Calculus Examples

$\frac{16 x^{2} + 25}{x}$

Step 1

Write $\frac{16 x^{2} + 25}{x}$ as a function.

$f (x) = \frac{16 x^{2} + 25}{x}$

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 $f (x) = 16 x^{2} + 25$ and $g (x) = x$ .

$\frac{x \frac{d}{d x} [16 x^{2} + 25] - (16 x^{2} + 25) \frac{d}{d x} [x]}{x^{2}}$

Step 2.2

Differentiate.

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Step 2.2.1

By the Sum Rule, the derivative of $16 x^{2} + 25$ with respect to $x$ is $\frac{d}{d x} [16 x^{2}] + \frac{d}{d x} [25]$ .

$\frac{x (\frac{d}{d x} [16 x^{2}] + \frac{d}{d x} [25]) - (16 x^{2} + 25) \frac{d}{d x} [x]}{x^{2}}$

Step 2.2.2

Since $16$ is constant with respect to $x$ , the derivative of $16 x^{2}$ with respect to $x$ is $16 \frac{d}{d x} [x^{2}]$ .

$\frac{x (16 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [25]) - (16 x^{2} + 25) \frac{d}{d x} [x]}{x^{2}}$

Step 2.2.3

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

$\frac{x (16 (2 x) + \frac{d}{d x} [25]) - (16 x^{2} + 25) \frac{d}{d x} [x]}{x^{2}}$

Step 2.2.4

Multiply $2$ by $16$ .

$\frac{x (32 x + \frac{d}{d x} [25]) - (16 x^{2} + 25) \frac{d}{d x} [x]}{x^{2}}$

Step 2.2.5

Since $25$ is constant with respect to $x$ , the derivative of $25$ with respect to $x$ is $0$ .

$\frac{x (32 x + 0) - (16 x^{2} + 25) \frac{d}{d x} [x]}{x^{2}}$

Step 2.2.6

Add $32 x$ and $0$ .

$\frac{x (32 x) - (16 x^{2} + 25) \frac{d}{d x} [x]}{x^{2}}$

Step 2.3

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

$\frac{32 (x^{1} x) - (16 x^{2} + 25) \frac{d}{d x} [x]}{x^{2}}$

Step 2.4

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

$\frac{32 (x^{1} x^{1}) - (16 x^{2} + 25) \frac{d}{d x} [x]}{x^{2}}$

Step 2.5

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

$\frac{32 x^{1 + 1} - (16 x^{2} + 25) \frac{d}{d x} [x]}{x^{2}}$

Step 2.6

Add $1$ and $1$ .

$\frac{32 x^{2} - (16 x^{2} + 25) \frac{d}{d x} [x]}{x^{2}}$

Step 2.7

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

$\frac{32 x^{2} - (16 x^{2} + 25) \cdot 1}{x^{2}}$

Step 2.8

Multiply $- 1$ by $1$ .

$\frac{32 x^{2} - (16 x^{2} + 25)}{x^{2}}$

Step 2.9

Simplify.

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Step 2.9.1

Apply the distributive property.

$\frac{32 x^{2} - (16 x^{2}) - 1 \cdot 25}{x^{2}}$

Step 2.9.2

Simplify the numerator.

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Step 2.9.2.1

Simplify each term.

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Step 2.9.2.1.1

Multiply $16$ by $- 1$ .

$\frac{32 x^{2} - 16 x^{2} - 1 \cdot 25}{x^{2}}$

Step 2.9.2.1.2

Multiply $- 1$ by $25$ .

$\frac{32 x^{2} - 16 x^{2} - 25}{x^{2}}$

Step 2.9.2.2

Subtract $16 x^{2}$ from $32 x^{2}$ .

$\frac{16 x^{2} - 25}{x^{2}}$

Step 2.9.3

Simplify the numerator.

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Step 2.9.3.1

Rewrite $16 x^{2}$ as ${(4 x)}^{2}$ .

$\frac{{(4 x)}^{2} - 25}{x^{2}}$

Step 2.9.3.2

Rewrite $25$ as $5^{2}$ .

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

Step 2.9.3.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 4 x$ and $b = 5$ .

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

Step 3

Find the second derivative of the function.

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Step 3.1

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

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, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 3.2.2

Multiply $2$ by $2$ .

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

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 $f (x) = 4 x + 5$ and $g (x) = 4 x - 5$ .

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

Step 3.4

Differentiate.

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Step 3.4.1

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

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

Step 3.4.2

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

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

Step 3.4.3

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

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

Step 3.4.4

Multiply $4$ by $1$ .

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

Step 3.4.5

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5$ with respect to $x$ is $0$ .

$f'' (x) = \frac{x^{2} ((4 x + 5) (4 + 0) + (4 x - 5) \frac{d}{d x} (4 x + 5)) - (4 x + 5) (4 x - 5) \frac{d}{d x} x^{2}}{x^{4}}$

Step 3.4.6

Simplify the expression.

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Step 3.4.6.1

Add $4$ and $0$ .

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

Step 3.4.6.2

Move $4$ to the left of $4 x + 5$ .

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

Step 3.4.7

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

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

Step 3.4.8

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

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

Step 3.4.9

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

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

Step 3.4.10

Multiply $4$ by $1$ .

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

Step 3.4.11

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$f'' (x) = \frac{x^{2} (4 (4 x + 5) + (4 x - 5) (4 + 0)) - (4 x + 5) (4 x - 5) \frac{d}{d x} x^{2}}{x^{4}}$

Step 3.4.12

Simplify the expression.

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Step 3.4.12.1

Add $4$ and $0$ .

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

Step 3.4.12.2

Move $4$ to the left of $4 x - 5$ .

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

Step 3.4.13

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

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

Step 3.4.14

Simplify with factoring out.

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Step 3.4.14.1

Multiply $2$ by $- 1$ .

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

Step 3.4.14.2

Factor $x$ out of $x^{2} (4 (4 x + 5) + 4 (4 x - 5)) - 2 (4 x + 5) (4 x - 5) x$ .

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Step 3.4.14.2.1

Factor $x$ out of $x^{2} (4 (4 x + 5) + 4 (4 x - 5))$ .

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

Step 3.4.14.2.2

Factor $x$ out of $- 2 (4 x + 5) (4 x - 5) x$ .

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

Step 3.4.14.2.3

Factor $x$ out of $x (x (4 (4 x + 5) + 4 (4 x - 5))) + x (- 2 (4 x + 5) (4 x - 5))$ .

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

Step 3.5

Cancel the common factors.

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Step 3.5.1

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

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

Step 3.5.2

Cancel the common factor.

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

Step 3.5.3

Rewrite the expression.

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

Step 3.6

Simplify.

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Step 3.6.1

Apply the distributive property.

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

Step 3.6.2

Apply the distributive property.

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

Step 3.6.3

Apply the distributive property.

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

Step 3.6.4

Apply the distributive property.

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

Step 3.6.5

Simplify the numerator.

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Step 3.6.5.1

Simplify each term.

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Step 3.6.5.1.1

Rewrite using the commutative property of multiplication.

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

Step 3.6.5.1.2

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

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Step 3.6.5.1.2.1

Move $x$ .

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

Step 3.6.5.1.2.2

Multiply $x$ by $x$ .

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

Step 3.6.5.1.3

Multiply $4$ by $4$ .

$f'' (x) = \frac{16 x^{2} + x (4 \cdot 5) + x (4 (4 x)) + x (4 \cdot - 5) + (- 2 (4 x) - 2 \cdot 5) (4 x - 5)}{x^{3}}$

Step 3.6.5.1.4

Multiply $4$ by $5$ .

$f'' (x) = \frac{16 x^{2} + x \cdot 20 + x (4 (4 x)) + x (4 \cdot - 5) + (- 2 (4 x) - 2 \cdot 5) (4 x - 5)}{x^{3}}$

Step 3.6.5.1.5

Move $20$ to the left of $x$ .

$f'' (x) = \frac{16 x^{2} + 20 \cdot x + x (4 (4 x)) + x (4 \cdot - 5) + (- 2 (4 x) - 2 \cdot 5) (4 x - 5)}{x^{3}}$

Step 3.6.5.1.6

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{16 x^{2} + 20 x + 4 \cdot (4 x \cdot x) + x (4 \cdot - 5) + (- 2 (4 x) - 2 \cdot 5) (4 x - 5)}{x^{3}}$

Step 3.6.5.1.7

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

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Step 3.6.5.1.7.1

Move $x$ .

$f'' (x) = \frac{16 x^{2} + 20 x + 4 \cdot (4 (x \cdot x)) + x (4 \cdot - 5) + (- 2 (4 x) - 2 \cdot 5) (4 x - 5)}{x^{3}}$

Step 3.6.5.1.7.2

Multiply $x$ by $x$ .

$f'' (x) = \frac{16 x^{2} + 20 x + 4 \cdot (4 x^{2}) + x (4 \cdot - 5) + (- 2 (4 x) - 2 \cdot 5) (4 x - 5)}{x^{3}}$

Step 3.6.5.1.8

Multiply $4$ by $4$ .

$f'' (x) = \frac{16 x^{2} + 20 x + 16 x^{2} + x (4 \cdot - 5) + (- 2 (4 x) - 2 \cdot 5) (4 x - 5)}{x^{3}}$

Step 3.6.5.1.9

Multiply $4$ by $- 5$ .

$f'' (x) = \frac{16 x^{2} + 20 x + 16 x^{2} + x \cdot - 20 + (- 2 (4 x) - 2 \cdot 5) (4 x - 5)}{x^{3}}$

Step 3.6.5.1.10

Move $- 20$ to the left of $x$ .

$f'' (x) = \frac{16 x^{2} + 20 x + 16 x^{2} - 20 \cdot x + (- 2 (4 x) - 2 \cdot 5) (4 x - 5)}{x^{3}}$

Step 3.6.5.1.11

Simplify each term.

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Step 3.6.5.1.11.1

Multiply $4$ by $- 2$ .

$f'' (x) = \frac{16 x^{2} + 20 x + 16 x^{2} - 20 x + (- 8 x - 2 \cdot 5) (4 x - 5)}{x^{3}}$

Step 3.6.5.1.11.2

Multiply $- 2$ by $5$ .

$f'' (x) = \frac{16 x^{2} + 20 x + 16 x^{2} - 20 x + (- 8 x - 10) (4 x - 5)}{x^{3}}$

Step 3.6.5.1.12

Expand $(- 8 x - 10) (4 x - 5)$ using the FOIL Method.

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Step 3.6.5.1.12.1

Apply the distributive property.

$f'' (x) = \frac{16 x^{2} + 20 x + 16 x^{2} - 20 x - 8 x (4 x - 5) - 10 (4 x - 5)}{x^{3}}$

Step 3.6.5.1.12.2

Apply the distributive property.

$f'' (x) = \frac{16 x^{2} + 20 x + 16 x^{2} - 20 x - 8 x (4 x) - 8 x \cdot - 5 - 10 (4 x - 5)}{x^{3}}$

Step 3.6.5.1.12.3

Apply the distributive property.

$f'' (x) = \frac{16 x^{2} + 20 x + 16 x^{2} - 20 x - 8 x (4 x) - 8 x \cdot - 5 - 10 (4 x) - 10 \cdot - 5}{x^{3}}$

Step 3.6.5.1.13

Simplify and combine like terms.

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Step 3.6.5.1.13.1

Simplify each term.

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Step 3.6.5.1.13.1.1

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{16 x^{2} + 20 x + 16 x^{2} - 20 x - 8 \cdot (4 x \cdot x) - 8 x \cdot - 5 - 10 (4 x) - 10 \cdot - 5}{x^{3}}$

Step 3.6.5.1.13.1.2

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

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Step 3.6.5.1.13.1.2.1

Move $x$ .

$f'' (x) = \frac{16 x^{2} + 20 x + 16 x^{2} - 20 x - 8 \cdot (4 (x \cdot x)) - 8 x \cdot - 5 - 10 (4 x) - 10 \cdot - 5}{x^{3}}$

Step 3.6.5.1.13.1.2.2

Multiply $x$ by $x$ .

$f'' (x) = \frac{16 x^{2} + 20 x + 16 x^{2} - 20 x - 8 \cdot (4 x^{2}) - 8 x \cdot - 5 - 10 (4 x) - 10 \cdot - 5}{x^{3}}$

Step 3.6.5.1.13.1.3

Multiply $- 8$ by $4$ .

$f'' (x) = \frac{16 x^{2} + 20 x + 16 x^{2} - 20 x - 32 x^{2} - 8 x \cdot - 5 - 10 (4 x) - 10 \cdot - 5}{x^{3}}$

Step 3.6.5.1.13.1.4

Multiply $- 5$ by $- 8$ .

$f'' (x) = \frac{16 x^{2} + 20 x + 16 x^{2} - 20 x - 32 x^{2} + 40 x - 10 (4 x) - 10 \cdot - 5}{x^{3}}$

Step 3.6.5.1.13.1.5

Multiply $4$ by $- 10$ .

$f'' (x) = \frac{16 x^{2} + 20 x + 16 x^{2} - 20 x - 32 x^{2} + 40 x - 40 x - 10 \cdot - 5}{x^{3}}$

Step 3.6.5.1.13.1.6

Multiply $- 10$ by $- 5$ .

$f'' (x) = \frac{16 x^{2} + 20 x + 16 x^{2} - 20 x - 32 x^{2} + 40 x - 40 x + 50}{x^{3}}$

Step 3.6.5.1.13.2

Subtract $40 x$ from $40 x$ .

$f'' (x) = \frac{16 x^{2} + 20 x + 16 x^{2} - 20 x - 32 x^{2} + 0 + 50}{x^{3}}$

Step 3.6.5.1.13.3

Add $- 32 x^{2}$ and $0$ .

$f'' (x) = \frac{16 x^{2} + 20 x + 16 x^{2} - 20 x - 32 x^{2} + 50}{x^{3}}$

Step 3.6.5.2

Combine the opposite terms in $16 x^{2} + 20 x + 16 x^{2} - 20 x - 32 x^{2} + 50$ .

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Step 3.6.5.2.1

Subtract $20 x$ from $20 x$ .

$f'' (x) = \frac{16 x^{2} + 16 x^{2} + 0 - 32 x^{2} + 50}{x^{3}}$

Step 3.6.5.2.2

Add $16 x^{2} + 16 x^{2}$ and $0$ .

$f'' (x) = \frac{16 x^{2} + 16 x^{2} - 32 x^{2} + 50}{x^{3}}$

Step 3.6.5.3

Add $16 x^{2}$ and $16 x^{2}$ .

$f'' (x) = \frac{32 x^{2} - 32 x^{2} + 50}{x^{3}}$

Step 3.6.5.4

Subtract $32 x^{2}$ from $32 x^{2}$ .

$f'' (x) = \frac{0 + 50}{x^{3}}$

Step 3.6.5.5

Add $0$ and $50$ .

$f'' (x) = \frac{50}{x^{3}}$

Step 4

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

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

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

$\frac{x \frac{d}{d x} [16 x^{2} + 25] - (16 x^{2} + 25) \frac{d}{d x} [x]}{x^{2}}$

Step 5.1.2

Differentiate.

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Step 5.1.2.1

By the Sum Rule, the derivative of $16 x^{2} + 25$ with respect to $x$ is $\frac{d}{d x} [16 x^{2}] + \frac{d}{d x} [25]$ .

$\frac{x (\frac{d}{d x} [16 x^{2}] + \frac{d}{d x} [25]) - (16 x^{2} + 25) \frac{d}{d x} [x]}{x^{2}}$

Step 5.1.2.2

Since $16$ is constant with respect to $x$ , the derivative of $16 x^{2}$ with respect to $x$ is $16 \frac{d}{d x} [x^{2}]$ .

$\frac{x (16 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [25]) - (16 x^{2} + 25) \frac{d}{d x} [x]}{x^{2}}$

Step 5.1.2.3

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

$\frac{x (16 (2 x) + \frac{d}{d x} [25]) - (16 x^{2} + 25) \frac{d}{d x} [x]}{x^{2}}$

Step 5.1.2.4

Multiply $2$ by $16$ .

$\frac{x (32 x + \frac{d}{d x} [25]) - (16 x^{2} + 25) \frac{d}{d x} [x]}{x^{2}}$

Step 5.1.2.5

Since $25$ is constant with respect to $x$ , the derivative of $25$ with respect to $x$ is $0$ .

$\frac{x (32 x + 0) - (16 x^{2} + 25) \frac{d}{d x} [x]}{x^{2}}$

Step 5.1.2.6

Add $32 x$ and $0$ .

$\frac{x (32 x) - (16 x^{2} + 25) \frac{d}{d x} [x]}{x^{2}}$

Step 5.1.3

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

$\frac{32 (x^{1} x) - (16 x^{2} + 25) \frac{d}{d x} [x]}{x^{2}}$

Step 5.1.4

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

$\frac{32 (x^{1} x^{1}) - (16 x^{2} + 25) \frac{d}{d x} [x]}{x^{2}}$

Step 5.1.5

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

$\frac{32 x^{1 + 1} - (16 x^{2} + 25) \frac{d}{d x} [x]}{x^{2}}$

Step 5.1.6

Add $1$ and $1$ .

$\frac{32 x^{2} - (16 x^{2} + 25) \frac{d}{d x} [x]}{x^{2}}$

Step 5.1.7

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

$\frac{32 x^{2} - (16 x^{2} + 25) \cdot 1}{x^{2}}$

Step 5.1.8

Multiply $- 1$ by $1$ .

$\frac{32 x^{2} - (16 x^{2} + 25)}{x^{2}}$

Step 5.1.9

Simplify.

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Step 5.1.9.1

Apply the distributive property.

$\frac{32 x^{2} - (16 x^{2}) - 1 \cdot 25}{x^{2}}$

Step 5.1.9.2

Simplify the numerator.

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Step 5.1.9.2.1

Simplify each term.

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Step 5.1.9.2.1.1

Multiply $16$ by $- 1$ .

$\frac{32 x^{2} - 16 x^{2} - 1 \cdot 25}{x^{2}}$

Step 5.1.9.2.1.2

Multiply $- 1$ by $25$ .

$\frac{32 x^{2} - 16 x^{2} - 25}{x^{2}}$

Step 5.1.9.2.2

Subtract $16 x^{2}$ from $32 x^{2}$ .

$\frac{16 x^{2} - 25}{x^{2}}$

Step 5.1.9.3

Simplify the numerator.

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Step 5.1.9.3.1

Rewrite $16 x^{2}$ as ${(4 x)}^{2}$ .

$\frac{{(4 x)}^{2} - 25}{x^{2}}$

Step 5.1.9.3.2

Rewrite $25$ as $5^{2}$ .

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

Step 5.1.9.3.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 4 x$ and $b = 5$ .

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

Step 5.2

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

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

Step 6

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

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Step 6.1

Set the first derivative equal to $0$ .

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

Step 6.2

Set the numerator equal to zero.

$(4 x + 5) (4 x - 5) = 0$

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 $0$ .

$4 x + 5 = 0$

$4 x - 5 = 0$

Step 6.3.2

Set $4 x + 5$ equal to $0$ and solve for $x$ .

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Step 6.3.2.1

Set $4 x + 5$ equal to $0$ .

$4 x + 5 = 0$

Step 6.3.2.2

Solve $4 x + 5 = 0$ for $x$ .

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Step 6.3.2.2.1

Subtract $5$ from both sides of the equation.

$4 x = - 5$

Step 6.3.2.2.2

Divide each term in $4 x = - 5$ by $4$ and simplify.

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Step 6.3.2.2.2.1

Divide each term in $4 x = - 5$ by $4$ .

$\frac{4 x}{4} = \frac{- 5}{4}$

Step 6.3.2.2.2.2

Simplify the left side.

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Step 6.3.2.2.2.2.1

Cancel the common factor of $4$ .

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Step 6.3.2.2.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{- 5}{4}$

Step 6.3.2.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 5}{4}$

Step 6.3.2.2.2.3

Simplify the right side.

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Step 6.3.2.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{5}{4}$

Step 6.3.3

Set $4 x - 5$ equal to $0$ and solve for $x$ .

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Step 6.3.3.1

Set $4 x - 5$ equal to $0$ .

$4 x - 5 = 0$

Step 6.3.3.2

Solve $4 x - 5 = 0$ for $x$ .

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Step 6.3.3.2.1

Add $5$ to both sides of the equation.

$4 x = 5$

Step 6.3.3.2.2

Divide each term in $4 x = 5$ by $4$ and simplify.

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Step 6.3.3.2.2.1

Divide each term in $4 x = 5$ by $4$ .

$\frac{4 x}{4} = \frac{5}{4}$

Step 6.3.3.2.2.2

Simplify the left side.

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Step 6.3.3.2.2.2.1

Cancel the common factor of $4$ .

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Step 6.3.3.2.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{5}{4}$

Step 6.3.3.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{5}{4}$

Step 6.3.4

The final solution is all the values that make $(4 x + 5) (4 x - 5) = 0$ true.

$x = - \frac{5}{4}, \frac{5}{4}$

Step 7

Find the values where the derivative is undefined.

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Step 7.1

Set the denominator in $\frac{(4 x + 5) (4 x - 5)}{x^{2}}$ 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.

$x = - \frac{5}{4}, \frac{5}{4}$

Step 9

Evaluate the second derivative at $x = - \frac{5}{4}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{50}{{(- \frac{5}{4})}^{3}}$

Step 10

Evaluate the second derivative.

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Step 10.1

Simplify the denominator.

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Step 10.1.1

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

$\frac{50}{{(- 1)}^{3} {(\frac{5}{4})}^{3}}$

Step 10.1.2

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

$\frac{50}{- {(\frac{5}{4})}^{3}}$

Step 10.1.3

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

$\frac{50}{- \frac{5^{3}}{4^{3}}}$

Step 10.1.4

Raise $5$ to the power of $3$ .

$\frac{50}{- \frac{125}{4^{3}}}$

Step 10.1.5

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

$\frac{50}{- \frac{125}{64}}$

Step 10.2

Multiply the numerator by the reciprocal of the denominator.

$50 (- \frac{64}{125})$

Step 10.3

Cancel the common factor of $25$ .

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Step 10.3.1

Move the leading negative in $- \frac{64}{125}$ into the numerator.

$50 (\frac{- 64}{125})$

Step 10.3.2

Factor $25$ out of $50$ .

$25 (2) \frac{- 64}{125}$

Step 10.3.3

Factor $25$ out of $125$ .

$25 \cdot 2 \frac{- 64}{25 \cdot 5}$

Step 10.3.4

Cancel the common factor.

$25 \cdot 2 \frac{- 64}{25 \cdot 5}$

Step 10.3.5

Rewrite the expression.

$2 (\frac{- 64}{5})$

Step 10.4

Combine $2$ and $\frac{- 64}{5}$ .

$\frac{2 \cdot - 64}{5}$

Step 10.5

Simplify the expression.

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Step 10.5.1

Multiply $2$ by $- 64$ .

$\frac{- 128}{5}$

Step 10.5.2

Move the negative in front of the fraction.

$- \frac{128}{5}$

Step 11

$x = - \frac{5}{4}$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = - \frac{5}{4}$ is a local maximum

Step 12

Find the y-value when $x = - \frac{5}{4}$ .

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Step 12.1

Replace the variable $x$ with $- \frac{5}{4}$ in the expression.

$f (- \frac{5}{4}) = \frac{16 {(- \frac{5}{4})}^{2} + 25}{- \frac{5}{4}}$

Step 12.2

Simplify the result.

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Step 12.2.1

Multiply the numerator by the reciprocal of the denominator.

$f (- \frac{5}{4}) = (16 {(- \frac{5}{4})}^{2} + 25) (- \frac{4}{5})$

Step 12.2.2

Simplify each term.

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Step 12.2.2.1

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

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Step 12.2.2.1.1

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

$f (- \frac{5}{4}) = (16 ({(- 1)}^{2} {(\frac{5}{4})}^{2}) + 25) (- \frac{4}{5})$

Step 12.2.2.1.2

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

$f (- \frac{5}{4}) = (16 ({(- 1)}^{2} (\frac{5^{2}}{4^{2}})) + 25) (- \frac{4}{5})$

Step 12.2.2.2

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

$f (- \frac{5}{4}) = (16 (1 (\frac{5^{2}}{4^{2}})) + 25) (- \frac{4}{5})$

Step 12.2.2.3

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

$f (- \frac{5}{4}) = (16 (\frac{5^{2}}{4^{2}}) + 25) (- \frac{4}{5})$

Step 12.2.2.4

Raise $5$ to the power of $2$ .

$f (- \frac{5}{4}) = (16 (\frac{25}{4^{2}}) + 25) (- \frac{4}{5})$

Step 12.2.2.5

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

$f (- \frac{5}{4}) = (16 (\frac{25}{16}) + 25) (- \frac{4}{5})$

Step 12.2.2.6

Cancel the common factor of $16$ .

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Step 12.2.2.6.1

Cancel the common factor.

$f (- \frac{5}{4}) = (16 (\frac{25}{16}) + 25) (- \frac{4}{5})$

Step 12.2.2.6.2

Rewrite the expression.

$f (- \frac{5}{4}) = (25 + 25) (- \frac{4}{5})$

Step 12.2.3

Reduce the expression by cancelling the common factors.

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Step 12.2.3.1

Add $25$ and $25$ .

$f (- \frac{5}{4}) = 50 (- \frac{4}{5})$

Step 12.2.3.2

Cancel the common factor of $5$ .

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Step 12.2.3.2.1

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

$f (- \frac{5}{4}) = 50 (\frac{- 4}{5})$

Step 12.2.3.2.2

Factor $5$ out of $50$ .

$f (- \frac{5}{4}) = 5 (10) (\frac{- 4}{5})$

Step 12.2.3.2.3

Cancel the common factor.

$f (- \frac{5}{4}) = 5 \cdot (10 (\frac{- 4}{5}))$

Step 12.2.3.2.4

Rewrite the expression.

$f (- \frac{5}{4}) = 10 \cdot - 4$

Step 12.2.3.3

Multiply $10$ by $- 4$ .

$f (- \frac{5}{4}) = - 40$

Step 12.2.4

The final answer is $- 40$ .

$y = - 40$

Step 13

Evaluate the second derivative at $x = \frac{5}{4}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{50}{{(\frac{5}{4})}^{3}}$

Step 14

Evaluate the second derivative.

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Step 14.1

Simplify the denominator.

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Step 14.1.1

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

$\frac{50}{\frac{5^{3}}{4^{3}}}$

Step 14.1.2

Raise $5$ to the power of $3$ .

$\frac{50}{\frac{125}{4^{3}}}$

Step 14.1.3

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

$\frac{50}{\frac{125}{64}}$

Step 14.2

Multiply the numerator by the reciprocal of the denominator.

$50 (\frac{64}{125})$

Step 14.3

Cancel the common factor of $25$ .

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Step 14.3.1

Factor $25$ out of $50$ .

$25 (2) \frac{64}{125}$

Step 14.3.2

Factor $25$ out of $125$ .

$25 \cdot 2 \frac{64}{25 \cdot 5}$

Step 14.3.3

Cancel the common factor.

$25 \cdot 2 \frac{64}{25 \cdot 5}$

Step 14.3.4

Rewrite the expression.

$2 (\frac{64}{5})$

Step 14.4

Combine $2$ and $\frac{64}{5}$ .

$\frac{2 \cdot 64}{5}$

Step 14.5

Multiply $2$ by $64$ .

$\frac{128}{5}$

Step 15

$x = \frac{5}{4}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = \frac{5}{4}$ is a local minimum

Step 16

Find the y-value when $x = \frac{5}{4}$ .

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Step 16.1

Replace the variable $x$ with $\frac{5}{4}$ in the expression.

$f (\frac{5}{4}) = \frac{16 {(\frac{5}{4})}^{2} + 25}{\frac{5}{4}}$

Step 16.2

Simplify the result.

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Step 16.2.1

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{5}{4}) = (16 {(\frac{5}{4})}^{2} + 25) (\frac{4}{5})$

Step 16.2.2

Simplify each term.

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Step 16.2.2.1

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

$f (\frac{5}{4}) = (16 (\frac{5^{2}}{4^{2}}) + 25) (\frac{4}{5})$

Step 16.2.2.2

Raise $5$ to the power of $2$ .

$f (\frac{5}{4}) = (16 (\frac{25}{4^{2}}) + 25) (\frac{4}{5})$

Step 16.2.2.3

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

$f (\frac{5}{4}) = (16 (\frac{25}{16}) + 25) (\frac{4}{5})$

Step 16.2.2.4

Cancel the common factor of $16$ .

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Step 16.2.2.4.1

Cancel the common factor.

$f (\frac{5}{4}) = (16 (\frac{25}{16}) + 25) (\frac{4}{5})$

Step 16.2.2.4.2

Rewrite the expression.

$f (\frac{5}{4}) = (25 + 25) (\frac{4}{5})$

Step 16.2.3

Reduce the expression by cancelling the common factors.

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Step 16.2.3.1

Add $25$ and $25$ .

$f (\frac{5}{4}) = 50 (\frac{4}{5})$

Step 16.2.3.2

Cancel the common factor of $5$ .

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Step 16.2.3.2.1

Factor $5$ out of $50$ .

$f (\frac{5}{4}) = 5 (10) (\frac{4}{5})$

Step 16.2.3.2.2

Cancel the common factor.

$f (\frac{5}{4}) = 5 \cdot (10 (\frac{4}{5}))$

Step 16.2.3.2.3

Rewrite the expression.

$f (\frac{5}{4}) = 10 \cdot 4$

Step 16.2.3.3

Multiply $10$ by $4$ .

$f (\frac{5}{4}) = 40$

Step 16.2.4

The final answer is $40$ .

$y = 40$

Step 17

These are the local extrema for $f (x) = \frac{16 x^{2} + 25}{x}$ .

$(- \frac{5}{4}, - 40)$ is a local maxima

$(\frac{5}{4}, 40)$ is a local minima

Step 18