Calculus Examples

$h (x) = x^{\frac{1}{5}} - 8$ , $[- 32, 1]$

Step 1

Find the critical points.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Find the first derivative.

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Step 1.1.1.1

By the Sum Rule, the derivative of $x^{\frac{1}{5}} - 8$ with respect to $x$ is $\frac{d}{d x} [x^{\frac{1}{5}}] + \frac{d}{d x} [- 8]$ .

$\frac{d}{d x} [x^{\frac{1}{5}}] + \frac{d}{d x} [- 8]$

Step 1.1.1.2

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

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Step 1.1.1.2.1

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

$\frac{1}{5} x^{\frac{1}{5} - 1} + \frac{d}{d x} [- 8]$

Step 1.1.1.2.2

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

$\frac{1}{5} x^{\frac{1}{5} - 1 \cdot \frac{5}{5}} + \frac{d}{d x} [- 8]$

Step 1.1.1.2.3

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

$\frac{1}{5} x^{\frac{1}{5} + \frac{- 1 \cdot 5}{5}} + \frac{d}{d x} [- 8]$

Step 1.1.1.2.4

Combine the numerators over the common denominator.

$\frac{1}{5} x^{\frac{1 - 1 \cdot 5}{5}} + \frac{d}{d x} [- 8]$

Step 1.1.1.2.5

Simplify the numerator.

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Step 1.1.1.2.5.1

Multiply $- 1$ by $5$ .

$\frac{1}{5} x^{\frac{1 - 5}{5}} + \frac{d}{d x} [- 8]$

Step 1.1.1.2.5.2

Subtract $5$ from $1$ .

$\frac{1}{5} x^{\frac{- 4}{5}} + \frac{d}{d x} [- 8]$

Step 1.1.1.2.6

Move the negative in front of the fraction.

$\frac{1}{5} x^{- \frac{4}{5}} + \frac{d}{d x} [- 8]$

Step 1.1.1.3

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8$ with respect to $x$ is $0$ .

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

Step 1.1.1.4

Simplify.

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Step 1.1.1.4.1

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

$\frac{1}{5} \cdot \frac{1}{x^{\frac{4}{5}}} + 0$

Step 1.1.1.4.2

Combine terms.

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Step 1.1.1.4.2.1

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

$\frac{1}{5 x^{\frac{4}{5}}} + 0$

Step 1.1.1.4.2.2

Add $\frac{1}{5 x^{\frac{4}{5}}}$ and $0$ .

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

Step 1.1.2

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

$\frac{1}{5 x^{\frac{4}{5}}}$

Step 1.2

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

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Step 1.2.1

Set the first derivative equal to $0$ .

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

Step 1.2.2

Set the numerator equal to zero.

$1 = 0$

Step 1.2.3

Since $1 \neq 0$ , there are no solutions.

No solution

Step 1.3

Find the values where the derivative is undefined.

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Step 1.3.1

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

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

Step 1.3.2

Set the denominator in $\frac{1}{5 \sqrt[5]{x^{4}}}$ equal to $0$ to find where the expression is undefined.

$5 \sqrt[5]{x^{4}} = 0$

Step 1.3.3

Solve for $x$ .

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Step 1.3.3.1

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

${(5 \sqrt[5]{x^{4}})}^{5} = 0^{5}$

Step 1.3.3.2

Simplify each side of the equation.

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Step 1.3.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{x^{4}}$ as $x^{\frac{4}{5}}$ .

${(5 x^{\frac{4}{5}})}^{5} = 0^{5}$

Step 1.3.3.2.2

Simplify the left side.

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Step 1.3.3.2.2.1

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

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Step 1.3.3.2.2.1.1

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

$5^{5} {(x^{\frac{4}{5}})}^{5} = 0^{5}$

Step 1.3.3.2.2.1.2

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

$3125 {(x^{\frac{4}{5}})}^{5} = 0^{5}$

Step 1.3.3.2.2.1.3

Multiply the exponents in ${(x^{\frac{4}{5}})}^{5}$ .

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Step 1.3.3.2.2.1.3.1

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

$3125 x^{\frac{4}{5} \cdot 5} = 0^{5}$

Step 1.3.3.2.2.1.3.2

Cancel the common factor of $5$ .

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Step 1.3.3.2.2.1.3.2.1

Cancel the common factor.

$3125 x^{\frac{4}{5} \cdot 5} = 0^{5}$

Step 1.3.3.2.2.1.3.2.2

Rewrite the expression.

$3125 x^{4} = 0^{5}$

Step 1.3.3.2.3

Simplify the right side.

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Step 1.3.3.2.3.1

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

$3125 x^{4} = 0$

Step 1.3.3.3

Solve for $x$ .

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Step 1.3.3.3.1

Divide each term in $3125 x^{4} = 0$ by $3125$ and simplify.

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Step 1.3.3.3.1.1

Divide each term in $3125 x^{4} = 0$ by $3125$ .

$\frac{3125 x^{4}}{3125} = \frac{0}{3125}$

Step 1.3.3.3.1.2

Simplify the left side.

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Step 1.3.3.3.1.2.1

Cancel the common factor of $3125$ .

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Step 1.3.3.3.1.2.1.1

Cancel the common factor.

$\frac{3125 x^{4}}{3125} = \frac{0}{3125}$

Step 1.3.3.3.1.2.1.2

Divide $x^{4}$ by $1$ .

$x^{4} = \frac{0}{3125}$

Step 1.3.3.3.1.3

Simplify the right side.

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Step 1.3.3.3.1.3.1

Divide $0$ by $3125$ .

$x^{4} = 0$

Step 1.3.3.3.2

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

$x = \pm \sqrt[4]{0}$

Step 1.3.3.3.3

Simplify $\pm \sqrt[4]{0}$ .

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Step 1.3.3.3.3.1

Rewrite $0$ as $0^{4}$ .

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

Step 1.3.3.3.3.2

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

$x = \pm 0$

Step 1.3.3.3.3.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 1.4

Evaluate $x^{\frac{1}{5}} - 8$ at each $x$ value where the derivative is $0$ or undefined.

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Step 1.4.1

Evaluate at $x = 0$ .

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Step 1.4.1.1

Substitute $0$ for $x$ .

${(0)}^{\frac{1}{5}} - 8$

Step 1.4.1.2

Simplify.

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Step 1.4.1.2.1

Simplify each term.

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Step 1.4.1.2.1.1

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

${(0^{5})}^{\frac{1}{5}} - 8$

Step 1.4.1.2.1.2

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

$0^{5 (\frac{1}{5})} - 8$

Step 1.4.1.2.1.3

Cancel the common factor of $5$ .

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Step 1.4.1.2.1.3.1

Cancel the common factor.

$0^{5 (\frac{1}{5})} - 8$

Step 1.4.1.2.1.3.2

Rewrite the expression.

$0^{1} - 8$

Step 1.4.1.2.1.4

Evaluate the exponent.

$0 - 8$

Step 1.4.1.2.2

Subtract $8$ from $0$ .

$- 8$

Step 1.4.2

List all of the points.

$(0, - 8)$

Step 2

Evaluate at the included endpoints.

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Step 2.1

Evaluate at $x = - 32$ .

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Step 2.1.1

Substitute $- 32$ for $x$ .

${(- 32)}^{\frac{1}{5}} - 8$

Step 2.1.2

Simplify.

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Step 2.1.2.1

Simplify each term.

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Step 2.1.2.1.1

Rewrite $- 32$ as ${(- 2)}^{5}$ .

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

Step 2.1.2.1.2

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

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

Step 2.1.2.1.3

Cancel the common factor of $5$ .

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Step 2.1.2.1.3.1

Cancel the common factor.

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

Step 2.1.2.1.3.2

Rewrite the expression.

${(- 2)}^{1} - 8$

Step 2.1.2.1.4

Evaluate the exponent.

$- 2 - 8$

Step 2.1.2.2

Subtract $8$ from $- 2$ .

$- 10$

Step 2.2

Evaluate at $x = 1$ .

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Step 2.2.1

Substitute $1$ for $x$ .

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

Step 2.2.2

Simplify.

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Step 2.2.2.1

One to any power is one.

$1 - 8$

Step 2.2.2.2

Subtract $8$ from $1$ .

$- 7$

Step 2.3

List all of the points.

$(- 32, - 10), (1, - 7)$

Step 3

Compare the $h (x)$ values found for each value of $x$ in order to determine the absolute maximum and minimum over the given interval. The maximum will occur at the highest $h (x)$ value and the minimum will occur at the lowest $h (x)$ value.

Absolute Maximum: $(1, - 7)$

Absolute Minimum: $(- 32, - 10)$

Step 4