Calculus Examples

$f (x) = - 4 x^{\frac{1}{4}} + \frac{\sqrt[4]{19}}{19} x$

Step 1

Find the first derivative of the function.

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Step 1.1

By the Sum Rule, the derivative of $- 4 x^{\frac{1}{4}} + \frac{\sqrt[4]{19}}{19} x$ with respect to $x$ is $\frac{d}{d x} [- 4 x^{\frac{1}{4}}] + \frac{d}{d x} [\frac{\sqrt[4]{19}}{19} x]$ .

$\frac{d}{d x} [- 4 x^{\frac{1}{4}}] + \frac{d}{d x} [\frac{\sqrt[4]{19}}{19} x]$

Step 1.2

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

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Step 1.2.1

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

$- 4 \frac{d}{d x} [x^{\frac{1}{4}}] + \frac{d}{d x} [\frac{\sqrt[4]{19}}{19} x]$

Step 1.2.2

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

$- 4 (\frac{1}{4} x^{\frac{1}{4} - 1}) + \frac{d}{d x} [\frac{\sqrt[4]{19}}{19} x]$

Step 1.2.3

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

$- 4 (\frac{1}{4} x^{\frac{1}{4} - 1 \cdot \frac{4}{4}}) + \frac{d}{d x} [\frac{\sqrt[4]{19}}{19} x]$

Step 1.2.4

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

$- 4 (\frac{1}{4} x^{\frac{1}{4} + \frac{- 1 \cdot 4}{4}}) + \frac{d}{d x} [\frac{\sqrt[4]{19}}{19} x]$

Step 1.2.5

Combine the numerators over the common denominator.

$- 4 (\frac{1}{4} x^{\frac{1 - 1 \cdot 4}{4}}) + \frac{d}{d x} [\frac{\sqrt[4]{19}}{19} x]$

Step 1.2.6

Simplify the numerator.

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Step 1.2.6.1

Multiply $- 1$ by $4$ .

$- 4 (\frac{1}{4} x^{\frac{1 - 4}{4}}) + \frac{d}{d x} [\frac{\sqrt[4]{19}}{19} x]$

Step 1.2.6.2

Subtract $4$ from $1$ .

$- 4 (\frac{1}{4} x^{\frac{- 3}{4}}) + \frac{d}{d x} [\frac{\sqrt[4]{19}}{19} x]$

Step 1.2.7

Move the negative in front of the fraction.

$- 4 (\frac{1}{4} x^{- \frac{3}{4}}) + \frac{d}{d x} [\frac{\sqrt[4]{19}}{19} x]$

Step 1.2.8

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

$- 4 \frac{x^{- \frac{3}{4}}}{4} + \frac{d}{d x} [\frac{\sqrt[4]{19}}{19} x]$

Step 1.2.9

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

$\frac{- 4 x^{- \frac{3}{4}}}{4} + \frac{d}{d x} [\frac{\sqrt[4]{19}}{19} x]$

Step 1.2.10

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

$\frac{- 4}{4 x^{\frac{3}{4}}} + \frac{d}{d x} [\frac{\sqrt[4]{19}}{19} x]$

Step 1.2.11

Factor $4$ out of $- 4$ .

$\frac{4 \cdot - 1}{4 x^{\frac{3}{4}}} + \frac{d}{d x} [\frac{\sqrt[4]{19}}{19} x]$

Step 1.2.12

Cancel the common factors.

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Step 1.2.12.1

Factor $4$ out of $4 x^{\frac{3}{4}}$ .

$\frac{4 \cdot - 1}{4 (x^{\frac{3}{4}})} + \frac{d}{d x} [\frac{\sqrt[4]{19}}{19} x]$

Step 1.2.12.2

Cancel the common factor.

$\frac{4 \cdot - 1}{4 x^{\frac{3}{4}}} + \frac{d}{d x} [\frac{\sqrt[4]{19}}{19} x]$

Step 1.2.12.3

Rewrite the expression.

$\frac{- 1}{x^{\frac{3}{4}}} + \frac{d}{d x} [\frac{\sqrt[4]{19}}{19} x]$

Step 1.2.13

Move the negative in front of the fraction.

$- \frac{1}{x^{\frac{3}{4}}} + \frac{d}{d x} [\frac{\sqrt[4]{19}}{19} x]$

Step 1.3

Evaluate $\frac{d}{d x} [\frac{\sqrt[4]{19}}{19} x]$ .

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Step 1.3.1

Since $\frac{\sqrt[4]{19}}{19}$ is constant with respect to $x$ , the derivative of $\frac{\sqrt[4]{19}}{19} x$ with respect to $x$ is $\frac{\sqrt[4]{19}}{19} \frac{d}{d x} [x]$ .

$- \frac{1}{x^{\frac{3}{4}}} + \frac{\sqrt[4]{19}}{19} \frac{d}{d x} [x]$

Step 1.3.2

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

$- \frac{1}{x^{\frac{3}{4}}} + \frac{\sqrt[4]{19}}{19} \cdot 1$

Step 1.3.3

Multiply $\frac{\sqrt[4]{19}}{19}$ by $1$ .

$- \frac{1}{x^{\frac{3}{4}}} + \frac{\sqrt[4]{19}}{19}$

Step 1.4

Reorder terms.

$\frac{\sqrt[4]{19}}{19} - \frac{1}{x^{\frac{3}{4}}}$

Step 2

Find the second derivative of the function.

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Step 2.1

Differentiate.

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Step 2.1.1

By the Sum Rule, the derivative of $\frac{\sqrt[4]{19}}{19} - \frac{1}{x^{\frac{3}{4}}}$ with respect to $x$ is $\frac{d}{d x} [\frac{\sqrt[4]{19}}{19}] + \frac{d}{d x} [- \frac{1}{x^{\frac{3}{4}}}]$ .

$f'' (x) = \frac{d}{d x} (\frac{\sqrt[4]{19}}{19}) + \frac{d}{d x} (- \frac{1}{x^{\frac{3}{4}}})$

Step 2.1.2

Since $\frac{\sqrt[4]{19}}{19}$ is constant with respect to $x$ , the derivative of $\frac{\sqrt[4]{19}}{19}$ with respect to $x$ is $0$ .

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

Step 2.2

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

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Step 2.2.1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = - 1$ and $g (x) = \frac{1}{x^{\frac{3}{4}}}$ .

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

Step 2.2.2

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

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

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 $g (x) = x^{\frac{3}{4}}$ .

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Step 2.2.3.1

To apply the Chain Rule, set $u$ as $x^{\frac{3}{4}}$ .

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

Step 2.2.3.2

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

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

Step 2.2.3.3

Replace all occurrences of $u$ with $x^{\frac{3}{4}}$ .

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

Step 2.2.4

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

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

Step 2.2.5

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

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

Step 2.2.6

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

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Step 2.2.6.1

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

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

Step 2.2.6.2

Cancel the common factor of $2$ .

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Step 2.2.6.2.1

Factor $2$ out of $4$ .

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

Step 2.2.6.2.2

Factor $2$ out of $- 2$ .

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

Step 2.2.6.2.3

Cancel the common factor.

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

Step 2.2.6.2.4

Rewrite the expression.

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

Step 2.2.6.3

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

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

Step 2.2.6.4

Multiply $3$ by $- 1$ .

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

Step 2.2.6.5

Move the negative in front of the fraction.

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

Step 2.2.7

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

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

Step 2.2.8

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

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

Step 2.2.9

Combine the numerators over the common denominator.

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

Step 2.2.10

Simplify the numerator.

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Step 2.2.10.1

Multiply $- 1$ by $4$ .

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

Step 2.2.10.2

Subtract $4$ from $3$ .

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

Step 2.2.11

Move the negative in front of the fraction.

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

Step 2.2.12

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

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

Step 2.2.13

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

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

Step 2.2.14

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

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Step 2.2.14.1

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

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

Step 2.2.14.2

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

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

Step 2.2.14.3

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

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

Step 2.2.14.4

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

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Step 2.2.14.4.1

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

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

Step 2.2.14.4.2

Multiply $2$ by $2$ .

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

Step 2.2.14.5

Combine the numerators over the common denominator.

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

Step 2.2.14.6

Simplify the numerator.

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Step 2.2.14.6.1

Multiply $- 3$ by $2$ .

$f'' (x) = 0 + \frac{3 x^{\frac{- 6 - 1}{4}}}{4} + \frac{1}{x^{\frac{3}{4}}} \cdot 0$

Step 2.2.14.6.2

Subtract $1$ from $- 6$ .

$f'' (x) = 0 + \frac{3 x^{\frac{- 7}{4}}}{4} + \frac{1}{x^{\frac{3}{4}}} \cdot 0$

Step 2.2.14.7

Move the negative in front of the fraction.

$f'' (x) = 0 + \frac{3 x^{- \frac{7}{4}}}{4} + \frac{1}{x^{\frac{3}{4}}} \cdot 0$

Step 2.2.15

Move $x^{- \frac{7}{4}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (x) = 0 + \frac{3}{4 x^{\frac{7}{4}}} + \frac{1}{x^{\frac{3}{4}}} \cdot 0$

Step 2.2.16

Multiply $- 1$ by $- 1$ .

$f'' (x) = 0 + 1 (\frac{3}{4 x^{\frac{7}{4}}}) + \frac{1}{x^{\frac{3}{4}}} \cdot 0$

Step 2.2.17

Multiply $\frac{3}{4 x^{\frac{7}{4}}}$ by $1$ .

$f'' (x) = 0 + \frac{3}{4 x^{\frac{7}{4}}} + \frac{1}{x^{\frac{3}{4}}} \cdot 0$

Step 2.2.18

Multiply $\frac{1}{x^{\frac{3}{4}}}$ by $0$ .

$f'' (x) = 0 + \frac{3}{4 x^{\frac{7}{4}}} + 0$

Step 2.2.19

Add $\frac{3}{4 x^{\frac{7}{4}}}$ and $0$ .

$f'' (x) = 0 + \frac{3}{4 x^{\frac{7}{4}}}$

Step 2.3

Add $0$ and $\frac{3}{4 x^{\frac{7}{4}}}$ .

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

Step 3

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

$\frac{\sqrt[4]{19}}{19} - \frac{1}{x^{\frac{3}{4}}} = 0$

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 $- 4 x^{\frac{1}{4}} + \frac{\sqrt[4]{19}}{19} x$ with respect to $x$ is $\frac{d}{d x} [- 4 x^{\frac{1}{4}}] + \frac{d}{d x} [\frac{\sqrt[4]{19}}{19} x]$ .

$\frac{d}{d x} [- 4 x^{\frac{1}{4}}] + \frac{d}{d x} [\frac{\sqrt[4]{19}}{19} x]$

Step 4.1.2

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

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Step 4.1.2.1

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

$- 4 \frac{d}{d x} [x^{\frac{1}{4}}] + \frac{d}{d x} [\frac{\sqrt[4]{19}}{19} x]$

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 $n = \frac{1}{4}$ .

$- 4 (\frac{1}{4} x^{\frac{1}{4} - 1}) + \frac{d}{d x} [\frac{\sqrt[4]{19}}{19} x]$

Step 4.1.2.3

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

$- 4 (\frac{1}{4} x^{\frac{1}{4} - 1 \cdot \frac{4}{4}}) + \frac{d}{d x} [\frac{\sqrt[4]{19}}{19} x]$

Step 4.1.2.4

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

$- 4 (\frac{1}{4} x^{\frac{1}{4} + \frac{- 1 \cdot 4}{4}}) + \frac{d}{d x} [\frac{\sqrt[4]{19}}{19} x]$

Step 4.1.2.5

Combine the numerators over the common denominator.

$- 4 (\frac{1}{4} x^{\frac{1 - 1 \cdot 4}{4}}) + \frac{d}{d x} [\frac{\sqrt[4]{19}}{19} x]$

Step 4.1.2.6

Simplify the numerator.

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Step 4.1.2.6.1

Multiply $- 1$ by $4$ .

$- 4 (\frac{1}{4} x^{\frac{1 - 4}{4}}) + \frac{d}{d x} [\frac{\sqrt[4]{19}}{19} x]$

Step 4.1.2.6.2

Subtract $4$ from $1$ .

$- 4 (\frac{1}{4} x^{\frac{- 3}{4}}) + \frac{d}{d x} [\frac{\sqrt[4]{19}}{19} x]$

Step 4.1.2.7

Move the negative in front of the fraction.

$- 4 (\frac{1}{4} x^{- \frac{3}{4}}) + \frac{d}{d x} [\frac{\sqrt[4]{19}}{19} x]$

Step 4.1.2.8

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

$- 4 \frac{x^{- \frac{3}{4}}}{4} + \frac{d}{d x} [\frac{\sqrt[4]{19}}{19} x]$

Step 4.1.2.9

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

$\frac{- 4 x^{- \frac{3}{4}}}{4} + \frac{d}{d x} [\frac{\sqrt[4]{19}}{19} x]$

Step 4.1.2.10

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

$\frac{- 4}{4 x^{\frac{3}{4}}} + \frac{d}{d x} [\frac{\sqrt[4]{19}}{19} x]$

Step 4.1.2.11

Factor $4$ out of $- 4$ .

$\frac{4 \cdot - 1}{4 x^{\frac{3}{4}}} + \frac{d}{d x} [\frac{\sqrt[4]{19}}{19} x]$

Step 4.1.2.12

Cancel the common factors.

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Step 4.1.2.12.1

Factor $4$ out of $4 x^{\frac{3}{4}}$ .

$\frac{4 \cdot - 1}{4 (x^{\frac{3}{4}})} + \frac{d}{d x} [\frac{\sqrt[4]{19}}{19} x]$

Step 4.1.2.12.2

Cancel the common factor.

$\frac{4 \cdot - 1}{4 x^{\frac{3}{4}}} + \frac{d}{d x} [\frac{\sqrt[4]{19}}{19} x]$

Step 4.1.2.12.3

Rewrite the expression.

$\frac{- 1}{x^{\frac{3}{4}}} + \frac{d}{d x} [\frac{\sqrt[4]{19}}{19} x]$

Step 4.1.2.13

Move the negative in front of the fraction.

$- \frac{1}{x^{\frac{3}{4}}} + \frac{d}{d x} [\frac{\sqrt[4]{19}}{19} x]$

Step 4.1.3

Evaluate $\frac{d}{d x} [\frac{\sqrt[4]{19}}{19} x]$ .

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Step 4.1.3.1

Since $\frac{\sqrt[4]{19}}{19}$ is constant with respect to $x$ , the derivative of $\frac{\sqrt[4]{19}}{19} x$ with respect to $x$ is $\frac{\sqrt[4]{19}}{19} \frac{d}{d x} [x]$ .

$- \frac{1}{x^{\frac{3}{4}}} + \frac{\sqrt[4]{19}}{19} \frac{d}{d x} [x]$

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 $n = 1$ .

$- \frac{1}{x^{\frac{3}{4}}} + \frac{\sqrt[4]{19}}{19} \cdot 1$

Step 4.1.3.3

Multiply $\frac{\sqrt[4]{19}}{19}$ by $1$ .

$- \frac{1}{x^{\frac{3}{4}}} + \frac{\sqrt[4]{19}}{19}$

Step 4.1.4

Reorder terms.

$f' (x) = \frac{\sqrt[4]{19}}{19} - \frac{1}{x^{\frac{3}{4}}}$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{\sqrt[4]{19}}{19} - \frac{1}{x^{\frac{3}{4}}}$ .

$\frac{\sqrt[4]{19}}{19} - \frac{1}{x^{\frac{3}{4}}}$

Step 5

Set the first derivative equal to $0$ then solve the equation $\frac{\sqrt[4]{19}}{19} - \frac{1}{x^{\frac{3}{4}}} = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$\frac{\sqrt[4]{19}}{19} - \frac{1}{x^{\frac{3}{4}}} = 0$

Step 5.2

Subtract $\frac{\sqrt[4]{19}}{19}$ from both sides of the equation.

$- \frac{1}{x^{\frac{3}{4}}} = - \frac{\sqrt[4]{19}}{19}$

Step 5.3

Find the LCD of the terms in the equation.

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Step 5.3.1

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

$x^{\frac{3}{4}}, 19$

Step 5.3.2

Since $x^{\frac{3}{4}}, 19$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $1, 19$ then find LCM for the variable part $x^{\frac{3}{4}}$ .

Step 5.3.3

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 5.3.4

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

Not prime

Step 5.3.5

Since $19$ has no factors besides $1$ and $19$ .

$19$ is a prime number

Step 5.3.6

The LCM of $1, 19$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$19$

Step 5.3.7

The LCM of $x^{\frac{3}{4}}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

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

Step 5.3.8

The LCM for $x^{\frac{3}{4}}, 19$ is the numeric part $19$ multiplied by the variable part.

$19 x^{\frac{3}{4}}$

Step 5.4

Multiply each term in $- \frac{1}{x^{\frac{3}{4}}} = - \frac{\sqrt[4]{19}}{19}$ by $19 x^{\frac{3}{4}}$ to eliminate the fractions.

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Step 5.4.1

Multiply each term in $- \frac{1}{x^{\frac{3}{4}}} = - \frac{\sqrt[4]{19}}{19}$ by $19 x^{\frac{3}{4}}$ .

$- \frac{1}{x^{\frac{3}{4}}} (19 x^{\frac{3}{4}}) = - \frac{\sqrt[4]{19}}{19} (19 x^{\frac{3}{4}})$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $x^{\frac{3}{4}}$ .

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Step 5.4.2.1.1

Move the leading negative in $- \frac{1}{x^{\frac{3}{4}}}$ into the numerator.

$\frac{- 1}{x^{\frac{3}{4}}} (19 x^{\frac{3}{4}}) = - \frac{\sqrt[4]{19}}{19} (19 x^{\frac{3}{4}})$

Step 5.4.2.1.2

Factor $x^{\frac{3}{4}}$ out of $19 x^{\frac{3}{4}}$ .

$\frac{- 1}{x^{\frac{3}{4}}} (x^{\frac{3}{4}} \cdot 19) = - \frac{\sqrt[4]{19}}{19} (19 x^{\frac{3}{4}})$

Step 5.4.2.1.3

Cancel the common factor.

$\frac{- 1}{x^{\frac{3}{4}}} (x^{\frac{3}{4}} \cdot 19) = - \frac{\sqrt[4]{19}}{19} (19 x^{\frac{3}{4}})$

Step 5.4.2.1.4

Rewrite the expression.

$- 1 \cdot 19 = - \frac{\sqrt[4]{19}}{19} (19 x^{\frac{3}{4}})$

Step 5.4.2.2

Multiply $- 1$ by $19$ .

$- 19 = - \frac{\sqrt[4]{19}}{19} (19 x^{\frac{3}{4}})$

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Cancel the common factor of $19$ .

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Step 5.4.3.1.1

Move the leading negative in $- \frac{\sqrt[4]{19}}{19}$ into the numerator.

$- 19 = \frac{- \sqrt[4]{19}}{19} (19 x^{\frac{3}{4}})$

Step 5.4.3.1.2

Factor $19$ out of $19 x^{\frac{3}{4}}$ .

$- 19 = \frac{- \sqrt[4]{19}}{19} (19 (x^{\frac{3}{4}}))$

Step 5.4.3.1.3

Cancel the common factor.

$- 19 = \frac{- \sqrt[4]{19}}{19} (19 x^{\frac{3}{4}})$

Step 5.4.3.1.4

Rewrite the expression.

$- 19 = - \sqrt[4]{19} x^{\frac{3}{4}}$

Step 5.5

Solve the equation.

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Step 5.5.1

Rewrite the equation as $- \sqrt[4]{19} x^{\frac{3}{4}} = - 19$ .

$- \sqrt[4]{19} x^{\frac{3}{4}} = - 19$

Step 5.5.2

Divide each term in $- \sqrt[4]{19} x^{\frac{3}{4}} = - 19$ by $- \sqrt[4]{19}$ and simplify.

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Step 5.5.2.1

Divide each term in $- \sqrt[4]{19} x^{\frac{3}{4}} = - 19$ by $- \sqrt[4]{19}$ .

$\frac{- \sqrt[4]{19} x^{\frac{3}{4}}}{- \sqrt[4]{19}} = \frac{- 19}{- \sqrt[4]{19}}$

Step 5.5.2.2

Simplify the left side.

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Step 5.5.2.2.1

Dividing two negative values results in a positive value.

$\frac{\sqrt[4]{19} x^{\frac{3}{4}}}{\sqrt[4]{19}} = \frac{- 19}{- \sqrt[4]{19}}$

Step 5.5.2.2.2

Cancel the common factor.

$\frac{\sqrt[4]{19} x^{\frac{3}{4}}}{\sqrt[4]{19}} = \frac{- 19}{- \sqrt[4]{19}}$

Step 5.5.2.2.3

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

$x^{\frac{3}{4}} = \frac{- 19}{- \sqrt[4]{19}}$

Step 5.5.2.3

Simplify the right side.

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Step 5.5.2.3.1

Dividing two negative values results in a positive value.

$x^{\frac{3}{4}} = \frac{19}{\sqrt[4]{19}}$

Step 5.5.2.3.2

Multiply $\frac{19}{\sqrt[4]{19}}$ by $\frac{{\sqrt[4]{19}}^{3}}{{\sqrt[4]{19}}^{3}}$ .

$x^{\frac{3}{4}} = \frac{19}{\sqrt[4]{19}} \cdot \frac{{\sqrt[4]{19}}^{3}}{{\sqrt[4]{19}}^{3}}$

Step 5.5.2.3.3

Combine and simplify the denominator.

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Step 5.5.2.3.3.1

Multiply $\frac{19}{\sqrt[4]{19}}$ by $\frac{{\sqrt[4]{19}}^{3}}{{\sqrt[4]{19}}^{3}}$ .

$x^{\frac{3}{4}} = \frac{19 {\sqrt[4]{19}}^{3}}{\sqrt[4]{19} {\sqrt[4]{19}}^{3}}$

Step 5.5.2.3.3.2

Raise $\sqrt[4]{19}$ to the power of $1$ .

$x^{\frac{3}{4}} = \frac{19 {\sqrt[4]{19}}^{3}}{{\sqrt[4]{19}}^{1} {\sqrt[4]{19}}^{3}}$

Step 5.5.2.3.3.3

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

$x^{\frac{3}{4}} = \frac{19 {\sqrt[4]{19}}^{3}}{{\sqrt[4]{19}}^{1 + 3}}$

Step 5.5.2.3.3.4

Add $1$ and $3$ .

$x^{\frac{3}{4}} = \frac{19 {\sqrt[4]{19}}^{3}}{{\sqrt[4]{19}}^{4}}$

Step 5.5.2.3.3.5

Rewrite ${\sqrt[4]{19}}^{4}$ as $19$ .

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Step 5.5.2.3.3.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{19}$ as $19^{\frac{1}{4}}$ .

$x^{\frac{3}{4}} = \frac{19 {\sqrt[4]{19}}^{3}}{{(19^{\frac{1}{4}})}^{4}}$

Step 5.5.2.3.3.5.2

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

$x^{\frac{3}{4}} = \frac{19 {\sqrt[4]{19}}^{3}}{19^{\frac{1}{4} \cdot 4}}$

Step 5.5.2.3.3.5.3

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

$x^{\frac{3}{4}} = \frac{19 {\sqrt[4]{19}}^{3}}{19^{\frac{4}{4}}}$

Step 5.5.2.3.3.5.4

Cancel the common factor of $4$ .

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Step 5.5.2.3.3.5.4.1

Cancel the common factor.

$x^{\frac{3}{4}} = \frac{19 {\sqrt[4]{19}}^{3}}{19^{\frac{4}{4}}}$

Step 5.5.2.3.3.5.4.2

Rewrite the expression.

$x^{\frac{3}{4}} = \frac{19 {\sqrt[4]{19}}^{3}}{19^{1}}$

Step 5.5.2.3.3.5.5

Evaluate the exponent.

$x^{\frac{3}{4}} = \frac{19 {\sqrt[4]{19}}^{3}}{19}$

Step 5.5.2.3.4

Cancel the common factor of $19$ .

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Step 5.5.2.3.4.1

Cancel the common factor.

$x^{\frac{3}{4}} = \frac{19 {\sqrt[4]{19}}^{3}}{19}$

Step 5.5.2.3.4.2

Divide ${\sqrt[4]{19}}^{3}$ by $1$ .

$x^{\frac{3}{4}} = {\sqrt[4]{19}}^{3}$

Step 5.5.2.3.5

Rewrite ${\sqrt[4]{19}}^{3}$ as $\sqrt[4]{19^{3}}$ .

$x^{\frac{3}{4}} = \sqrt[4]{19^{3}}$

Step 5.5.2.3.6

Raise $19$ to the power of $3$ .

$x^{\frac{3}{4}} = \sqrt[4]{6859}$

Step 5.5.3

Raise each side of the equation to the power of $\frac{4}{3}$ to eliminate the fractional exponent on the left side.

${(x^{\frac{3}{4}})}^{\frac{4}{3}} = {\sqrt[4]{6859}}^{\frac{4}{3}}$

Step 5.5.4

Simplify the exponent.

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Step 5.5.4.1

Simplify the left side.

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Step 5.5.4.1.1

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

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Step 5.5.4.1.1.1

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

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Step 5.5.4.1.1.1.1

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

$x^{\frac{3}{4} \cdot \frac{4}{3}} = {\sqrt[4]{6859}}^{\frac{4}{3}}$

Step 5.5.4.1.1.1.2

Cancel the common factor of $3$ .

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Step 5.5.4.1.1.1.2.1

Cancel the common factor.

$x^{\frac{3}{4} \cdot \frac{4}{3}} = {\sqrt[4]{6859}}^{\frac{4}{3}}$

Step 5.5.4.1.1.1.2.2

Rewrite the expression.

$x^{\frac{1}{4} \cdot 4} = {\sqrt[4]{6859}}^{\frac{4}{3}}$

Step 5.5.4.1.1.1.3

Cancel the common factor of $4$ .

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Step 5.5.4.1.1.1.3.1

Cancel the common factor.

$x^{\frac{1}{4} \cdot 4} = {\sqrt[4]{6859}}^{\frac{4}{3}}$

Step 5.5.4.1.1.1.3.2

Rewrite the expression.

$x^{1} = {\sqrt[4]{6859}}^{\frac{4}{3}}$

Step 5.5.4.1.1.2

Simplify.

$x = {\sqrt[4]{6859}}^{\frac{4}{3}}$

Step 5.5.4.2

Simplify the right side.

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Step 5.5.4.2.1

Simplify ${\sqrt[4]{6859}}^{\frac{4}{3}}$ .

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Step 5.5.4.2.1.1

Rewrite ${\sqrt[4]{6859}}^{\frac{4}{3}}$ as $\sqrt[3]{6859}$ .

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Step 5.5.4.2.1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{6859}$ as $6859^{\frac{1}{4}}$ .

$x = {(6859^{\frac{1}{4}})}^{\frac{4}{3}}$

Step 5.5.4.2.1.1.2

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

$x = 6859^{\frac{1}{4} \cdot \frac{4}{3}}$

Step 5.5.4.2.1.1.3

Multiply $\frac{1}{4}$ by $\frac{4}{3}$ .

$x = 6859^{\frac{4}{4 \cdot 3}}$

Step 5.5.4.2.1.1.4

Multiply $4$ by $3$ .

$x = 6859^{\frac{4}{12}}$

Step 5.5.4.2.1.1.5

Cancel the common factor of $4$ and $12$ .

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Step 5.5.4.2.1.1.5.1

Factor $4$ out of $4$ .

$x = 6859^{\frac{4 (1)}{12}}$

Step 5.5.4.2.1.1.5.2

Cancel the common factors.

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Step 5.5.4.2.1.1.5.2.1

Factor $4$ out of $12$ .

$x = 6859^{\frac{4 \cdot 1}{4 \cdot 3}}$

Step 5.5.4.2.1.1.5.2.2

Cancel the common factor.

$x = 6859^{\frac{4 \cdot 1}{4 \cdot 3}}$

Step 5.5.4.2.1.1.5.2.3

Rewrite the expression.

$x = 6859^{\frac{1}{3}}$

Step 5.5.4.2.1.1.6

Rewrite $6859^{\frac{1}{3}}$ as $\sqrt[3]{6859}$ .

$x = \sqrt[3]{6859}$

Step 5.5.4.2.1.2

Rewrite $6859$ as $19^{3}$ .

$x = \sqrt[3]{19^{3}}$

Step 5.5.4.2.1.3

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

$x = 19$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

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

$\frac{\sqrt[4]{19}}{19} - \frac{1}{\sqrt[4]{x^{3}}}$

Step 6.2

Set the denominator in $\frac{1}{\sqrt[4]{x^{3}}}$ equal to $0$ to find where the expression is undefined.

$\sqrt[4]{x^{3}} = 0$

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 $4$ .

${\sqrt[4]{x^{3}}}^{4} = 0^{4}$

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 $\sqrt[4]{x^{3}}$ as $x^{\frac{3}{4}}$ .

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

Step 6.3.2.2

Simplify the left side.

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Step 6.3.2.2.1

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

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Step 6.3.2.2.1.1

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

$x^{\frac{3}{4} \cdot 4} = 0^{4}$

Step 6.3.2.2.1.2

Cancel the common factor of $4$ .

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Step 6.3.2.2.1.2.1

Cancel the common factor.

$x^{\frac{3}{4} \cdot 4} = 0^{4}$

Step 6.3.2.2.1.2.2

Rewrite the expression.

$x^{3} = 0^{4}$

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 $0$ .

$x^{3} = 0$

Step 6.3.3

Solve for $x$ .

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Step 6.3.3.1

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

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

Step 6.3.3.2

Simplify $\sqrt[3]{0}$ .

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Step 6.3.3.2.1

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

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

Step 6.3.3.2.2

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

$x = 0$

Step 6.4

Set the radicand in $\sqrt[4]{x^{3}}$ less than $0$ to find where the expression is undefined.

$x^{3} < 0$

Step 6.5

Solve for $x$ .

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Step 6.5.1

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

$\sqrt[3]{x^{3}} < \sqrt[3]{0}$

Step 6.5.2

Simplify the equation.

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Step 6.5.2.1

Simplify the left side.

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Step 6.5.2.1.1

Pull terms out from under the radical.

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

Step 6.5.2.2

Simplify the right side.

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Step 6.5.2.2.1

Simplify $\sqrt[3]{0}$ .

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Step 6.5.2.2.1.1

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

$x < \sqrt[3]{0^{3}}$

Step 6.5.2.2.1.2

Pull terms out from under the radical.

$x < 0$

Step 6.6

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x \leq 0$

$(- \infty, 0]$

$x \leq 0$

$(- \infty, 0]$

Step 7

Critical points to evaluate.

$x = 19, 0$

Step 8

Evaluate the second derivative at $x = 19$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{3}{4 {(19)}^{\frac{7}{4}}}$

Step 9

Remove parentheses.

$\frac{3}{4 \cdot 19^{\frac{7}{4}}}$

Step 10

$x = 19$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = 19$ is a local minimum

Step 11

Find the y-value when $x = 19$ .

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Step 11.1

Replace the variable $x$ with $19$ in the expression.

$f (19) = - 4 {(19)}^{\frac{1}{4}} + \frac{\sqrt[4]{19}}{19} \cdot (19)$

Step 11.2

Simplify the result.

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Step 11.2.1

Cancel the common factor of $19$ .

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Step 11.2.1.1

Cancel the common factor.

$f (19) = - 4 \cdot 19^{\frac{1}{4}} + \frac{\sqrt[4]{19}}{19} \cdot 19$

Step 11.2.1.2

Rewrite the expression.

$f (19) = - 4 \cdot 19^{\frac{1}{4}} + \sqrt[4]{19}$

Step 11.2.2

The final answer is $- 4 \cdot 19^{\frac{1}{4}} + \sqrt[4]{19}$ .

$y = - 4 \cdot 19^{\frac{1}{4}} + \sqrt[4]{19}$

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.

$\frac{3}{4 {(0)}^{\frac{7}{4}}}$

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 $0^{4}$ .

$\frac{3}{4 {(0^{4})}^{\frac{7}{4}}}$

Step 13.1.2

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

$\frac{3}{4 \cdot 0^{4 (\frac{7}{4})}}$

Step 13.2

Cancel the common factor of $4$ .

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Step 13.2.1

Cancel the common factor.

$\frac{3}{4 \cdot 0^{4 (\frac{7}{4})}}$

Step 13.2.2

Rewrite the expression.

$\frac{3}{4 \cdot 0^{7}}$

Step 13.3

Simplify the expression.

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Step 13.3.1

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

$\frac{3}{4 \cdot 0}$

Step 13.3.2

Multiply $4$ by $0$ .

$\frac{3}{0}$

Step 13.3.3

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

Undefined

$\frac{3}{0}$

Step 13.4

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

Undefined

Step 14

Since the first derivative test failed, there are no local extrema.

No Local Extrema

Step 15