Calculus Examples

$\frac{x^{2} + 64}{x}$

Step 1

Write $\frac{x^{2} + 64}{x}$ as a function.

$f (x) = \frac{x^{2} + 64}{x}$

Step 2

Find the first derivative of the function.

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Step 2.1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x^{2} + 64$ and $g (x) = x$ .

$\frac{x \frac{d}{d x} [x^{2} + 64] - (x^{2} + 64) \frac{d}{d x} [x]}{x^{2}}$

Step 2.2

Differentiate.

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Step 2.2.1

By the Sum Rule, the derivative of $x^{2} + 64$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [64]$ .

$\frac{x (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [64]) - (x^{2} + 64) \frac{d}{d x} [x]}{x^{2}}$

Step 2.2.2

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

$\frac{x (2 x + \frac{d}{d x} [64]) - (x^{2} + 64) \frac{d}{d x} [x]}{x^{2}}$

Step 2.2.3

Since $64$ is constant with respect to $x$ , the derivative of $64$ with respect to $x$ is $0$ .

$\frac{x (2 x + 0) - (x^{2} + 64) \frac{d}{d x} [x]}{x^{2}}$

Step 2.2.4

Add $2 x$ and $0$ .

$\frac{x (2 x) - (x^{2} + 64) \frac{d}{d x} [x]}{x^{2}}$

Step 2.3

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

$\frac{2 (x^{1} x) - (x^{2} + 64) \frac{d}{d x} [x]}{x^{2}}$

Step 2.4

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

$\frac{2 (x^{1} x^{1}) - (x^{2} + 64) \frac{d}{d x} [x]}{x^{2}}$

Step 2.5

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

$\frac{2 x^{1 + 1} - (x^{2} + 64) \frac{d}{d x} [x]}{x^{2}}$

Step 2.6

Add $1$ and $1$ .

$\frac{2 x^{2} - (x^{2} + 64) \frac{d}{d x} [x]}{x^{2}}$

Step 2.7

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

$\frac{2 x^{2} - (x^{2} + 64) \cdot 1}{x^{2}}$

Step 2.8

Multiply $- 1$ by $1$ .

$\frac{2 x^{2} - (x^{2} + 64)}{x^{2}}$

Step 2.9

Simplify.

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Step 2.9.1

Apply the distributive property.

$\frac{2 x^{2} - x^{2} - 1 \cdot 64}{x^{2}}$

Step 2.9.2

Simplify the numerator.

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Step 2.9.2.1

Multiply $- 1$ by $64$ .

$\frac{2 x^{2} - x^{2} - 64}{x^{2}}$

Step 2.9.2.2

Subtract $x^{2}$ from $2 x^{2}$ .

$\frac{x^{2} - 64}{x^{2}}$

Step 2.9.3

Simplify the numerator.

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Step 2.9.3.1

Rewrite $64$ as $8^{2}$ .

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

Step 2.9.3.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 8$ .

$\frac{(x + 8) (x - 8)}{x^{2}}$

Step 3

Find the second derivative of the function.

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Step 3.1

$f'' (x) = \frac{x^{2} \frac{d}{d x} ((x + 8) (x - 8)) - (x + 8) (x - 8) \frac{d}{d x} x^{2}}{{(x^{2})}^{2}}$

Step 3.2

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

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Step 3.2.1

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

$f'' (x) = \frac{x^{2} \frac{d}{d x} ((x + 8) (x - 8)) - (x + 8) (x - 8) \frac{d}{d x} x^{2}}{x^{2 \cdot 2}}$

Step 3.2.2

Multiply $2$ by $2$ .

$f'' (x) = \frac{x^{2} \frac{d}{d x} ((x + 8) (x - 8)) - (x + 8) (x - 8) \frac{d}{d x} x^{2}}{x^{4}}$

Step 3.3

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x + 8$ and $g (x) = x - 8$ .

$f'' (x) = \frac{x^{2} ((x + 8) \frac{d}{d x} (x - 8) + (x - 8) \frac{d}{d x} (x + 8)) - (x + 8) (x - 8) \frac{d}{d x} x^{2}}{x^{4}}$

Step 3.4

Differentiate.

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Step 3.4.1

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

$f'' (x) = \frac{x^{2} ((x + 8) (\frac{d}{d x} (x) + \frac{d}{d x} (- 8)) + (x - 8) \frac{d}{d x} (x + 8)) - (x + 8) (x - 8) \frac{d}{d x} x^{2}}{x^{4}}$

Step 3.4.2

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

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

Step 3.4.3

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

$f'' (x) = \frac{x^{2} ((x + 8) (1 + 0) + (x - 8) \frac{d}{d x} (x + 8)) - (x + 8) (x - 8) \frac{d}{d x} x^{2}}{x^{4}}$

Step 3.4.4

Simplify the expression.

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Step 3.4.4.1

Add $1$ and $0$ .

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

Step 3.4.4.2

Multiply $x + 8$ by $1$ .

$f'' (x) = \frac{x^{2} (x + 8 + (x - 8) \frac{d}{d x} (x + 8)) - (x + 8) (x - 8) \frac{d}{d x} x^{2}}{x^{4}}$

Step 3.4.5

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

$f'' (x) = \frac{x^{2} (x + 8 + (x - 8) (\frac{d}{d x} (x) + \frac{d}{d x} (8))) - (x + 8) (x - 8) \frac{d}{d x} x^{2}}{x^{4}}$

Step 3.4.6

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

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

Step 3.4.7

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

$f'' (x) = \frac{x^{2} (x + 8 + (x - 8) (1 + 0)) - (x + 8) (x - 8) \frac{d}{d x} x^{2}}{x^{4}}$

Step 3.4.8

Simplify by adding terms.

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Step 3.4.8.1

Add $1$ and $0$ .

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

Step 3.4.8.2

Multiply $x - 8$ by $1$ .

$f'' (x) = \frac{x^{2} (x + 8 + x - 8) - (x + 8) (x - 8) \frac{d}{d x} x^{2}}{x^{4}}$

Step 3.4.8.3

Add $x$ and $x$ .

$f'' (x) = \frac{x^{2} (2 x + 8 - 8) - (x + 8) (x - 8) \frac{d}{d x} x^{2}}{x^{4}}$

Step 3.4.8.4

Simplify by subtracting numbers.

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Step 3.4.8.4.1

Subtract $8$ from $8$ .

$f'' (x) = \frac{x^{2} (2 x + 0) - (x + 8) (x - 8) \frac{d}{d x} x^{2}}{x^{4}}$

Step 3.4.8.4.2

Add $2 x$ and $0$ .

$f'' (x) = \frac{x^{2} (2 x) - (x + 8) (x - 8) \frac{d}{d x} x^{2}}{x^{4}}$

Step 3.5

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 3.5.1

Move $x$ .

$f'' (x) = \frac{x \cdot x^{2} \cdot 2 - (x + 8) (x - 8) \frac{d}{d x} x^{2}}{x^{4}}$

Step 3.5.2

Multiply $x$ by $x^{2}$ .

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Step 3.5.2.1

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

$f'' (x) = \frac{x \cdot x^{2} \cdot 2 - (x + 8) (x - 8) \frac{d}{d x} x^{2}}{x^{4}}$

Step 3.5.2.2

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

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

Step 3.5.3

Add $1$ and $2$ .

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

Step 3.6

Move $2$ to the left of $x^{3}$ .

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

Step 3.7

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

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

Step 3.8

Multiply $2$ by $- 1$ .

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

Step 3.9

Simplify.

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Step 3.9.1

Apply the distributive property.

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

Step 3.9.2

Simplify the numerator.

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Step 3.9.2.1

Simplify each term.

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Step 3.9.2.1.1

Multiply $- 2$ by $8$ .

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

Step 3.9.2.1.2

Expand $(- 2 x - 16) (x - 8)$ using the FOIL Method.

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Step 3.9.2.1.2.1

Apply the distributive property.

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

Step 3.9.2.1.2.2

Apply the distributive property.

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

Step 3.9.2.1.2.3

Apply the distributive property.

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

Step 3.9.2.1.3

Simplify and combine like terms.

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Step 3.9.2.1.3.1

Simplify each term.

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Step 3.9.2.1.3.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 3.9.2.1.3.1.1.1

Move $x$ .

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

Step 3.9.2.1.3.1.1.2

Multiply $x$ by $x$ .

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

Step 3.9.2.1.3.1.2

Multiply $- 8$ by $- 2$ .

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

Step 3.9.2.1.3.1.3

Multiply $- 16$ by $- 8$ .

$f'' (x) = \frac{2 x^{3} + (- 2 x^{2} + 16 x - 16 x + 128) x}{x^{4}}$

Step 3.9.2.1.3.2

Subtract $16 x$ from $16 x$ .

$f'' (x) = \frac{2 x^{3} + (- 2 x^{2} + 0 + 128) x}{x^{4}}$

Step 3.9.2.1.3.3

Add $- 2 x^{2}$ and $0$ .

$f'' (x) = \frac{2 x^{3} + (- 2 x^{2} + 128) x}{x^{4}}$

Step 3.9.2.1.4

Apply the distributive property.

$f'' (x) = \frac{2 x^{3} - 2 x^{2} x + 128 x}{x^{4}}$

Step 3.9.2.1.5

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 3.9.2.1.5.1

Move $x$ .

$f'' (x) = \frac{2 x^{3} - 2 (x \cdot x^{2}) + 128 x}{x^{4}}$

Step 3.9.2.1.5.2

Multiply $x$ by $x^{2}$ .

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Step 3.9.2.1.5.2.1

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

$f'' (x) = \frac{2 x^{3} - 2 (x \cdot x^{2}) + 128 x}{x^{4}}$

Step 3.9.2.1.5.2.2

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

$f'' (x) = \frac{2 x^{3} - 2 x^{1 + 2} + 128 x}{x^{4}}$

Step 3.9.2.1.5.3

Add $1$ and $2$ .

$f'' (x) = \frac{2 x^{3} - 2 x^{3} + 128 x}{x^{4}}$

Step 3.9.2.2

Subtract $2 x^{3}$ from $2 x^{3}$ .

$f'' (x) = \frac{0 + 128 x}{x^{4}}$

Step 3.9.2.3

Add $0$ and $128 x$ .

$f'' (x) = \frac{128 x}{x^{4}}$

Step 3.9.3

Cancel the common factor of $x$ and $x^{4}$ .

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Step 3.9.3.1

Factor $x$ out of $128 x$ .

$f'' (x) = \frac{x \cdot 128}{x^{4}}$

Step 3.9.3.2

Cancel the common factors.

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Step 3.9.3.2.1

Factor $x$ out of $x^{4}$ .

$f'' (x) = \frac{x \cdot 128}{x \cdot x^{3}}$

Step 3.9.3.2.2

Cancel the common factor.

$f'' (x) = \frac{x \cdot 128}{x \cdot x^{3}}$

Step 3.9.3.2.3

Rewrite the expression.

$f'' (x) = \frac{128}{x^{3}}$

Step 4

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

$\frac{(x + 8) (x - 8)}{x^{2}} = 0$

Step 5

Find the first derivative.

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Step 5.1

Find the first derivative.

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Step 5.1.1

$\frac{x \frac{d}{d x} [x^{2} + 64] - (x^{2} + 64) \frac{d}{d x} [x]}{x^{2}}$

Step 5.1.2

Differentiate.

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Step 5.1.2.1

By the Sum Rule, the derivative of $x^{2} + 64$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [64]$ .

$\frac{x (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [64]) - (x^{2} + 64) \frac{d}{d x} [x]}{x^{2}}$

Step 5.1.2.2

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

$\frac{x (2 x + \frac{d}{d x} [64]) - (x^{2} + 64) \frac{d}{d x} [x]}{x^{2}}$

Step 5.1.2.3

Since $64$ is constant with respect to $x$ , the derivative of $64$ with respect to $x$ is $0$ .

$\frac{x (2 x + 0) - (x^{2} + 64) \frac{d}{d x} [x]}{x^{2}}$

Step 5.1.2.4

Add $2 x$ and $0$ .

$\frac{x (2 x) - (x^{2} + 64) \frac{d}{d x} [x]}{x^{2}}$

Step 5.1.3

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

$\frac{2 (x^{1} x) - (x^{2} + 64) \frac{d}{d x} [x]}{x^{2}}$

Step 5.1.4

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

$\frac{2 (x^{1} x^{1}) - (x^{2} + 64) \frac{d}{d x} [x]}{x^{2}}$

Step 5.1.5

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

$\frac{2 x^{1 + 1} - (x^{2} + 64) \frac{d}{d x} [x]}{x^{2}}$

Step 5.1.6

Add $1$ and $1$ .

$\frac{2 x^{2} - (x^{2} + 64) \frac{d}{d x} [x]}{x^{2}}$

Step 5.1.7

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

$\frac{2 x^{2} - (x^{2} + 64) \cdot 1}{x^{2}}$

Step 5.1.8

Multiply $- 1$ by $1$ .

$\frac{2 x^{2} - (x^{2} + 64)}{x^{2}}$

Step 5.1.9

Simplify.

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Step 5.1.9.1

Apply the distributive property.

$\frac{2 x^{2} - x^{2} - 1 \cdot 64}{x^{2}}$

Step 5.1.9.2

Simplify the numerator.

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Step 5.1.9.2.1

Multiply $- 1$ by $64$ .

$\frac{2 x^{2} - x^{2} - 64}{x^{2}}$

Step 5.1.9.2.2

Subtract $x^{2}$ from $2 x^{2}$ .

$\frac{x^{2} - 64}{x^{2}}$

Step 5.1.9.3

Simplify the numerator.

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Step 5.1.9.3.1

Rewrite $64$ as $8^{2}$ .

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

Step 5.1.9.3.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 8$ .

$f' (x) = \frac{(x + 8) (x - 8)}{x^{2}}$

Step 5.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{(x + 8) (x - 8)}{x^{2}}$ .

$\frac{(x + 8) (x - 8)}{x^{2}}$

Step 6

Set the first derivative equal to $0$ then solve the equation $\frac{(x + 8) (x - 8)}{x^{2}} = 0$ .

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Step 6.1

Set the first derivative equal to $0$ .

$\frac{(x + 8) (x - 8)}{x^{2}} = 0$

Step 6.2

Set the numerator equal to zero.

$(x + 8) (x - 8) = 0$

Step 6.3

Solve the equation for $x$ .

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Step 6.3.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x + 8 = 0$

$x - 8 = 0$

Step 6.3.2

Set $x + 8$ equal to $0$ and solve for $x$ .

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Step 6.3.2.1

Set $x + 8$ equal to $0$ .

$x + 8 = 0$

Step 6.3.2.2

Subtract $8$ from both sides of the equation.

$x = - 8$

Step 6.3.3

Set $x - 8$ equal to $0$ and solve for $x$ .

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Step 6.3.3.1

Set $x - 8$ equal to $0$ .

$x - 8 = 0$

Step 6.3.3.2

Add $8$ to both sides of the equation.

$x = 8$

Step 6.3.4

The final solution is all the values that make $(x + 8) (x - 8) = 0$ true.

$x = - 8, 8$

Step 7

Find the values where the derivative is undefined.

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Step 7.1

Set the denominator in $\frac{(x + 8) (x - 8)}{x^{2}}$ equal to $0$ to find where the expression is undefined.

$x^{2} = 0$

Step 7.2

Solve for $x$ .

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Step 7.2.1

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

$x = \pm \sqrt{0}$

Step 7.2.2

Simplify $\pm \sqrt{0}$ .

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Step 7.2.2.1

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

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

Step 7.2.2.2

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

$x = \pm 0$

Step 7.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 8

Critical points to evaluate.

$x = - 8, 8$

Step 9

Evaluate the second derivative at $x = - 8$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{128}{{(- 8)}^{3}}$

Step 10

Evaluate the second derivative.

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Step 10.1

Raise $- 8$ to the power of $3$ .

$\frac{128}{- 512}$

Step 10.2

Cancel the common factor of $128$ and $- 512$ .

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Step 10.2.1

Factor $128$ out of $128$ .

$\frac{128 (1)}{- 512}$

Step 10.2.2

Cancel the common factors.

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Step 10.2.2.1

Factor $128$ out of $- 512$ .

$\frac{128 \cdot 1}{128 \cdot - 4}$

Step 10.2.2.2

Cancel the common factor.

$\frac{128 \cdot 1}{128 \cdot - 4}$

Step 10.2.2.3

Rewrite the expression.

$\frac{1}{- 4}$

Step 10.3

Move the negative in front of the fraction.

$- \frac{1}{4}$

Step 11

$x = - 8$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = - 8$ is a local maximum

Step 12

Find the y-value when $x = - 8$ .

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Step 12.1

Replace the variable $x$ with $- 8$ in the expression.

$f (- 8) = \frac{{(- 8)}^{2} + 64}{- 8}$

Step 12.2

Simplify the result.

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Step 12.2.1

Simplify the numerator.

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Step 12.2.1.1

Raise $- 8$ to the power of $2$ .

$f (- 8) = \frac{64 + 64}{- 8}$

Step 12.2.1.2

Add $64$ and $64$ .

$f (- 8) = \frac{128}{- 8}$

Step 12.2.2

Divide $128$ by $- 8$ .

$f (- 8) = - 16$

Step 12.2.3

The final answer is $- 16$ .

$y = - 16$

Step 13

Evaluate the second derivative at $x = 8$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{128}{{(8)}^{3}}$

Step 14

Evaluate the second derivative.

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Step 14.1

Raise $8$ to the power of $3$ .

$\frac{128}{512}$

Step 14.2

Cancel the common factor of $128$ and $512$ .

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Step 14.2.1

Factor $128$ out of $128$ .

$\frac{128 (1)}{512}$

Step 14.2.2

Cancel the common factors.

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Step 14.2.2.1

Factor $128$ out of $512$ .

$\frac{128 \cdot 1}{128 \cdot 4}$

Step 14.2.2.2

Cancel the common factor.

$\frac{128 \cdot 1}{128 \cdot 4}$

Step 14.2.2.3

Rewrite the expression.

$\frac{1}{4}$

Step 15

$x = 8$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = 8$ is a local minimum

Step 16

Find the y-value when $x = 8$ .

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Step 16.1

Replace the variable $x$ with $8$ in the expression.

$f (8) = \frac{{(8)}^{2} + 64}{8}$

Step 16.2

Simplify the result.

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Step 16.2.1

Simplify the numerator.

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Step 16.2.1.1

Raise $8$ to the power of $2$ .

$f (8) = \frac{64 + 64}{8}$

Step 16.2.1.2

Add $64$ and $64$ .

$f (8) = \frac{128}{8}$

Step 16.2.2

Divide $128$ by $8$ .

$f (8) = 16$

Step 16.2.3

The final answer is $16$ .

$y = 16$

Step 17

These are the local extrema for $f (x) = \frac{x^{2} + 64}{x}$ .

$(- 8, - 16)$ is a local maxima

$(8, 16)$ is a local minima

Step 18