Calculus Examples

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

Step 1

Find the first derivative of the function.

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Step 1.1

By the Sum Rule, the derivative of $4 x^{\frac{3}{5}} - x^{\frac{4}{5}}$ with respect to $x$ is $\frac{d}{d x} [4 x^{\frac{3}{5}}] + \frac{d}{d x} [- x^{\frac{4}{5}}]$ .

$\frac{d}{d x} [4 x^{\frac{3}{5}}] + \frac{d}{d x} [- x^{\frac{4}{5}}]$

Step 1.2

Evaluate $\frac{d}{d x} [4 x^{\frac{3}{5}}]$ .

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Step 1.2.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{\frac{3}{5}}$ with respect to $x$ is $4 \frac{d}{d x} [x^{\frac{3}{5}}]$ .

$4 \frac{d}{d x} [x^{\frac{3}{5}}] + \frac{d}{d x} [- x^{\frac{4}{5}}]$

Step 1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{3}{5}$ .

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

Step 1.2.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

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

Step 1.2.4

Combine $- 1$ and $\frac{5}{5}$ .

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

Step 1.2.5

Combine the numerators over the common denominator.

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

Step 1.2.6

Simplify the numerator.

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Step 1.2.6.1

Multiply $- 1$ by $5$ .

$4 (\frac{3}{5} x^{\frac{3 - 5}{5}}) + \frac{d}{d x} [- x^{\frac{4}{5}}]$

Step 1.2.6.2

Subtract $5$ from $3$ .

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

Step 1.2.7

Move the negative in front of the fraction.

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

Step 1.2.8

Combine $\frac{3}{5}$ and $x^{- \frac{2}{5}}$ .

$4 \frac{3 x^{- \frac{2}{5}}}{5} + \frac{d}{d x} [- x^{\frac{4}{5}}]$

Step 1.2.9

Combine $4$ and $\frac{3 x^{- \frac{2}{5}}}{5}$ .

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

Step 1.2.10

Multiply $3$ by $4$ .

$\frac{12 x^{- \frac{2}{5}}}{5} + \frac{d}{d x} [- x^{\frac{4}{5}}]$

Step 1.2.11

Move $x^{- \frac{2}{5}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{12}{5 x^{\frac{2}{5}}} + \frac{d}{d x} [- x^{\frac{4}{5}}]$

Step 1.3

Evaluate $\frac{d}{d x} [- x^{\frac{4}{5}}]$ .

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Step 1.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{\frac{4}{5}}$ with respect to $x$ is $- \frac{d}{d x} [x^{\frac{4}{5}}]$ .

$\frac{12}{5 x^{\frac{2}{5}}} - \frac{d}{d x} [x^{\frac{4}{5}}]$

Step 1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{4}{5}$ .

$\frac{12}{5 x^{\frac{2}{5}}} - (\frac{4}{5} x^{\frac{4}{5} - 1})$

Step 1.3.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{12}{5 x^{\frac{2}{5}}} - (\frac{4}{5} x^{\frac{4}{5} - 1 \cdot \frac{5}{5}})$

Step 1.3.4

Combine $- 1$ and $\frac{5}{5}$ .

$\frac{12}{5 x^{\frac{2}{5}}} - (\frac{4}{5} x^{\frac{4}{5} + \frac{- 1 \cdot 5}{5}})$

Step 1.3.5

Combine the numerators over the common denominator.

$\frac{12}{5 x^{\frac{2}{5}}} - (\frac{4}{5} x^{\frac{4 - 1 \cdot 5}{5}})$

Step 1.3.6

Simplify the numerator.

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Step 1.3.6.1

Multiply $- 1$ by $5$ .

$\frac{12}{5 x^{\frac{2}{5}}} - (\frac{4}{5} x^{\frac{4 - 5}{5}})$

Step 1.3.6.2

Subtract $5$ from $4$ .

$\frac{12}{5 x^{\frac{2}{5}}} - (\frac{4}{5} x^{\frac{- 1}{5}})$

Step 1.3.7

Move the negative in front of the fraction.

$\frac{12}{5 x^{\frac{2}{5}}} - (\frac{4}{5} x^{- \frac{1}{5}})$

Step 1.3.8

Combine $\frac{4}{5}$ and $x^{- \frac{1}{5}}$ .

$\frac{12}{5 x^{\frac{2}{5}}} - \frac{4 x^{- \frac{1}{5}}}{5}$

Step 1.3.9

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

$\frac{12}{5 x^{\frac{2}{5}}} - \frac{4}{5 x^{\frac{1}{5}}}$

Step 2

Find the second derivative of the function.

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Step 2.1

By the Sum Rule, the derivative of $\frac{12}{5 x^{\frac{2}{5}}} - \frac{4}{5 x^{\frac{1}{5}}}$ with respect to $x$ is $\frac{d}{d x} [\frac{12}{5 x^{\frac{2}{5}}}] + \frac{d}{d x} [- \frac{4}{5 x^{\frac{1}{5}}}]$ .

$f'' (x) = \frac{d}{d x} (\frac{12}{5 x^{\frac{2}{5}}}) + \frac{d}{d x} (- \frac{4}{5 x^{\frac{1}{5}}})$

Step 2.2

Evaluate $\frac{d}{d x} [\frac{12}{5 x^{\frac{2}{5}}}]$ .

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Step 2.2.1

Since $\frac{12}{5}$ is constant with respect to $x$ , the derivative of $\frac{12}{5 x^{\frac{2}{5}}}$ with respect to $x$ is $\frac{12}{5} \frac{d}{d x} [\frac{1}{x^{\frac{2}{5}}}]$ .

$f'' (x) = \frac{12}{5} \cdot \frac{d}{d x} (\frac{1}{x^{\frac{2}{5}}}) + \frac{d}{d x} (- \frac{4}{5 x^{\frac{1}{5}}})$

Step 2.2.2

Rewrite $\frac{1}{x^{\frac{2}{5}}}$ as ${(x^{\frac{2}{5}})}^{- 1}$ .

$f'' (x) = \frac{12}{5} \cdot \frac{d}{d x} ({(x^{\frac{2}{5}})}^{- 1}) + \frac{d}{d x} (- \frac{4}{5 x^{\frac{1}{5}}})$

Step 2.2.3

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 $g (x) = x^{\frac{2}{5}}$ .

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Step 2.2.3.1

To apply the Chain Rule, set $u_{1}$ as $x^{\frac{2}{5}}$ .

$f'' (x) = \frac{12}{5} \cdot (\frac{d}{d u (1)} ({u_{1}}^{- 1}) \frac{d}{d x} (x^{\frac{2}{5}})) + \frac{d}{d x} (- \frac{4}{5 x^{\frac{1}{5}}})$

Step 2.2.3.2

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

$f'' (x) = \frac{12}{5} \cdot (- {u_{1}}^{- 2} \frac{d}{d x} x^{\frac{2}{5}}) + \frac{d}{d x} (- \frac{4}{5 x^{\frac{1}{5}}})$

Step 2.2.3.3

Replace all occurrences of $u_{1}$ with $x^{\frac{2}{5}}$ .

$f'' (x) = \frac{12}{5} \cdot (- {(x^{\frac{2}{5}})}^{- 2} \frac{d}{d x} x^{\frac{2}{5}}) + \frac{d}{d x} (- \frac{4}{5 x^{\frac{1}{5}}})$

Step 2.2.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{2}{5}$ .

$f'' (x) = \frac{12}{5} \cdot (- {(x^{\frac{2}{5}})}^{- 2} (\frac{2}{5} x^{\frac{2}{5} - 1})) + \frac{d}{d x} (- \frac{4}{5 x^{\frac{1}{5}}})$

Step 2.2.5

Multiply the exponents in ${(x^{\frac{2}{5}})}^{- 2}$ .

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Step 2.2.5.1

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

$f'' (x) = \frac{12}{5} \cdot (- x^{\frac{2}{5} \cdot - 2} (\frac{2}{5} x^{\frac{2}{5} - 1})) + \frac{d}{d x} (- \frac{4}{5 x^{\frac{1}{5}}})$

Step 2.2.5.2

Multiply $\frac{2}{5} \cdot - 2$ .

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Step 2.2.5.2.1

Combine $\frac{2}{5}$ and $- 2$ .

$f'' (x) = \frac{12}{5} \cdot (- x^{\frac{2 \cdot - 2}{5}} (\frac{2}{5} x^{\frac{2}{5} - 1})) + \frac{d}{d x} (- \frac{4}{5 x^{\frac{1}{5}}})$

Step 2.2.5.2.2

Multiply $2$ by $- 2$ .

$f'' (x) = \frac{12}{5} \cdot (- x^{\frac{- 4}{5}} (\frac{2}{5} x^{\frac{2}{5} - 1})) + \frac{d}{d x} (- \frac{4}{5 x^{\frac{1}{5}}})$

Step 2.2.5.3

Move the negative in front of the fraction.

$f'' (x) = \frac{12}{5} \cdot (- x^{- \frac{4}{5}} (\frac{2}{5} x^{\frac{2}{5} - 1})) + \frac{d}{d x} (- \frac{4}{5 x^{\frac{1}{5}}})$

Step 2.2.6

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$f'' (x) = \frac{12}{5} \cdot (- x^{- \frac{4}{5}} (\frac{2}{5} x^{\frac{2}{5} - 1 \cdot \frac{5}{5}})) + \frac{d}{d x} (- \frac{4}{5 x^{\frac{1}{5}}})$

Step 2.2.7

Combine $- 1$ and $\frac{5}{5}$ .

$f'' (x) = \frac{12}{5} \cdot (- x^{- \frac{4}{5}} (\frac{2}{5} x^{\frac{2}{5} + \frac{- 1 \cdot 5}{5}})) + \frac{d}{d x} (- \frac{4}{5 x^{\frac{1}{5}}})$

Step 2.2.8

Combine the numerators over the common denominator.

$f'' (x) = \frac{12}{5} \cdot (- x^{- \frac{4}{5}} (\frac{2}{5} x^{\frac{2 - 1 \cdot 5}{5}})) + \frac{d}{d x} (- \frac{4}{5 x^{\frac{1}{5}}})$

Step 2.2.9

Simplify the numerator.

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Step 2.2.9.1

Multiply $- 1$ by $5$ .

$f'' (x) = \frac{12}{5} \cdot (- x^{- \frac{4}{5}} (\frac{2}{5} x^{\frac{2 - 5}{5}})) + \frac{d}{d x} (- \frac{4}{5 x^{\frac{1}{5}}})$

Step 2.2.9.2

Subtract $5$ from $2$ .

$f'' (x) = \frac{12}{5} \cdot (- x^{- \frac{4}{5}} (\frac{2}{5} x^{\frac{- 3}{5}})) + \frac{d}{d x} (- \frac{4}{5 x^{\frac{1}{5}}})$

Step 2.2.10

Move the negative in front of the fraction.

$f'' (x) = \frac{12}{5} \cdot (- x^{- \frac{4}{5}} (\frac{2}{5} x^{- \frac{3}{5}})) + \frac{d}{d x} (- \frac{4}{5 x^{\frac{1}{5}}})$

Step 2.2.11

Combine $\frac{2}{5}$ and $x^{- \frac{3}{5}}$ .

$f'' (x) = \frac{12}{5} \cdot (- x^{- \frac{4}{5}} \frac{2 x^{- \frac{3}{5}}}{5}) + \frac{d}{d x} (- \frac{4}{5 x^{\frac{1}{5}}})$

Step 2.2.12

Combine $\frac{2 x^{- \frac{3}{5}}}{5}$ and $x^{- \frac{4}{5}}$ .

$f'' (x) = \frac{12}{5} \cdot (- \frac{2 x^{- \frac{3}{5}} x^{- \frac{4}{5}}}{5}) + \frac{d}{d x} (- \frac{4}{5 x^{\frac{1}{5}}})$

Step 2.2.13

Multiply $x^{- \frac{3}{5}}$ by $x^{- \frac{4}{5}}$ by adding the exponents.

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Step 2.2.13.1

Move $x^{- \frac{4}{5}}$ .

$f'' (x) = \frac{12}{5} \cdot (- \frac{2 (x^{- \frac{4}{5}} x^{- \frac{3}{5}})}{5}) + \frac{d}{d x} (- \frac{4}{5 x^{\frac{1}{5}}})$

Step 2.2.13.2

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

$f'' (x) = \frac{12}{5} \cdot (- \frac{2 x^{- \frac{4}{5} - \frac{3}{5}}}{5}) + \frac{d}{d x} (- \frac{4}{5 x^{\frac{1}{5}}})$

Step 2.2.13.3

Combine the numerators over the common denominator.

$f'' (x) = \frac{12}{5} \cdot (- \frac{2 x^{\frac{- 4 - 3}{5}}}{5}) + \frac{d}{d x} (- \frac{4}{5 x^{\frac{1}{5}}})$

Step 2.2.13.4

Subtract $3$ from $- 4$ .

$f'' (x) = \frac{12}{5} \cdot (- \frac{2 x^{\frac{- 7}{5}}}{5}) + \frac{d}{d x} (- \frac{4}{5 x^{\frac{1}{5}}})$

Step 2.2.13.5

Move the negative in front of the fraction.

$f'' (x) = \frac{12}{5} \cdot (- \frac{2 x^{- \frac{7}{5}}}{5}) + \frac{d}{d x} (- \frac{4}{5 x^{\frac{1}{5}}})$

Step 2.2.14

Move $x^{- \frac{7}{5}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (x) = \frac{12}{5} \cdot (- \frac{2}{5 x^{\frac{7}{5}}}) + \frac{d}{d x} (- \frac{4}{5 x^{\frac{1}{5}}})$

Step 2.2.15

Multiply $\frac{12}{5}$ by $\frac{2}{5 x^{\frac{7}{5}}}$ .

$f'' (x) = - \frac{12 \cdot 2}{5 (5 x^{\frac{7}{5}})} + \frac{d}{d x} (- \frac{4}{5 x^{\frac{1}{5}}})$

Step 2.2.16

Multiply $12$ by $2$ .

$f'' (x) = - \frac{24}{5 (5 x^{\frac{7}{5}})} + \frac{d}{d x} (- \frac{4}{5 x^{\frac{1}{5}}})$

Step 2.2.17

Multiply $5$ by $5$ .

$f'' (x) = - \frac{24}{25 x^{\frac{7}{5}}} + \frac{d}{d x} (- \frac{4}{5 x^{\frac{1}{5}}})$

Step 2.3

Evaluate $\frac{d}{d x} [- \frac{4}{5 x^{\frac{1}{5}}}]$ .

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Step 2.3.1

Since $- \frac{4}{5}$ is constant with respect to $x$ , the derivative of $- \frac{4}{5 x^{\frac{1}{5}}}$ with respect to $x$ is $- \frac{4}{5} \frac{d}{d x} [\frac{1}{x^{\frac{1}{5}}}]$ .

$f'' (x) = - \frac{24}{25 x^{\frac{7}{5}}} - \frac{4}{5} \cdot \frac{d}{d x} \frac{1}{x^{\frac{1}{5}}}$

Step 2.3.2

Rewrite $\frac{1}{x^{\frac{1}{5}}}$ as ${(x^{\frac{1}{5}})}^{- 1}$ .

$f'' (x) = - \frac{24}{25 x^{\frac{7}{5}}} - \frac{4}{5} \cdot \frac{d}{d x} {(x^{\frac{1}{5}})}^{- 1}$

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 $g (x) = x^{\frac{1}{5}}$ .

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Step 2.3.3.1

To apply the Chain Rule, set $u_{2}$ as $x^{\frac{1}{5}}$ .

$f'' (x) = - \frac{24}{25 x^{\frac{7}{5}}} - \frac{4}{5} \cdot (\frac{d}{d u (2)} ({u_{2}}^{- 1}) \frac{d}{d x} (x^{\frac{1}{5}}))$

Step 2.3.3.2

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

$f'' (x) = - \frac{24}{25 x^{\frac{7}{5}}} - \frac{4}{5} \cdot (- {u_{2}}^{- 2} \frac{d}{d x} x^{\frac{1}{5}})$

Step 2.3.3.3

Replace all occurrences of $u_{2}$ with $x^{\frac{1}{5}}$ .

$f'' (x) = - \frac{24}{25 x^{\frac{7}{5}}} - \frac{4}{5} \cdot (- {(x^{\frac{1}{5}})}^{- 2} \frac{d}{d x} x^{\frac{1}{5}})$

Step 2.3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{1}{5}$ .

$f'' (x) = - \frac{24}{25 x^{\frac{7}{5}}} - \frac{4}{5} \cdot (- {(x^{\frac{1}{5}})}^{- 2} (\frac{1}{5} x^{\frac{1}{5} - 1}))$

Step 2.3.5

Multiply the exponents in ${(x^{\frac{1}{5}})}^{- 2}$ .

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Step 2.3.5.1

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

$f'' (x) = - \frac{24}{25 x^{\frac{7}{5}}} - \frac{4}{5} \cdot (- x^{\frac{1}{5} \cdot - 2} (\frac{1}{5} x^{\frac{1}{5} - 1}))$

Step 2.3.5.2

Combine $\frac{1}{5}$ and $- 2$ .

$f'' (x) = - \frac{24}{25 x^{\frac{7}{5}}} - \frac{4}{5} \cdot (- x^{\frac{- 2}{5}} (\frac{1}{5} x^{\frac{1}{5} - 1}))$

Step 2.3.5.3

Move the negative in front of the fraction.

$f'' (x) = - \frac{24}{25 x^{\frac{7}{5}}} - \frac{4}{5} \cdot (- x^{- \frac{2}{5}} (\frac{1}{5} x^{\frac{1}{5} - 1}))$

Step 2.3.6

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$f'' (x) = - \frac{24}{25 x^{\frac{7}{5}}} - \frac{4}{5} \cdot (- x^{- \frac{2}{5}} (\frac{1}{5} x^{\frac{1}{5} - 1 \cdot \frac{5}{5}}))$

Step 2.3.7

Combine $- 1$ and $\frac{5}{5}$ .

$f'' (x) = - \frac{24}{25 x^{\frac{7}{5}}} - \frac{4}{5} \cdot (- x^{- \frac{2}{5}} (\frac{1}{5} x^{\frac{1}{5} + \frac{- 1 \cdot 5}{5}}))$

Step 2.3.8

Combine the numerators over the common denominator.

$f'' (x) = - \frac{24}{25 x^{\frac{7}{5}}} - \frac{4}{5} \cdot (- x^{- \frac{2}{5}} (\frac{1}{5} x^{\frac{1 - 1 \cdot 5}{5}}))$

Step 2.3.9

Simplify the numerator.

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Step 2.3.9.1

Multiply $- 1$ by $5$ .

$f'' (x) = - \frac{24}{25 x^{\frac{7}{5}}} - \frac{4}{5} \cdot (- x^{- \frac{2}{5}} (\frac{1}{5} x^{\frac{1 - 5}{5}}))$

Step 2.3.9.2

Subtract $5$ from $1$ .

$f'' (x) = - \frac{24}{25 x^{\frac{7}{5}}} - \frac{4}{5} \cdot (- x^{- \frac{2}{5}} (\frac{1}{5} x^{\frac{- 4}{5}}))$

Step 2.3.10

Move the negative in front of the fraction.

$f'' (x) = - \frac{24}{25 x^{\frac{7}{5}}} - \frac{4}{5} \cdot (- x^{- \frac{2}{5}} (\frac{1}{5} x^{- \frac{4}{5}}))$

Step 2.3.11

Combine $\frac{1}{5}$ and $x^{- \frac{4}{5}}$ .

$f'' (x) = - \frac{24}{25 x^{\frac{7}{5}}} - \frac{4}{5} \cdot (- x^{- \frac{2}{5}} \frac{x^{- \frac{4}{5}}}{5})$

Step 2.3.12

Combine $\frac{x^{- \frac{4}{5}}}{5}$ and $x^{- \frac{2}{5}}$ .

$f'' (x) = - \frac{24}{25 x^{\frac{7}{5}}} - \frac{4}{5} \cdot (- \frac{x^{- \frac{4}{5}} x^{- \frac{2}{5}}}{5})$

Step 2.3.13

Multiply $x^{- \frac{4}{5}}$ by $x^{- \frac{2}{5}}$ by adding the exponents.

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Step 2.3.13.1

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

$f'' (x) = - \frac{24}{25 x^{\frac{7}{5}}} - \frac{4}{5} \cdot (- \frac{x^{- \frac{4}{5} - \frac{2}{5}}}{5})$

Step 2.3.13.2

Combine the numerators over the common denominator.

$f'' (x) = - \frac{24}{25 x^{\frac{7}{5}}} - \frac{4}{5} \cdot (- \frac{x^{\frac{- 4 - 2}{5}}}{5})$

Step 2.3.13.3

Subtract $2$ from $- 4$ .

$f'' (x) = - \frac{24}{25 x^{\frac{7}{5}}} - \frac{4}{5} \cdot (- \frac{x^{\frac{- 6}{5}}}{5})$

Step 2.3.13.4

Move the negative in front of the fraction.

$f'' (x) = - \frac{24}{25 x^{\frac{7}{5}}} - \frac{4}{5} \cdot (- \frac{x^{- \frac{6}{5}}}{5})$

Step 2.3.14

Move $x^{- \frac{6}{5}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (x) = - \frac{24}{25 x^{\frac{7}{5}}} - \frac{4}{5} \cdot (- \frac{1}{5 x^{\frac{6}{5}}})$

Step 2.3.15

Multiply $- 1$ by $- 1$ .

$f'' (x) = - \frac{24}{25 x^{\frac{7}{5}}} + 1 (\frac{4}{5}) \cdot \frac{1}{5 x^{\frac{6}{5}}}$

Step 2.3.16

Multiply $\frac{4}{5}$ by $1$ .

$f'' (x) = - \frac{24}{25 x^{\frac{7}{5}}} + \frac{4}{5} \cdot \frac{1}{5 x^{\frac{6}{5}}}$

Step 2.3.17

Multiply $\frac{4}{5}$ by $\frac{1}{5 x^{\frac{6}{5}}}$ .

$f'' (x) = - \frac{24}{25 x^{\frac{7}{5}}} + \frac{4}{5 (5 x^{\frac{6}{5}})}$

Step 2.3.18

Multiply $5$ by $5$ .

$f'' (x) = - \frac{24}{25 x^{\frac{7}{5}}} + \frac{4}{25 x^{\frac{6}{5}}}$

Step 2.4

Reorder terms.

$f'' (x) = \frac{4}{25 x^{\frac{6}{5}}} - \frac{24}{25 x^{\frac{7}{5}}}$

Step 3

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

$\frac{12}{5 x^{\frac{2}{5}}} - \frac{4}{5 x^{\frac{1}{5}}} = 0$

Step 4

Find the first derivative.

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Step 4.1

Find the first derivative.

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Step 4.1.1

By the Sum Rule, the derivative of $4 x^{\frac{3}{5}} - x^{\frac{4}{5}}$ with respect to $x$ is $\frac{d}{d x} [4 x^{\frac{3}{5}}] + \frac{d}{d x} [- x^{\frac{4}{5}}]$ .

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

Step 4.1.2

Evaluate $\frac{d}{d x} [4 x^{\frac{3}{5}}]$ .

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Step 4.1.2.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{\frac{3}{5}}$ with respect to $x$ is $4 \frac{d}{d x} [x^{\frac{3}{5}}]$ .

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

Step 4.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{3}{5}$ .

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

Step 4.1.2.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

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

Step 4.1.2.4

Combine $- 1$ and $\frac{5}{5}$ .

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

Step 4.1.2.5

Combine the numerators over the common denominator.

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

Step 4.1.2.6

Simplify the numerator.

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Step 4.1.2.6.1

Multiply $- 1$ by $5$ .

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

Step 4.1.2.6.2

Subtract $5$ from $3$ .

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

Step 4.1.2.7

Move the negative in front of the fraction.

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

Step 4.1.2.8

Combine $\frac{3}{5}$ and $x^{- \frac{2}{5}}$ .

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

Step 4.1.2.9

Combine $4$ and $\frac{3 x^{- \frac{2}{5}}}{5}$ .

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

Step 4.1.2.10

Multiply $3$ by $4$ .

$f' (x) = \frac{12 x^{- \frac{2}{5}}}{5} + \frac{d}{d x} (- x^{\frac{4}{5}})$

Step 4.1.2.11

Move $x^{- \frac{2}{5}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f' (x) = \frac{12}{5 x^{\frac{2}{5}}} + \frac{d}{d x} (- x^{\frac{4}{5}})$

Step 4.1.3

Evaluate $\frac{d}{d x} [- x^{\frac{4}{5}}]$ .

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Step 4.1.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{\frac{4}{5}}$ with respect to $x$ is $- \frac{d}{d x} [x^{\frac{4}{5}}]$ .

$f' (x) = \frac{12}{5 x^{\frac{2}{5}}} - \frac{d}{d x} x^{\frac{4}{5}}$

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 $n = \frac{4}{5}$ .

$f' (x) = \frac{12}{5 x^{\frac{2}{5}}} - (\frac{4}{5} x^{\frac{4}{5} - 1})$

Step 4.1.3.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$f' (x) = \frac{12}{5 x^{\frac{2}{5}}} - (\frac{4}{5} x^{\frac{4}{5} - 1 \cdot \frac{5}{5}})$

Step 4.1.3.4

Combine $- 1$ and $\frac{5}{5}$ .

$f' (x) = \frac{12}{5 x^{\frac{2}{5}}} - (\frac{4}{5} x^{\frac{4}{5} + \frac{- 1 \cdot 5}{5}})$

Step 4.1.3.5

Combine the numerators over the common denominator.

$f' (x) = \frac{12}{5 x^{\frac{2}{5}}} - (\frac{4}{5} x^{\frac{4 - 1 \cdot 5}{5}})$

Step 4.1.3.6

Simplify the numerator.

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Step 4.1.3.6.1

Multiply $- 1$ by $5$ .

$f' (x) = \frac{12}{5 x^{\frac{2}{5}}} - (\frac{4}{5} x^{\frac{4 - 5}{5}})$

Step 4.1.3.6.2

Subtract $5$ from $4$ .

$f' (x) = \frac{12}{5 x^{\frac{2}{5}}} - (\frac{4}{5} x^{\frac{- 1}{5}})$

Step 4.1.3.7

Move the negative in front of the fraction.

$f' (x) = \frac{12}{5 x^{\frac{2}{5}}} - (\frac{4}{5} x^{- \frac{1}{5}})$

Step 4.1.3.8

Combine $\frac{4}{5}$ and $x^{- \frac{1}{5}}$ .

$f' (x) = \frac{12}{5 x^{\frac{2}{5}}} - \frac{4 x^{- \frac{1}{5}}}{5}$

Step 4.1.3.9

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

$f' (x) = \frac{12}{5 x^{\frac{2}{5}}} - \frac{4}{5 x^{\frac{1}{5}}}$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{12}{5 x^{\frac{2}{5}}} - \frac{4}{5 x^{\frac{1}{5}}}$ .

$\frac{12}{5 x^{\frac{2}{5}}} - \frac{4}{5 x^{\frac{1}{5}}}$

Step 5

Set the first derivative equal to $0$ then solve the equation $\frac{12}{5 x^{\frac{2}{5}}} - \frac{4}{5 x^{\frac{1}{5}}} = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$\frac{12}{5 x^{\frac{2}{5}}} - \frac{4}{5 x^{\frac{1}{5}}} = 0$

Step 5.2

Find the LCD of the terms in the equation.

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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.

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

Step 5.2.2

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

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 $5$ has no factors besides $1$ and $5$ .

$5$ 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 $5, 5, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$5$

Step 5.2.7

The LCM of $x^{\frac{2}{5}}, x^{\frac{1}{5}}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$x^{\frac{2}{5}}$

Step 5.2.8

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

$5 x^{\frac{2}{5}}$

Step 5.3

Multiply each term in $\frac{12}{5 x^{\frac{2}{5}}} - \frac{4}{5 x^{\frac{1}{5}}} = 0$ by $5 x^{\frac{2}{5}}$ to eliminate the fractions.

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Step 5.3.1

Multiply each term in $\frac{12}{5 x^{\frac{2}{5}}} - \frac{4}{5 x^{\frac{1}{5}}} = 0$ by $5 x^{\frac{2}{5}}$ .

$\frac{12}{5 x^{\frac{2}{5}}} (5 x^{\frac{2}{5}}) - \frac{4}{5 x^{\frac{1}{5}}} (5 x^{\frac{2}{5}}) = 0 (5 x^{\frac{2}{5}})$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Simplify each term.

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Step 5.3.2.1.1

Rewrite using the commutative property of multiplication.

$5 \frac{12}{5 x^{\frac{2}{5}}} x^{\frac{2}{5}} - \frac{4}{5 x^{\frac{1}{5}}} (5 x^{\frac{2}{5}}) = 0 (5 x^{\frac{2}{5}})$

Step 5.3.2.1.2

Cancel the common factor of $5$ .

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Step 5.3.2.1.2.1

Cancel the common factor.

$5 \frac{12}{5 x^{\frac{2}{5}}} x^{\frac{2}{5}} - \frac{4}{5 x^{\frac{1}{5}}} (5 x^{\frac{2}{5}}) = 0 (5 x^{\frac{2}{5}})$

Step 5.3.2.1.2.2

Rewrite the expression.

$\frac{12}{x^{\frac{2}{5}}} x^{\frac{2}{5}} - \frac{4}{5 x^{\frac{1}{5}}} (5 x^{\frac{2}{5}}) = 0 (5 x^{\frac{2}{5}})$

Step 5.3.2.1.3

Cancel the common factor of $x^{\frac{2}{5}}$ .

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Step 5.3.2.1.3.1

Cancel the common factor.

$\frac{12}{x^{\frac{2}{5}}} x^{\frac{2}{5}} - \frac{4}{5 x^{\frac{1}{5}}} (5 x^{\frac{2}{5}}) = 0 (5 x^{\frac{2}{5}})$

Step 5.3.2.1.3.2

Rewrite the expression.

$12 - \frac{4}{5 x^{\frac{1}{5}}} (5 x^{\frac{2}{5}}) = 0 (5 x^{\frac{2}{5}})$

Step 5.3.2.1.4

Cancel the common factor of $x^{\frac{1}{5}} \cdot 5$ .

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Step 5.3.2.1.4.1

Move the leading negative in $- \frac{4}{5 x^{\frac{1}{5}}}$ into the numerator.

$12 + \frac{- 4}{5 x^{\frac{1}{5}}} (5 x^{\frac{2}{5}}) = 0 (5 x^{\frac{2}{5}})$

Step 5.3.2.1.4.2

Factor $x^{\frac{1}{5}} \cdot 5$ out of $5 x^{\frac{1}{5}}$ .

$12 + \frac{- 4}{x^{\frac{1}{5}} \cdot 5 (1)} (5 x^{\frac{2}{5}}) = 0 (5 x^{\frac{2}{5}})$

Step 5.3.2.1.4.3

Factor $x^{\frac{1}{5}} \cdot 5$ out of $5 x^{\frac{2}{5}}$ .

$12 + \frac{- 4}{x^{\frac{1}{5}} \cdot 5 (1)} (x^{\frac{1}{5}} \cdot 5 (x^{\frac{1}{5}})) = 0 (5 x^{\frac{2}{5}})$

Step 5.3.2.1.4.4

Cancel the common factor.

$12 + \frac{- 4}{x^{\frac{1}{5}} \cdot 5 \cdot 1} (x^{\frac{1}{5}} \cdot 5 x^{\frac{1}{5}}) = 0 (5 x^{\frac{2}{5}})$

Step 5.3.2.1.4.5

Rewrite the expression.

$12 - 4 x^{\frac{1}{5}} = 0 (5 x^{\frac{2}{5}})$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Multiply $0 (5 x^{\frac{2}{5}})$ .

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Step 5.3.3.1.1

Multiply $5$ by $0$ .

$12 - 4 x^{\frac{1}{5}} = 0 x^{\frac{2}{5}}$

Step 5.3.3.1.2

Multiply $0$ by $x^{\frac{2}{5}}$ .

$12 - 4 x^{\frac{1}{5}} = 0$

Step 5.4

Solve the equation.

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Step 5.4.1

Subtract $12$ from both sides of the equation.

$- 4 x^{\frac{1}{5}} = - 12$

Step 5.4.2

Raise each side of the equation to the power of $5$ to eliminate the fractional exponent on the left side.

${(- 4 x^{\frac{1}{5}})}^{5} = {(- 12)}^{5}$

Step 5.4.3

Simplify the exponent.

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Step 5.4.3.1

Simplify the left side.

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Step 5.4.3.1.1

Simplify ${(- 4 x^{\frac{1}{5}})}^{5}$ .

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Step 5.4.3.1.1.1

Apply the product rule to $- 4 x^{\frac{1}{5}}$ .

${(- 4)}^{5} {(x^{\frac{1}{5}})}^{5} = {(- 12)}^{5}$

Step 5.4.3.1.1.2

Raise $- 4$ to the power of $5$ .

$- 1024 {(x^{\frac{1}{5}})}^{5} = {(- 12)}^{5}$

Step 5.4.3.1.1.3

Multiply the exponents in ${(x^{\frac{1}{5}})}^{5}$ .

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Step 5.4.3.1.1.3.1

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

$- 1024 x^{\frac{1}{5} \cdot 5} = {(- 12)}^{5}$

Step 5.4.3.1.1.3.2

Cancel the common factor of $5$ .

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Step 5.4.3.1.1.3.2.1

Cancel the common factor.

$- 1024 x^{\frac{1}{5} \cdot 5} = {(- 12)}^{5}$

Step 5.4.3.1.1.3.2.2

Rewrite the expression.

$- 1024 x^{1} = {(- 12)}^{5}$

Step 5.4.3.1.1.4

Simplify.

$- 1024 x = {(- 12)}^{5}$

Step 5.4.3.2

Simplify the right side.

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Step 5.4.3.2.1

Raise $- 12$ to the power of $5$ .

$- 1024 x = - 248832$

Step 5.4.4

Divide each term in $- 1024 x = - 248832$ by $- 1024$ and simplify.

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Step 5.4.4.1

Divide each term in $- 1024 x = - 248832$ by $- 1024$ .

$\frac{- 1024 x}{- 1024} = \frac{- 248832}{- 1024}$

Step 5.4.4.2

Simplify the left side.

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Step 5.4.4.2.1

Cancel the common factor of $- 1024$ .

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Step 5.4.4.2.1.1

Cancel the common factor.

$\frac{- 1024 x}{- 1024} = \frac{- 248832}{- 1024}$

Step 5.4.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 248832}{- 1024}$

Step 5.4.4.3

Simplify the right side.

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Step 5.4.4.3.1

Divide $- 248832$ by $- 1024$ .

$x = 243$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

Convert expressions with fractional exponents to radicals.

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Step 6.1.1

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

$\frac{12}{5 \sqrt[5]{x^{2}}} - \frac{4}{5 x^{\frac{1}{5}}}$

Step 6.1.2

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

$\frac{12}{5 \sqrt[5]{x^{2}}} - \frac{4}{5 \sqrt[5]{x^{1}}}$

Step 6.1.3

Anything raised to $1$ is the base itself.

$\frac{12}{5 \sqrt[5]{x^{2}}} - \frac{4}{5 \sqrt[5]{x}}$

Step 6.2

Set the denominator in $\frac{12}{5 \sqrt[5]{x^{2}}}$ equal to $0$ to find where the expression is undefined.

$5 \sqrt[5]{x^{2}} = 0$

Step 6.3

Solve for $x$ .

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Step 6.3.1

To remove the radical on the left side of the equation, raise both sides of the equation to the power of $5$ .

${(5 \sqrt[5]{x^{2}})}^{5} = 0^{5}$

Step 6.3.2

Simplify each side of the equation.

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Step 6.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{x^{2}}$ as $x^{\frac{2}{5}}$ .

${(5 x^{\frac{2}{5}})}^{5} = 0^{5}$

Step 6.3.2.2

Simplify the left side.

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Step 6.3.2.2.1

Simplify ${(5 x^{\frac{2}{5}})}^{5}$ .

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Step 6.3.2.2.1.1

Apply the product rule to $5 x^{\frac{2}{5}}$ .

$5^{5} {(x^{\frac{2}{5}})}^{5} = 0^{5}$

Step 6.3.2.2.1.2

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

$3125 {(x^{\frac{2}{5}})}^{5} = 0^{5}$

Step 6.3.2.2.1.3

Multiply the exponents in ${(x^{\frac{2}{5}})}^{5}$ .

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Step 6.3.2.2.1.3.1

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

$3125 x^{\frac{2}{5} \cdot 5} = 0^{5}$

Step 6.3.2.2.1.3.2

Cancel the common factor of $5$ .

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Step 6.3.2.2.1.3.2.1

Cancel the common factor.

$3125 x^{\frac{2}{5} \cdot 5} = 0^{5}$

Step 6.3.2.2.1.3.2.2

Rewrite the expression.

$3125 x^{2} = 0^{5}$

Step 6.3.2.3

Simplify the right side.

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Step 6.3.2.3.1

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

$3125 x^{2} = 0$

Step 6.3.3

Solve for $x$ .

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Step 6.3.3.1

Divide each term in $3125 x^{2} = 0$ by $3125$ and simplify.

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Step 6.3.3.1.1

Divide each term in $3125 x^{2} = 0$ by $3125$ .

$\frac{3125 x^{2}}{3125} = \frac{0}{3125}$

Step 6.3.3.1.2

Simplify the left side.

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Step 6.3.3.1.2.1

Cancel the common factor of $3125$ .

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Step 6.3.3.1.2.1.1

Cancel the common factor.

$\frac{3125 x^{2}}{3125} = \frac{0}{3125}$

Step 6.3.3.1.2.1.2

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

$x^{2} = \frac{0}{3125}$

Step 6.3.3.1.3

Simplify the right side.

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Step 6.3.3.1.3.1

Divide $0$ by $3125$ .

$x^{2} = 0$

Step 6.3.3.2

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

$x = \pm \sqrt{0}$

Step 6.3.3.3

Simplify $\pm \sqrt{0}$ .

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Step 6.3.3.3.1

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

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

Step 6.3.3.3.2

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

$x = \pm 0$

Step 6.3.3.3.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 6.4

Set the denominator in $\frac{4}{5 \sqrt[5]{x}}$ equal to $0$ to find where the expression is undefined.

$5 \sqrt[5]{x} = 0$

Step 6.5

Solve for $x$ .

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Step 6.5.1

To remove the radical on the left side of the equation, raise both sides of the equation to the power of $5$ .

${(5 \sqrt[5]{x})}^{5} = 0^{5}$

Step 6.5.2

Simplify each side of the equation.

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Step 6.5.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{x}$ as $x^{\frac{1}{5}}$ .

${(5 x^{\frac{1}{5}})}^{5} = 0^{5}$

Step 6.5.2.2

Simplify the left side.

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Step 6.5.2.2.1

Simplify ${(5 x^{\frac{1}{5}})}^{5}$ .

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Step 6.5.2.2.1.1

Apply the product rule to $5 x^{\frac{1}{5}}$ .

$5^{5} {(x^{\frac{1}{5}})}^{5} = 0^{5}$

Step 6.5.2.2.1.2

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

$3125 {(x^{\frac{1}{5}})}^{5} = 0^{5}$

Step 6.5.2.2.1.3

Multiply the exponents in ${(x^{\frac{1}{5}})}^{5}$ .

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Step 6.5.2.2.1.3.1

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

$3125 x^{\frac{1}{5} \cdot 5} = 0^{5}$

Step 6.5.2.2.1.3.2

Cancel the common factor of $5$ .

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Step 6.5.2.2.1.3.2.1

Cancel the common factor.

$3125 x^{\frac{1}{5} \cdot 5} = 0^{5}$

Step 6.5.2.2.1.3.2.2

Rewrite the expression.

$3125 x^{1} = 0^{5}$

Step 6.5.2.2.1.4

Simplify.

$3125 x = 0^{5}$

Step 6.5.2.3

Simplify the right side.

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Step 6.5.2.3.1

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

$3125 x = 0$

Step 6.5.3

Divide each term in $3125 x = 0$ by $3125$ and simplify.

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Step 6.5.3.1

Divide each term in $3125 x = 0$ by $3125$ .

$\frac{3125 x}{3125} = \frac{0}{3125}$

Step 6.5.3.2

Simplify the left side.

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Step 6.5.3.2.1

Cancel the common factor of $3125$ .

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Step 6.5.3.2.1.1

Cancel the common factor.

$\frac{3125 x}{3125} = \frac{0}{3125}$

Step 6.5.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{3125}$

Step 6.5.3.3

Simplify the right side.

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Step 6.5.3.3.1

Divide $0$ by $3125$ .

$x = 0$

Step 7

Critical points to evaluate.

$x = 243, 0$

Step 8

Evaluate the second derivative at $x = 243$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{4}{25 {(243)}^{\frac{6}{5}}} - \frac{24}{25 {(243)}^{\frac{7}{5}}}$

Step 9

Evaluate the second derivative.

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Step 9.1

Simplify each term.

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Step 9.1.1

Simplify the denominator.

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Step 9.1.1.1

Rewrite $243$ as $3^{5}$ .

$\frac{4}{25 \cdot {(3^{5})}^{\frac{6}{5}}} - \frac{24}{25 {(243)}^{\frac{7}{5}}}$

Step 9.1.1.2

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

$\frac{4}{25 \cdot 3^{5 (\frac{6}{5})}} - \frac{24}{25 {(243)}^{\frac{7}{5}}}$

Step 9.1.1.3

Cancel the common factor of $5$ .

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Step 9.1.1.3.1

Cancel the common factor.

$\frac{4}{25 \cdot 3^{5 (\frac{6}{5})}} - \frac{24}{25 {(243)}^{\frac{7}{5}}}$

Step 9.1.1.3.2

Rewrite the expression.

$\frac{4}{25 \cdot 3^{6}} - \frac{24}{25 {(243)}^{\frac{7}{5}}}$

Step 9.1.1.4

Raise $3$ to the power of $6$ .

$\frac{4}{25 \cdot 729} - \frac{24}{25 {(243)}^{\frac{7}{5}}}$

Step 9.1.2

Multiply $25$ by $729$ .

$\frac{4}{18225} - \frac{24}{25 {(243)}^{\frac{7}{5}}}$

Step 9.1.3

Simplify the denominator.

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Step 9.1.3.1

Rewrite $243$ as $3^{5}$ .

$\frac{4}{18225} - \frac{24}{25 \cdot {(3^{5})}^{\frac{7}{5}}}$

Step 9.1.3.2

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

$\frac{4}{18225} - \frac{24}{25 \cdot 3^{5 (\frac{7}{5})}}$

Step 9.1.3.3

Cancel the common factor of $5$ .

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Step 9.1.3.3.1

Cancel the common factor.

$\frac{4}{18225} - \frac{24}{25 \cdot 3^{5 (\frac{7}{5})}}$

Step 9.1.3.3.2

Rewrite the expression.

$\frac{4}{18225} - \frac{24}{25 \cdot 3^{7}}$

Step 9.1.3.4

Raise $3$ to the power of $7$ .

$\frac{4}{18225} - \frac{24}{25 \cdot 2187}$

Step 9.1.4

Multiply $25$ by $2187$ .

$\frac{4}{18225} - \frac{24}{54675}$

Step 9.1.5

Cancel the common factor of $24$ and $54675$ .

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Step 9.1.5.1

Factor $3$ out of $24$ .

$\frac{4}{18225} - \frac{3 (8)}{54675}$

Step 9.1.5.2

Cancel the common factors.

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Step 9.1.5.2.1

Factor $3$ out of $54675$ .

$\frac{4}{18225} - \frac{3 \cdot 8}{3 \cdot 18225}$

Step 9.1.5.2.2

Cancel the common factor.

$\frac{4}{18225} - \frac{3 \cdot 8}{3 \cdot 18225}$

Step 9.1.5.2.3

Rewrite the expression.

$\frac{4}{18225} - \frac{8}{18225}$

Step 9.2

Combine fractions.

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Step 9.2.1

Combine the numerators over the common denominator.

$\frac{4 - 8}{18225}$

Step 9.2.2

Simplify the expression.

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Step 9.2.2.1

Subtract $8$ from $4$ .

$\frac{- 4}{18225}$

Step 9.2.2.2

Move the negative in front of the fraction.

$- \frac{4}{18225}$

Step 10

$x = 243$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = 243$ is a local maximum

Step 11

Find the y-value when $x = 243$ .

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Step 11.1

Replace the variable $x$ with $243$ in the expression.

$f (243) = 4 {(243)}^{\frac{3}{5}} - {(243)}^{\frac{4}{5}}$

Step 11.2

Simplify the result.

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Step 11.2.1

Simplify each term.

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Step 11.2.1.1

Rewrite $243$ as $3^{5}$ .

$f (243) = 4 {(3^{5})}^{\frac{3}{5}} - {(243)}^{\frac{4}{5}}$

Step 11.2.1.2

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

$f (243) = 4 \cdot 3^{5 (\frac{3}{5})} - {(243)}^{\frac{4}{5}}$

Step 11.2.1.3

Cancel the common factor of $5$ .

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Step 11.2.1.3.1

Cancel the common factor.

$f (243) = 4 \cdot 3^{5 (\frac{3}{5})} - {(243)}^{\frac{4}{5}}$

Step 11.2.1.3.2

Rewrite the expression.

$f (243) = 4 \cdot 3^{3} - {(243)}^{\frac{4}{5}}$

Step 11.2.1.4

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

$f (243) = 4 \cdot 27 - {(243)}^{\frac{4}{5}}$

Step 11.2.1.5

Multiply $4$ by $27$ .

$f (243) = 108 - {(243)}^{\frac{4}{5}}$

Step 11.2.1.6

Rewrite $243$ as $3^{5}$ .

$f (243) = 108 - {(3^{5})}^{\frac{4}{5}}$

Step 11.2.1.7

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

$f (243) = 108 - 3^{5 (\frac{4}{5})}$

Step 11.2.1.8

Cancel the common factor of $5$ .

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Step 11.2.1.8.1

Cancel the common factor.

$f (243) = 108 - 3^{5 (\frac{4}{5})}$

Step 11.2.1.8.2

Rewrite the expression.

$f (243) = 108 - 3^{4}$

Step 11.2.1.9

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

$f (243) = 108 - 1 \cdot 81$

Step 11.2.1.10

Multiply $- 1$ by $81$ .

$f (243) = 108 - 81$

Step 11.2.2

Subtract $81$ from $108$ .

$f (243) = 27$

Step 11.2.3

The final answer is $27$ .

$y = 27$

Step 12

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.

$\frac{4}{25 {(0)}^{\frac{6}{5}}} - \frac{24}{25 {(0)}^{\frac{7}{5}}}$

Step 13

Evaluate the second derivative.

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Step 13.1

Simplify the expression.

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Step 13.1.1

Rewrite $0$ as $0^{5}$ .

$\frac{4}{25 {(0^{5})}^{\frac{6}{5}}} - \frac{24}{25 {(0)}^{\frac{7}{5}}}$

Step 13.1.2

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

$\frac{4}{25 \cdot 0^{5 (\frac{6}{5})}} - \frac{24}{25 {(0)}^{\frac{7}{5}}}$

Step 13.2

Cancel the common factor of $5$ .

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Step 13.2.1

Cancel the common factor.

$\frac{4}{25 \cdot 0^{5 (\frac{6}{5})}} - \frac{24}{25 {(0)}^{\frac{7}{5}}}$

Step 13.2.2

Rewrite the expression.

$\frac{4}{25 \cdot 0^{6}} - \frac{24}{25 {(0)}^{\frac{7}{5}}}$

Step 13.3

Simplify the expression.

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Step 13.3.1

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

$\frac{4}{25 \cdot 0} - \frac{24}{25 {(0)}^{\frac{7}{5}}}$

Step 13.3.2

Multiply $25$ by $0$ .

$\frac{4}{0} - \frac{24}{25 {(0)}^{\frac{7}{5}}}$

Step 13.3.3

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

Undefined

$\frac{4}{0} - \frac{24}{25 {(0)}^{\frac{7}{5}}}$

Step 13.4

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

Undefined

Step 14

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

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Step 14.1

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

$(- \infty, 0) \cup (0, 243) \cup (243, \infty)$

Step 14.2

Substitute any number, such as $- 2$ , from the interval $(- \infty, 0)$ in the first derivative $\frac{12}{5 x^{\frac{2}{5}}} - \frac{4}{5 x^{\frac{1}{5}}}$ to check if the result is negative or positive.

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Step 14.2.1

Replace the variable $x$ with $- 2$ in the expression.

$f' (- 2) = \frac{12}{5 {(- 2)}^{\frac{2}{5}}} - \frac{4}{5 {(- 2)}^{\frac{1}{5}}}$

Step 14.2.2

Simplify the result.

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Step 14.2.2.1

To write $- \frac{4}{5 {(- 2)}^{\frac{1}{5}}}$ as a fraction with a common denominator, multiply by $\frac{{(- 2)}^{\frac{1}{5}}}{{(- 2)}^{\frac{1}{5}}}$ .

$f' (- 2) = \frac{12}{5 {(- 2)}^{\frac{2}{5}}} - \frac{4}{5 {(- 2)}^{\frac{1}{5}}} \cdot \frac{{(- 2)}^{\frac{1}{5}}}{{(- 2)}^{\frac{1}{5}}}$

Step 14.2.2.2

Write each expression with a common denominator of $5 {(- 2)}^{\frac{2}{5}}$ , by multiplying each by an appropriate factor of $1$ .

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Step 14.2.2.2.1

Multiply $\frac{4}{5 {(- 2)}^{\frac{1}{5}}}$ by $\frac{{(- 2)}^{\frac{1}{5}}}{{(- 2)}^{\frac{1}{5}}}$ .

$f' (- 2) = \frac{12}{5 {(- 2)}^{\frac{2}{5}}} - \frac{4 {(- 2)}^{\frac{1}{5}}}{5 {(- 2)}^{\frac{1}{5}} {(- 2)}^{\frac{1}{5}}}$

Step 14.2.2.2.2

Multiply ${(- 2)}^{\frac{1}{5}}$ by ${(- 2)}^{\frac{1}{5}}$ by adding the exponents.

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Step 14.2.2.2.2.1

Move ${(- 2)}^{\frac{1}{5}}$ .

$f' (- 2) = \frac{12}{5 {(- 2)}^{\frac{2}{5}}} - \frac{4 {(- 2)}^{\frac{1}{5}}}{5 ({(- 2)}^{\frac{1}{5}} {(- 2)}^{\frac{1}{5}})}$

Step 14.2.2.2.2.2

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

$f' (- 2) = \frac{12}{5 {(- 2)}^{\frac{2}{5}}} - \frac{4 {(- 2)}^{\frac{1}{5}}}{5 {(- 2)}^{\frac{1}{5} + \frac{1}{5}}}$

Step 14.2.2.2.2.3

Combine the numerators over the common denominator.

$f' (- 2) = \frac{12}{5 {(- 2)}^{\frac{2}{5}}} - \frac{4 {(- 2)}^{\frac{1}{5}}}{5 {(- 2)}^{\frac{1 + 1}{5}}}$

Step 14.2.2.2.2.4

Add $1$ and $1$ .

$f' (- 2) = \frac{12}{5 {(- 2)}^{\frac{2}{5}}} - \frac{4 {(- 2)}^{\frac{1}{5}}}{5 {(- 2)}^{\frac{2}{5}}}$

Step 14.2.2.3

Combine the numerators over the common denominator.

$f' (- 2) = \frac{12 - 4 {(- 2)}^{\frac{1}{5}}}{5 {(- 2)}^{\frac{2}{5}}}$

Step 14.2.2.4

The final answer is $\frac{12 - 4 {(- 2)}^{\frac{1}{5}}}{5 {(- 2)}^{\frac{2}{5}}}$ .

$\frac{12 - 4 {(- 2)}^{\frac{1}{5}}}{5 {(- 2)}^{\frac{2}{5}}}$

Step 14.3

Substitute any number, such as $2$ , from the interval $(0, 243)$ in the first derivative $\frac{12}{5 x^{\frac{2}{5}}} - \frac{4}{5 x^{\frac{1}{5}}}$ to check if the result is negative or positive.

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Step 14.3.1

Replace the variable $x$ with $2$ in the expression.

$f' (2) = \frac{12}{5 {(2)}^{\frac{2}{5}}} - \frac{4}{5 {(2)}^{\frac{1}{5}}}$

Step 14.3.2

Simplify the result.

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Step 14.3.2.1

Remove parentheses.

$f' (2) = \frac{12}{5 \cdot 2^{\frac{2}{5}}} - \frac{4}{5 \cdot 2^{\frac{1}{5}}}$

Step 14.3.2.2

To write $- \frac{4}{5 \cdot 2^{\frac{1}{5}}}$ as a fraction with a common denominator, multiply by $\frac{2^{\frac{1}{5}}}{2^{\frac{1}{5}}}$ .

$f' (2) = \frac{12}{5 \cdot 2^{\frac{2}{5}}} - \frac{4}{5 \cdot 2^{\frac{1}{5}}} \cdot \frac{2^{\frac{1}{5}}}{2^{\frac{1}{5}}}$

Step 14.3.2.3

Write each expression with a common denominator of $5 \cdot 2^{\frac{2}{5}}$ , by multiplying each by an appropriate factor of $1$ .

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Step 14.3.2.3.1

Multiply $\frac{4}{5 \cdot 2^{\frac{1}{5}}}$ by $\frac{2^{\frac{1}{5}}}{2^{\frac{1}{5}}}$ .

$f' (2) = \frac{12}{5 \cdot 2^{\frac{2}{5}}} - \frac{4 \cdot 2^{\frac{1}{5}}}{5 \cdot 2^{\frac{1}{5}} \cdot 2^{\frac{1}{5}}}$

Step 14.3.2.3.2

Multiply $2^{\frac{1}{5}}$ by $2^{\frac{1}{5}}$ by adding the exponents.

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Step 14.3.2.3.2.1

Move $2^{\frac{1}{5}}$ .

$f' (2) = \frac{12}{5 \cdot 2^{\frac{2}{5}}} - \frac{4 \cdot 2^{\frac{1}{5}}}{5 \cdot (2^{\frac{1}{5}} \cdot 2^{\frac{1}{5}})}$

Step 14.3.2.3.2.2

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

$f' (2) = \frac{12}{5 \cdot 2^{\frac{2}{5}}} - \frac{4 \cdot 2^{\frac{1}{5}}}{5 \cdot 2^{\frac{1}{5} + \frac{1}{5}}}$

Step 14.3.2.3.2.3

Combine the numerators over the common denominator.

$f' (2) = \frac{12}{5 \cdot 2^{\frac{2}{5}}} - \frac{4 \cdot 2^{\frac{1}{5}}}{5 \cdot 2^{\frac{1 + 1}{5}}}$

Step 14.3.2.3.2.4

Add $1$ and $1$ .

$f' (2) = \frac{12}{5 \cdot 2^{\frac{2}{5}}} - \frac{4 \cdot 2^{\frac{1}{5}}}{5 \cdot 2^{\frac{2}{5}}}$

Step 14.3.2.4

Combine the numerators over the common denominator.

$f' (2) = \frac{12 - 4 \cdot 2^{\frac{1}{5}}}{5 \cdot 2^{\frac{2}{5}}}$

Step 14.3.2.5

Multiply $- 4 \cdot 2^{\frac{1}{5}}$ .

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Step 14.3.2.5.1

Factor out negative.

$f' (2) = \frac{12 - (4 \cdot 2^{\frac{1}{5}})}{5 \cdot 2^{\frac{2}{5}}}$

Step 14.3.2.5.2

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

$f' (2) = \frac{12 - (2^{2} \cdot 2^{\frac{1}{5}})}{5 \cdot 2^{\frac{2}{5}}}$

Step 14.3.2.5.3

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

$f' (2) = \frac{12 - 2^{2 + \frac{1}{5}}}{5 \cdot 2^{\frac{2}{5}}}$

Step 14.3.2.5.4

To write $2$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$f' (2) = \frac{12 - 2^{2 \cdot \frac{5}{5} + \frac{1}{5}}}{5 \cdot 2^{\frac{2}{5}}}$

Step 14.3.2.5.5

Combine $2$ and $\frac{5}{5}$ .

$f' (2) = \frac{12 - 2^{\frac{2 \cdot 5}{5} + \frac{1}{5}}}{5 \cdot 2^{\frac{2}{5}}}$

Step 14.3.2.5.6

Combine the numerators over the common denominator.

$f' (2) = \frac{12 - 2^{\frac{2 \cdot 5 + 1}{5}}}{5 \cdot 2^{\frac{2}{5}}}$

Step 14.3.2.5.7

Simplify the numerator.

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Step 14.3.2.5.7.1

Multiply $2$ by $5$ .

$f' (2) = \frac{12 - 2^{\frac{10 + 1}{5}}}{5 \cdot 2^{\frac{2}{5}}}$

Step 14.3.2.5.7.2

Add $10$ and $1$ .

$f' (2) = \frac{12 - 2^{\frac{11}{5}}}{5 \cdot 2^{\frac{2}{5}}}$

Step 14.3.2.6

The final answer is $\frac{12 - 2^{\frac{11}{5}}}{5 \cdot 2^{\frac{2}{5}}}$ .

$\frac{12 - 2^{\frac{11}{5}}}{5 \cdot 2^{\frac{2}{5}}}$

Step 14.4

Substitute any number, such as $246$ , from the interval $(243, \infty)$ in the first derivative $\frac{12}{5 x^{\frac{2}{5}}} - \frac{4}{5 x^{\frac{1}{5}}}$ to check if the result is negative or positive.

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Step 14.4.1

Replace the variable $x$ with $246$ in the expression.

$f' (246) = \frac{12}{5 {(246)}^{\frac{2}{5}}} - \frac{4}{5 {(246)}^{\frac{1}{5}}}$

Step 14.4.2

Simplify the result.

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Step 14.4.2.1

Remove parentheses.

$f' (246) = \frac{12}{5 \cdot 246^{\frac{2}{5}}} - \frac{4}{5 \cdot 246^{\frac{1}{5}}}$

Step 14.4.2.2

To write $- \frac{4}{5 \cdot 246^{\frac{1}{5}}}$ as a fraction with a common denominator, multiply by $\frac{246^{\frac{1}{5}}}{246^{\frac{1}{5}}}$ .

$f' (246) = \frac{12}{5 \cdot 246^{\frac{2}{5}}} - \frac{4}{5 \cdot 246^{\frac{1}{5}}} \cdot \frac{246^{\frac{1}{5}}}{246^{\frac{1}{5}}}$

Step 14.4.2.3

Write each expression with a common denominator of $5 \cdot 246^{\frac{2}{5}}$ , by multiplying each by an appropriate factor of $1$ .

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Step 14.4.2.3.1

Multiply $\frac{4}{5 \cdot 246^{\frac{1}{5}}}$ by $\frac{246^{\frac{1}{5}}}{246^{\frac{1}{5}}}$ .

$f' (246) = \frac{12}{5 \cdot 246^{\frac{2}{5}}} - \frac{4 \cdot 246^{\frac{1}{5}}}{5 \cdot 246^{\frac{1}{5}} \cdot 246^{\frac{1}{5}}}$

Step 14.4.2.3.2

Multiply $246^{\frac{1}{5}}$ by $246^{\frac{1}{5}}$ by adding the exponents.

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Step 14.4.2.3.2.1

Move $246^{\frac{1}{5}}$ .

$f' (246) = \frac{12}{5 \cdot 246^{\frac{2}{5}}} - \frac{4 \cdot 246^{\frac{1}{5}}}{5 \cdot (246^{\frac{1}{5}} \cdot 246^{\frac{1}{5}})}$

Step 14.4.2.3.2.2

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

$f' (246) = \frac{12}{5 \cdot 246^{\frac{2}{5}}} - \frac{4 \cdot 246^{\frac{1}{5}}}{5 \cdot 246^{\frac{1}{5} + \frac{1}{5}}}$

Step 14.4.2.3.2.3

Combine the numerators over the common denominator.

$f' (246) = \frac{12}{5 \cdot 246^{\frac{2}{5}}} - \frac{4 \cdot 246^{\frac{1}{5}}}{5 \cdot 246^{\frac{1 + 1}{5}}}$

Step 14.4.2.3.2.4

Add $1$ and $1$ .

$f' (246) = \frac{12}{5 \cdot 246^{\frac{2}{5}}} - \frac{4 \cdot 246^{\frac{1}{5}}}{5 \cdot 246^{\frac{2}{5}}}$

Step 14.4.2.4

Combine the numerators over the common denominator.

$f' (246) = \frac{12 - 4 \cdot 246^{\frac{1}{5}}}{5 \cdot 246^{\frac{2}{5}}}$

Step 14.4.2.5

The final answer is $\frac{12 - 4 \cdot 246^{\frac{1}{5}}}{5 \cdot 246^{\frac{2}{5}}}$ .

$\frac{12 - 4 \cdot 246^{\frac{1}{5}}}{5 \cdot 246^{\frac{2}{5}}}$

Step 14.5

Since the first derivative did not change signs around $x = 0$ , this is not a local maximum or minimum.

Not a local maximum or minimum

Step 14.6

Since the first derivative changed signs from positive to negative around $x = 243$ , then $x = 243$ is a local maximum.

$x = 243$ is a local maximum

Step 15