expression.

$\frac{8}{9} + \frac{2}{9 (\frac{1}{4})}$

Step 13.1.6

Combine $9$ and $\frac{1}{4}$ .

$\frac{8}{9} + \frac{2}{\frac{9}{4}}$

Step 13.1.7

Multiply the numerator by the reciprocal of the denominator.

$\frac{8}{9} + 2 (\frac{4}{9})$

Step 13.1.8

Multiply $2 (\frac{4}{9})$ .

Tap for more steps...

Step 13.1.8.1

Combine $2$ and $\frac{4}{9}$ .

$\frac{8}{9} + \frac{2 \cdot 4}{9}$

Step 13.1.8.2

Multiply $2$ by $4$ .

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

Step 13.2

Combine fractions.

Tap for more steps...

Step 13.2.1

Combine the numerators over the common denominator.

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

Step 13.2.2

Add $8$ and $8$ .

$\frac{16}{9}$

Step 14

$x = - \frac{2^{\frac{3}{2}}}{8}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = - \frac{2^{\frac{3}{2}}}{8}$ is a local minimum

Step 15

Find the y-value when $x = - \frac{2^{\frac{3}{2}}}{8}$ .

Tap for more steps...

Step 15.1

Replace the variable $x$ with $- \frac{2^{\frac{3}{2}}}{8}$ in the expression.

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

Step 15.2

Simplify the result.

Tap for more steps...

Step 15.2.1

Simplify each term.

Tap for more steps...

Step 15.2.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 15.2.1.1.1

Apply the product rule to $- \frac{2^{\frac{3}{2}}}{8}$ .

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

Step 15.2.1.1.2

Apply the product rule to $\frac{2^{\frac{3}{2}}}{8}$ .

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

Step 15.2.1.2

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

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

Step 15.2.1.3

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

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

Step 15.2.1.4

Cancel the common factor of $3$ .

Tap for more steps...

Step 15.2.1.4.1

Cancel the common factor.

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

Step 15.2.1.4.2

Rewrite the expression.

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

Step 15.2.1.5

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

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

Step 15.2.1.6

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

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

Step 15.2.1.7

Simplify the numerator.

Tap for more steps...

Step 15.2.1.7.1

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

Tap for more steps...

Step 15.2.1.7.1.1

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

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

Step 15.2.1.7.1.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 15.2.1.7.1.2.1

Cancel the common factor.

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

Step 15.2.1.7.1.2.2

Rewrite the expression.

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

Step 15.2.1.7.1.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 15.2.1.7.1.3.1

Factor $2$ out of $4$ .

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

Step 15.2.1.7.1.3.2

Cancel the common factor.

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

Step 15.2.1.7.1.3.3

Rewrite the expression.

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

Step 15.2.1.7.2

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

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

Step 15.2.1.8

Simplify the denominator.

Tap for more steps...

Step 15.2.1.8.1

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

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

Step 15.2.1.8.2

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

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

Step 15.2.1.8.3

Cancel the common factor of $3$ .

Tap for more steps...

Step 15.2.1.8.3.1

Cancel the common factor.

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

Step 15.2.1.8.3.2

Rewrite the expression.

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

Step 15.2.1.8.4

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

$f (- \frac{2^{\frac{3}{2}}}{8}) = \frac{4}{16} - {(- \frac{2^{\frac{3}{2}}}{8})}^{\frac{2}{3}}$

Step 15.2.1.9

Cancel the common factor of $4$ and $16$ .

Tap for more steps...

Step 15.2.1.9.1

Factor $4$ out of $4$ .

$f (- \frac{2^{\frac{3}{2}}}{8}) = \frac{4 (1)}{16} - {(- \frac{2^{\frac{3}{2}}}{8})}^{\frac{2}{3}}$

Step 15.2.1.9.2

Cancel the common factors.

Tap for more steps...

Step 15.2.1.9.2.1

Factor $4$ out of $16$ .

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

Step 15.2.1.9.2.2

Cancel the common factor.

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

Step 15.2.1.9.2.3

Rewrite the expression.

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

Step 15.2.1.10

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 15.2.1.10.1

Apply the product rule to $- \frac{2^{\frac{3}{2}}}{8}$ .

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

Step 15.2.1.10.2

Apply the product rule to $\frac{2^{\frac{3}{2}}}{8}$ .

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

Step 15.2.1.11

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

Tap for more steps...

Step 15.2.1.11.1

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

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

Step 15.2.1.11.2

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

Tap for more steps...

Step 15.2.1.11.2.1

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

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

Step 15.2.1.11.2.2

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

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

Step 15.2.1.11.3

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

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

Step 15.2.1.11.4

Combine the numerators over the common denominator.

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

Step 15.2.1.11.5

Add $2$ and $3$ .

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

Step 15.2.1.12

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

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

Step 15.2.1.13

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

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

Step 15.2.1.14

Cancel the common factor of $3$ .

Tap for more steps...

Step 15.2.1.14.1

Cancel the common factor.

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

Step 15.2.1.14.2

Rewrite the expression.

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

Step 15.2.1.15

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

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

Step 15.2.1.16

Simplify the numerator.

Tap for more steps...

Step 15.2.1.16.1

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

Tap for more steps...

Step 15.2.1.16.1.1

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

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

Step 15.2.1.16.1.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 15.2.1.16.1.2.1

Cancel the common factor.

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

Step 15.2.1.16.1.2.2

Rewrite the expression.

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

Step 15.2.1.16.1.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 15.2.1.16.1.3.1

Cancel the common factor.

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

Step 15.2.1.16.1.3.2

Rewrite the expression.

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

Step 15.2.1.16.2

Evaluate the exponent.

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

Step 15.2.1.17

Simplify the denominator.

Tap for more steps...

Step 15.2.1.17.1

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

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

Step 15.2.1.17.2

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

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

Step 15.2.1.17.3

Cancel the common factor of $3$ .

Tap for more steps...

Step 15.2.1.17.3.1

Cancel the common factor.

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

Step 15.2.1.17.3.2

Rewrite the expression.

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

Step 15.2.1.17.4

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

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

Step 15.2.1.18

Cancel the common factor of $2$ and $4$ .

Tap for more steps...

Step 15.2.1.18.1

Factor $2$ out of $2$ .

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

Step 15.2.1.18.2

Cancel the common factors.

Tap for more steps...

Step 15.2.1.18.2.1

Factor $2$ out of $4$ .

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

Step 15.2.1.18.2.2

Cancel the common factor.

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

Step 15.2.1.18.2.3

Rewrite the expression.

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

Step 15.2.2

To write $- \frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 15.2.3

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

Tap for more steps...

Step 15.2.3.1

Multiply $\frac{1}{2}$ by $\frac{2}{2}$ .

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

Step 15.2.3.2

Multiply $2$ by $2$ .

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

Step 15.2.4

Combine the numerators over the common denominator.

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

Step 15.2.5

Subtract $2$ from $1$ .

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

Step 15.2.6

Move the negative in front of the fraction.

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

Step 15.2.7

The final answer is $- \frac{1}{4}$ .

$y = - \frac{1}{4}$

Step 16

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.

$\frac{4}{9 {(0)}^{\frac{2}{3}}} + \frac{2}{9 {(0)}^{\frac{4}{3}}}$

Step 17

Evaluate the second derivative.

Tap for more steps...

Step 17.1

Simplify the expression.

Tap for more steps...

Step 17.1.1

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

$\frac{4}{9 {(0^{3})}^{\frac{2}{3}}} + \frac{2}{9 {(0)}^{\frac{4}{3}}}$

Step 17.1.2

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

$\frac{4}{9 \cdot 0^{3 (\frac{2}{3})}} + \frac{2}{9 {(0)}^{\frac{4}{3}}}$

Step 17.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 17.2.1

Cancel the common factor.

$\frac{4}{9 \cdot 0^{3 (\frac{2}{3})}} + \frac{2}{9 {(0)}^{\frac{4}{3}}}$

Step 17.2.2

Rewrite the expression.

$\frac{4}{9 \cdot 0^{2}} + \frac{2}{9 {(0)}^{\frac{4}{3}}}$

Step 17.3

Simplify the expression.

Tap for more steps...

Step 17.3.1

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

$\frac{4}{9 \cdot 0} + \frac{2}{9 {(0)}^{\frac{4}{3}}}$

Step 17.3.2

Multiply $9$ by $0$ .

$\frac{4}{0} + \frac{2}{9 {(0)}^{\frac{4}{3}}}$

Step 17.3.3

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

Undefined

$\frac{4}{0} + \frac{2}{9 {(0)}^{\frac{4}{3}}}$

Step 17.4

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

Undefined

Step 18

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

Tap for more steps...

Step 18.1

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

$(- \infty, - \frac{2^{\frac{3}{2}}}{8}) \cup (- \frac{2^{\frac{3}{2}}}{8}, 0) \cup (0, \frac{2^{\frac{3}{2}}}{8}) \cup (\frac{2^{\frac{3}{2}}}{8}, \infty)$

Step 18.2

Substitute any number, such as $- 3$ , from the interval $(- \infty, - \frac{2^{\frac{3}{2}}}{8})$ in the first derivative $\frac{4 x^{\frac{1}{3}}}{3} - \frac{2}{3 x^{\frac{1}{3}}}$ to check if the result is negative or positive.

Tap for more steps...

Step 18.2.1

Replace the variable $x$ with $- 3$ in the expression.

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

Step 18.2.2

Simplify the result.

Tap for more steps...

Step 18.2.2.1

Simplify each term.

Tap for more steps...

Step 18.2.2.1.1

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

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

Step 18.2.2.1.2

Factor $- 3$ out of $4 {(- 3)}^{\frac{1}{3}}$ .

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

Step 18.2.2.1.3

Move the negative one from the denominator of $\frac{4 {(- 3)}^{\frac{- 2}{3}}}{- 1}$ .

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

Step 18.2.2.1.4

Rewrite $- 1 \cdot (4 {(- 3)}^{\frac{- 2}{3}})$ as $- (4 {(- 3)}^{\frac{- 2}{3}})$ .

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

Step 18.2.2.1.5

Move the negative in front of the fraction.

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

Step 18.2.2.1.6

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 18.2.2.1.7

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

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

Step 18.2.2.2

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

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

Step 18.2.2.3

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

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

Step 18.2.2.4

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

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

Step 18.2.2.5

Write each expression with a common denominator of $- - 3 {(- 3)}^{\frac{2}{3}}$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 18.2.2.5.1

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

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

Step 18.2.2.5.2

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

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

Step 18.2.2.5.3

Multiply $- 1$ by $- 3$ .

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

Step 18.2.2.5.4

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

Tap for more steps...

Step 18.2.2.5.4.1

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

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

Step 18.2.2.5.4.2

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

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

Step 18.2.2.5.4.3

Combine the numerators over the common denominator.

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

Step 18.2.2.5.4.4

Add $1$ and $1$ .

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

Step 18.2.2.5.5

Reorder the factors of ${(- 3)}^{\frac{2}{3}} \cdot 3$ .

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

Step 18.2.2.6

Simplify the expression.

Tap for more steps...

Step 18.2.2.6.1

Combine the numerators over the common denominator.

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

Step 18.2.2.6.2

Multiply $- 4$ by $3$ .

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

Step 18.2.2.7

Rewrite $- 12$ as $- 1 (12)$ .

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

Step 18.2.2.8

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

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

Step 18.2.2.9

Factor $- 1$ out of $- 1 (12) - (2 {(- 3)}^{\frac{1}{3}})$ .

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

Step 18.2.2.10

Move the negative in front of the fraction.

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

Step 18.2.2.11

The final answer is $- \frac{12 + 2 {(- 3)}^{\frac{1}{3}}}{3 {(- 3)}^{\frac{2}{3}}}$ .

$- \frac{12 + 2 {(- 3)}^{\frac{1}{3}}}{3 {(- 3)}^{\frac{2}{3}}}$

Step 18.3

Substitute any number, such as $- 0.18$ , from the interval $(- \frac{2^{\frac{3}{2}}}{8}, 0)$ in the first derivative $\frac{4 x^{\frac{1}{3}}}{3} - \frac{2}{3 x^{\frac{1}{3}}}$ to check if the result is negative or positive.

Tap for more steps...

Step 18.3.1

Replace the variable $x$ with $- 0.18$ in the expression.

$f' (- 0.18) = \frac{4 {(- 0.18)}^{\frac{1}{3}}}{3} - \frac{2}{3 {(- 0.18)}^{\frac{1}{3}}}$

Step 18.3.2

The final answer is $\frac{4 {(- 0.18)}^{\frac{1}{3}}}{3} - \frac{2}{3 {(- 0.18)}^{\frac{1}{3}}}$ .

$\frac{4 {(- 0.18)}^{\frac{1}{3}}}{3} - \frac{2}{3 {(- 0.18)}^{\frac{1}{3}}}$

Step 18.4

Substitute any number, such as $0.18$ , from the interval $(0, \frac{2^{\frac{3}{2}}}{8})$ in the first derivative $\frac{4 x^{\frac{1}{3}}}{3} - \frac{2}{3 x^{\frac{1}{3}}}$ to check if the result is negative or positive.

Tap for more steps...

Step 18.4.1

Replace the variable $x$ with $0.18$ in the expression.

$f' (0.18) = \frac{4 {(0.18)}^{\frac{1}{3}}}{3} - \frac{2}{3 {(0.18)}^{\frac{1}{3}}}$

Step 18.4.2

Simplify the result.

Tap for more steps...

Step 18.4.2.1

Simplify each term.

Tap for more steps...

Step 18.4.2.1.1

Raise $0.18$ to the power of $\frac{1}{3}$ .

$f' (0.18) = \frac{4 \cdot 0.56462161}{3} - \frac{2}{3 {(0.18)}^{\frac{1}{3}}}$

Step 18.4.2.1.2

Multiply $4$ by $0.56462161$ .

$f' (0.18) = \frac{2.25848646}{3} - \frac{2}{3 {(0.18)}^{\frac{1}{3}}}$

Step 18.4.2.1.3

Divide $2.25848646$ by $3$ .

$f' (0.18) = 0.75282882 - \frac{2}{3 {(0.18)}^{\frac{1}{3}}}$

Step 18.4.2.1.4

Raise $0.18$ to the power of $\frac{1}{3}$ .

$f' (0.18) = 0.75282882 - \frac{2}{3 \cdot 0.56462161}$

Step 18.4.2.1.5

Multiply $3$ by $0.56462161$ .

$f' (0.18) = 0.75282882 - \frac{2}{1.69386485}$

Step 18.4.2.1.6

Divide $2$ by $1.69386485$ .

$f' (0.18) = 0.75282882 - 1 \cdot 1.18073174$

Step 18.4.2.1.7

Multiply $- 1$ by $1.18073174$ .

$f' (0.18) = 0.75282882 - 1.18073174$

Step 18.4.2.2

Subtract $1.18073174$ from $0.75282882$ .

$f' (0.18) = - 0.42790292$

Step 18.4.2.3

The final answer is $- 0.42790292$ .

$- 0.42790292$

Step 18.5

Substitute any number, such as $3$ , from the interval $(\frac{2^{\frac{3}{2}}}{8}, \infty)$ in the first derivative $\frac{4 x^{\frac{1}{3}}}{3} - \frac{2}{3 x^{\frac{1}{3}}}$ to check if the result is negative or positive.

Tap for more steps...

Step 18.5.1

Replace the variable $x$ with $3$ in the expression.

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

Step 18.5.2

Simplify the result.

Tap for more steps...

Step 18.5.2.1

Simplify each term.

Tap for more steps...

Step 18.5.2.1.1

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

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

Step 18.5.2.1.2

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

Tap for more steps...

Step 18.5.2.1.2.1

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

Tap for more steps...

Step 18.5.2.1.2.1.1

Raise $3$ to the power of $1$ .

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

Step 18.5.2.1.2.1.2

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

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

Step 18.5.2.1.2.2

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

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

Step 18.5.2.1.2.3

Combine the numerators over the common denominator.

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

Step 18.5.2.1.2.4

Subtract $1$ from $3$ .

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

Step 18.5.2.1.3

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

Tap for more steps...

Step 18.5.2.1.3.1

Multiply $3$ by ${(3)}^{\frac{1}{3}}$ .

Tap for more steps...

Step 18.5.2.1.3.1.1

Raise $3$ to the power of $1$ .

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

Step 18.5.2.1.3.1.2

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

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

Step 18.5.2.1.3.2

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

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

Step 18.5.2.1.3.3

Combine the numerators over the common denominator.

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

Step 18.5.2.1.3.4

Add $3$ and $1$ .

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

Step 18.5.2.2

To write $\frac{4}{3^{\frac{2}{3}}}$ as a fraction with a common denominator, multiply by $\frac{3^{\frac{2}{3}}}{3^{\frac{2}{3}}}$ .

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

Step 18.5.2.3

Write each expression with a common denominator of $3^{\frac{4}{3}}$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 18.5.2.3.1

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

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

Step 18.5.2.3.2

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

Tap for more steps...

Step 18.5.2.3.2.1

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

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

Step 18.5.2.3.2.2

Combine the numerators over the common denominator.

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

Step 18.5.2.3.2.3

Add $2$ and $2$ .

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

Step 18.5.2.4

Combine the numerators over the common denominator.

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

Step 18.5.2.5

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

$\frac{4 \cdot 3^{\frac{2}{3}} - 2}{3^{\frac{4}{3}}}$

Step 18.6

Since the first derivative changed signs from negative to positive around $x = - \frac{2^{\frac{3}{2}}}{8}$ , then $x = - \frac{2^{\frac{3}{2}}}{8}$ is a local minimum.

$x = - \frac{2^{\frac{3}{2}}}{8}$ is a local minimum

Step 18.7

Since the first derivative changed signs from positive to negative around $x = 0$ , then $x = 0$ is a local maximum.

$x = 0$ is a local maximum

Step 18.8

Since the first derivative changed signs from negative to positive around $x = \frac{2^{\frac{3}{2}}}{8}$ , then $x = \frac{2^{\frac{3}{2}}}{8}$ is a local minimum.

$x = \frac{2^{\frac{3}{2}}}{8}$ is a local minimum

Step 18.9

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

$x = - \frac{2^{\frac{3}{2}}}{8}$ is a local minimum

$x = 0$ is a local maximum

$x = \frac{2^{\frac{3}{2}}}{8}$ is a local minimum

$x = - \frac{2^{\frac{3}{2}}}{8}$ is a local minimum

$x = 0$ is a local maximum

$x = \frac{2^{\frac{3}{2}}}{8}$ is a local minimum

Step 19