Calculus Examples

$f (x) = 4 \ln (x) - 17 \arctan (x)$

Step 1

Find the first derivative of the function.

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Step 1.1

By the Sum Rule, the derivative of $4 \ln (x) - 17 \arctan (x)$ with respect to $x$ is $\frac{d}{d x} [4 \ln (x)] + \frac{d}{d x} [- 17 \arctan (x)]$ .

$\frac{d}{d x} [4 \ln (x)] + \frac{d}{d x} [- 17 \arctan (x)]$

Step 1.2

Evaluate $\frac{d}{d x} [4 \ln (x)]$ .

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Step 1.2.1

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

$4 \frac{d}{d x} [\ln (x)] + \frac{d}{d x} [- 17 \arctan (x)]$

Step 1.2.2

The derivative of $\ln (x)$ with respect to $x$ is $\frac{1}{x}$ .

$4 \frac{1}{x} + \frac{d}{d x} [- 17 \arctan (x)]$

Step 1.2.3

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

$\frac{4}{x} + \frac{d}{d x} [- 17 \arctan (x)]$

Step 1.3

Evaluate $\frac{d}{d x} [- 17 \arctan (x)]$ .

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Step 1.3.1

Since $- 17$ is constant with respect to $x$ , the derivative of $- 17 \arctan (x)$ with respect to $x$ is $- 17 \frac{d}{d x} [\arctan (x)]$ .

$\frac{4}{x} - 17 \frac{d}{d x} [\arctan (x)]$

Step 1.3.2

The derivative of $\arctan (x)$ with respect to $x$ is $\frac{1}{1 + x^{2}}$ .

$\frac{4}{x} - 17 \frac{1}{1 + x^{2}}$

Step 1.3.3

Combine $- 17$ and $\frac{1}{1 + x^{2}}$ .

$\frac{4}{x} + \frac{- 17}{1 + x^{2}}$

Step 1.3.4

Move the negative in front of the fraction.

$\frac{4}{x} - \frac{17}{1 + x^{2}}$

Step 1.4

Simplify.

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Step 1.4.1

Combine terms.

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Step 1.4.1.1

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

$\frac{4}{x} \cdot \frac{1 + x^{2}}{1 + x^{2}} - \frac{17}{1 + x^{2}}$

Step 1.4.1.2

To write $- \frac{17}{1 + x^{2}}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{4}{x} \cdot \frac{1 + x^{2}}{1 + x^{2}} - \frac{17}{1 + x^{2}} \cdot \frac{x}{x}$

Step 1.4.1.3

Write each expression with a common denominator of $x (1 + x^{2})$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.4.1.3.1

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

$\frac{4 (1 + x^{2})}{x (1 + x^{2})} - \frac{17}{1 + x^{2}} \cdot \frac{x}{x}$

Step 1.4.1.3.2

Multiply $\frac{17}{1 + x^{2}}$ by $\frac{x}{x}$ .

$\frac{4 (1 + x^{2})}{x (1 + x^{2})} - \frac{17 x}{(1 + x^{2}) x}$

Step 1.4.1.3.3

Reorder the factors of $(1 + x^{2}) x$ .

$\frac{4 (1 + x^{2})}{x (1 + x^{2})} - \frac{17 x}{x (1 + x^{2})}$

Step 1.4.1.4

Combine the numerators over the common denominator.

$\frac{4 (1 + x^{2}) - 17 x}{x (1 + x^{2})}$

Step 1.4.2

Reorder terms.

$\frac{- 17 x + 4 (1 + x^{2})}{x (1 + x^{2})}$

Step 2

Find the second 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 $f (x) = - 17 x + 4 (1 + x^{2})$ and $g (x) = x (1 + x^{2})$ .

$f'' (x) = \frac{x (1 + x^{2}) \frac{d}{d x} (- 17 x + 4 (1 + x^{2})) - (- 17 x + 4 (1 + x^{2})) \frac{d}{d x} (x (1 + x^{2}))}{{(x (1 + x^{2}))}^{2}}$

Step 2.2

Differentiate.

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Step 2.2.1

By the Sum Rule, the derivative of $- 17 x + 4 (1 + x^{2})$ with respect to $x$ is $\frac{d}{d x} [- 17 x] + \frac{d}{d x} [4 (1 + x^{2})]$ .

$f'' (x) = \frac{x (1 + x^{2}) (\frac{d}{d x} (- 17 x) + \frac{d}{d x} (4 (1 + x^{2}))) - (- 17 x + 4 (1 + x^{2})) \frac{d}{d x} (x (1 + x^{2}))}{{(x (1 + x^{2}))}^{2}}$

Step 2.2.2

Since $- 17$ is constant with respect to $x$ , the derivative of $- 17 x$ with respect to $x$ is $- 17 \frac{d}{d x} [x]$ .

$f'' (x) = \frac{x (1 + x^{2}) (- 17 \frac{d}{d x} x + \frac{d}{d x} (4 (1 + x^{2}))) - (- 17 x + 4 (1 + x^{2})) \frac{d}{d x} (x (1 + x^{2}))}{{(x (1 + x^{2}))}^{2}}$

Step 2.2.3

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

$f'' (x) = \frac{x (1 + x^{2}) (- 17 \cdot 1 + \frac{d}{d x} (4 (1 + x^{2}))) - (- 17 x + 4 (1 + x^{2})) \frac{d}{d x} (x (1 + x^{2}))}{{(x (1 + x^{2}))}^{2}}$

Step 2.2.4

Multiply $- 17$ by $1$ .

$f'' (x) = \frac{x (1 + x^{2}) (- 17 + \frac{d}{d x} (4 (1 + x^{2}))) - (- 17 x + 4 (1 + x^{2})) \frac{d}{d x} (x (1 + x^{2}))}{{(x (1 + x^{2}))}^{2}}$

Step 2.2.5

Since $4$ is constant with respect to $x$ , the derivative of $4 (1 + x^{2})$ with respect to $x$ is $4 \frac{d}{d x} [1 + x^{2}]$ .

$f'' (x) = \frac{x (1 + x^{2}) (- 17 + 4 \frac{d}{d x} (1 + x^{2})) - (- 17 x + 4 (1 + x^{2})) \frac{d}{d x} (x (1 + x^{2}))}{{(x (1 + x^{2}))}^{2}}$

Step 2.2.6

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

$f'' (x) = \frac{x (1 + x^{2}) (- 17 + 4 (\frac{d}{d x} (1) + \frac{d}{d x} (x^{2}))) - (- 17 x + 4 (1 + x^{2})) \frac{d}{d x} (x (1 + x^{2}))}{{(x (1 + x^{2}))}^{2}}$

Step 2.2.7

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

$f'' (x) = \frac{x (1 + x^{2}) (- 17 + 4 (0 + \frac{d}{d x} (x^{2}))) - (- 17 x + 4 (1 + x^{2})) \frac{d}{d x} (x (1 + x^{2}))}{{(x (1 + x^{2}))}^{2}}$

Step 2.2.8

Add $0$ and $\frac{d}{d x} [x^{2}]$ .

$f'' (x) = \frac{x (1 + x^{2}) (- 17 + 4 \frac{d}{d x} (x^{2})) - (- 17 x + 4 (1 + x^{2})) \frac{d}{d x} (x (1 + x^{2}))}{{(x (1 + x^{2}))}^{2}}$

Step 2.2.9

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

$f'' (x) = \frac{x (1 + x^{2}) (- 17 + 4 (2 x)) - (- 17 x + 4 (1 + x^{2})) \frac{d}{d x} (x (1 + x^{2}))}{{(x (1 + x^{2}))}^{2}}$

Step 2.2.10

Multiply $2$ by $4$ .

$f'' (x) = \frac{x (1 + x^{2}) (- 17 + 8 x) - (- 17 x + 4 (1 + x^{2})) \frac{d}{d x} (x (1 + x^{2}))}{{(x (1 + x^{2}))}^{2}}$

Step 2.3

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 $f (x) = x$ and $g (x) = 1 + x^{2}$ .

$f'' (x) = \frac{x (1 + x^{2}) (- 17 + 8 x) - (- 17 x + 4 (1 + x^{2})) (x \frac{d}{d x} (1 + x^{2}) + (1 + x^{2}) \frac{d}{d x} (x))}{{(x (1 + x^{2}))}^{2}}$

Step 2.4

Differentiate.

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Step 2.4.1

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

$f'' (x) = \frac{x (1 + x^{2}) (- 17 + 8 x) - (- 17 x + 4 (1 + x^{2})) (x (\frac{d}{d x} (1) + \frac{d}{d x} (x^{2})) + (1 + x^{2}) \frac{d}{d x} (x))}{{(x (1 + x^{2}))}^{2}}$

Step 2.4.2

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

$f'' (x) = \frac{x (1 + x^{2}) (- 17 + 8 x) - (- 17 x + 4 (1 + x^{2})) (x (0 + \frac{d}{d x} (x^{2})) + (1 + x^{2}) \frac{d}{d x} (x))}{{(x (1 + x^{2}))}^{2}}$

Step 2.4.3

Add $0$ and $\frac{d}{d x} [x^{2}]$ .

$f'' (x) = \frac{x (1 + x^{2}) (- 17 + 8 x) - (- 17 x + 4 (1 + x^{2})) (x \frac{d}{d x} (x^{2}) + (1 + x^{2}) \frac{d}{d x} (x))}{{(x (1 + x^{2}))}^{2}}$

Step 2.4.4

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

$f'' (x) = \frac{x (1 + x^{2}) (- 17 + 8 x) - (- 17 x + 4 (1 + x^{2})) (x (2 x) + (1 + x^{2}) \frac{d}{d x} (x))}{{(x (1 + x^{2}))}^{2}}$

Step 2.5

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

$f'' (x) = \frac{x (1 + x^{2}) (- 17 + 8 x) - (- 17 x + 4 (1 + x^{2})) (2 (x \cdot x) + (1 + x^{2}) \frac{d}{d x} (x))}{{(x (1 + x^{2}))}^{2}}$

Step 2.6

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

$f'' (x) = \frac{x (1 + x^{2}) (- 17 + 8 x) - (- 17 x + 4 (1 + x^{2})) (2 (x \cdot x) + (1 + x^{2}) \frac{d}{d x} (x))}{{(x (1 + x^{2}))}^{2}}$

Step 2.7

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

$f'' (x) = \frac{x (1 + x^{2}) (- 17 + 8 x) - (- 17 x + 4 (1 + x^{2})) (2 x^{1 + 1} + (1 + x^{2}) \frac{d}{d x} (x))}{{(x (1 + x^{2}))}^{2}}$

Step 2.8

Add $1$ and $1$ .

$f'' (x) = \frac{x (1 + x^{2}) (- 17 + 8 x) - (- 17 x + 4 (1 + x^{2})) (2 x^{2} + (1 + x^{2}) \frac{d}{d x} (x))}{{(x (1 + x^{2}))}^{2}}$

Step 2.9

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

$f'' (x) = \frac{x (1 + x^{2}) (- 17 + 8 x) - (- 17 x + 4 (1 + x^{2})) (2 x^{2} + (1 + x^{2}) \cdot 1)}{{(x (1 + x^{2}))}^{2}}$

Step 2.10

Simplify by adding terms.

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Step 2.10.1

Multiply $1 + x^{2}$ by $1$ .

$f'' (x) = \frac{x (1 + x^{2}) (- 17 + 8 x) - (- 17 x + 4 (1 + x^{2})) (2 x^{2} + 1 + x^{2})}{{(x (1 + x^{2}))}^{2}}$

Step 2.10.2

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

$f'' (x) = \frac{x (1 + x^{2}) (- 17 + 8 x) - (- 17 x + 4 (1 + x^{2})) (3 x^{2} + 1)}{{(x (1 + x^{2}))}^{2}}$

Step 2.11

Simplify.

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Step 2.11.1

Apply the product rule to $x (1 + x^{2})$ .

$f'' (x) = \frac{x (1 + x^{2}) (- 17 + 8 x) - (- 17 x + 4 (1 + x^{2})) (3 x^{2} + 1)}{x^{2} {(1 + x^{2})}^{2}}$

Step 2.11.2

Apply the distributive property.

$f'' (x) = \frac{(x \cdot 1 + x \cdot x^{2}) (- 17 + 8 x) - (- 17 x + 4 (1 + x^{2})) (3 x^{2} + 1)}{x^{2} {(1 + x^{2})}^{2}}$

Step 2.11.3

Apply the distributive property.

$f'' (x) = \frac{(x \cdot 1 + x \cdot x^{2}) (- 17 + 8 x) - (- 17 x + 4 \cdot 1 + 4 x^{2}) (3 x^{2} + 1)}{x^{2} {(1 + x^{2})}^{2}}$

Step 2.11.4

Apply the distributive property.

$f'' (x) = \frac{(x \cdot 1 + x \cdot x^{2}) (- 17 + 8 x) + (- (- 17 x) - (4 \cdot 1) - (4 x^{2})) (3 x^{2} + 1)}{x^{2} {(1 + x^{2})}^{2}}$

Step 2.11.5

Simplify the numerator.

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Step 2.11.5.1

Simplify each term.

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Step 2.11.5.1.1

Simplify each term.

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Step 2.11.5.1.1.1

Multiply $x$ by $1$ .

$f'' (x) = \frac{(x + x \cdot x^{2}) (- 17 + 8 x) + (- (- 17 x) - (4 \cdot 1) - (4 x^{2})) (3 x^{2} + 1)}{x^{2} {(1 + x^{2})}^{2}}$

Step 2.11.5.1.1.2

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

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Step 2.11.5.1.1.2.1

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

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Step 2.11.5.1.1.2.1.1

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

$f'' (x) = \frac{(x + x \cdot x^{2}) (- 17 + 8 x) + (- (- 17 x) - (4 \cdot 1) - (4 x^{2})) (3 x^{2} + 1)}{x^{2} {(1 + x^{2})}^{2}}$

Step 2.11.5.1.1.2.1.2

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

$f'' (x) = \frac{(x + x^{1 + 2}) (- 17 + 8 x) + (- (- 17 x) - (4 \cdot 1) - (4 x^{2})) (3 x^{2} + 1)}{x^{2} {(1 + x^{2})}^{2}}$

Step 2.11.5.1.1.2.2

Add $1$ and $2$ .

$f'' (x) = \frac{(x + x^{3}) (- 17 + 8 x) + (- (- 17 x) - (4 \cdot 1) - (4 x^{2})) (3 x^{2} + 1)}{x^{2} {(1 + x^{2})}^{2}}$

Step 2.11.5.1.2

Expand $(x + x^{3}) (- 17 + 8 x)$ using the FOIL Method.

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Step 2.11.5.1.2.1

Apply the distributive property.

$f'' (x) = \frac{x (- 17 + 8 x) + x^{3} (- 17 + 8 x) + (- (- 17 x) - (4 \cdot 1) - (4 x^{2})) (3 x^{2} + 1)}{x^{2} {(1 + x^{2})}^{2}}$

Step 2.11.5.1.2.2

Apply the distributive property.

$f'' (x) = \frac{x \cdot - 17 + x (8 x) + x^{3} (- 17 + 8 x) + (- (- 17 x) - (4 \cdot 1) - (4 x^{2})) (3 x^{2} + 1)}{x^{2} {(1 + x^{2})}^{2}}$

Step 2.11.5.1.2.3

Apply the distributive property.

$f'' (x) = \frac{x \cdot - 17 + x (8 x) + x^{3} \cdot - 17 + x^{3} (8 x) + (- (- 17 x) - (4 \cdot 1) - (4 x^{2})) (3 x^{2} + 1)}{x^{2} {(1 + x^{2})}^{2}}$

Step 2.11.5.1.3

Simplify each term.

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Step 2.11.5.1.3.1

Move $- 17$ to the left of $x$ .

$f'' (x) = \frac{- 17 \cdot x + x (8 x) + x^{3} \cdot - 17 + x^{3} (8 x) + (- (- 17 x) - (4 \cdot 1) - (4 x^{2})) (3 x^{2} + 1)}{x^{2} {(1 + x^{2})}^{2}}$

Step 2.11.5.1.3.2

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{- 17 x + 8 x \cdot x + x^{3} \cdot - 17 + x^{3} (8 x) + (- (- 17 x) - (4 \cdot 1) - (4 x^{2})) (3 x^{2} + 1)}{x^{2} {(1 + x^{2})}^{2}}$

Step 2.11.5.1.3.3

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

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Step 2.11.5.1.3.3.1

Move $x$ .

$f'' (x) = \frac{- 17 x + 8 (x \cdot x) + x^{3} \cdot - 17 + x^{3} (8 x) + (- (- 17 x) - (4 \cdot 1) - (4 x^{2})) (3 x^{2} + 1)}{x^{2} {(1 + x^{2})}^{2}}$

Step 2.11.5.1.3.3.2

Multiply $x$ by $x$ .

$f'' (x) = \frac{- 17 x + 8 x^{2} + x^{3} \cdot - 17 + x^{3} (8 x) + (- (- 17 x) - (4 \cdot 1) - (4 x^{2})) (3 x^{2} + 1)}{x^{2} {(1 + x^{2})}^{2}}$

Step 2.11.5.1.3.4

Move $- 17$ to the left of $x^{3}$ .

$f'' (x) = \frac{- 17 x + 8 x^{2} - 17 \cdot x^{3} + x^{3} (8 x) + (- (- 17 x) - (4 \cdot 1) - (4 x^{2})) (3 x^{2} + 1)}{x^{2} {(1 + x^{2})}^{2}}$

Step 2.11.5.1.3.5

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{- 17 x + 8 x^{2} - 17 x^{3} + 8 x^{3} x + (- (- 17 x) - (4 \cdot 1) - (4 x^{2})) (3 x^{2} + 1)}{x^{2} {(1 + x^{2})}^{2}}$

Step 2.11.5.1.3.6

Multiply $x^{3}$ by $x$ by adding the exponents.

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Step 2.11.5.1.3.6.1

Move $x$ .

$f'' (x) = \frac{- 17 x + 8 x^{2} - 17 x^{3} + 8 (x \cdot x^{3}) + (- (- 17 x) - (4 \cdot 1) - (4 x^{2})) (3 x^{2} + 1)}{x^{2} {(1 + x^{2})}^{2}}$

Step 2.11.5.1.3.6.2

Multiply $x$ by $x^{3}$ .

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Step 2.11.5.1.3.6.2.1

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

$f'' (x) = \frac{- 17 x + 8 x^{2} - 17 x^{3} + 8 (x \cdot x^{3}) + (- (- 17 x) - (4 \cdot 1) - (4 x^{2})) (3 x^{2} + 1)}{x^{2} {(1 + x^{2})}^{2}}$

Step 2.11.5.1.3.6.2.2

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

$f'' (x) = \frac{- 17 x + 8 x^{2} - 17 x^{3} + 8 x^{1 + 3} + (- (- 17 x) - (4 \cdot 1) - (4 x^{2})) (3 x^{2} + 1)}{x^{2} {(1 + x^{2})}^{2}}$

Step 2.11.5.1.3.6.3

Add $1$ and $3$ .

$f'' (x) = \frac{- 17 x + 8 x^{2} - 17 x^{3} + 8 x^{4} + (- (- 17 x) - (4 \cdot 1) - (4 x^{2})) (3 x^{2} + 1)}{x^{2} {(1 + x^{2})}^{2}}$

Step 2.11.5.1.4

Simplify each term.

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Step 2.11.5.1.4.1

Multiply $- 17$ by $- 1$ .

$f'' (x) = \frac{- 17 x + 8 x^{2} - 17 x^{3} + 8 x^{4} + (17 x - (4 \cdot 1) - (4 x^{2})) (3 x^{2} + 1)}{x^{2} {(1 + x^{2})}^{2}}$

Step 2.11.5.1.4.2

Multiply $- (4 \cdot 1)$ .

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Step 2.11.5.1.4.2.1

Multiply $4$ by $1$ .

$f'' (x) = \frac{- 17 x + 8 x^{2} - 17 x^{3} + 8 x^{4} + (17 x - 1 \cdot 4 - (4 x^{2})) (3 x^{2} + 1)}{x^{2} {(1 + x^{2})}^{2}}$

Step 2.11.5.1.4.2.2

Multiply $- 1$ by $4$ .

$f'' (x) = \frac{- 17 x + 8 x^{2} - 17 x^{3} + 8 x^{4} + (17 x - 4 - (4 x^{2})) (3 x^{2} + 1)}{x^{2} {(1 + x^{2})}^{2}}$

Step 2.11.5.1.4.3

Multiply $4$ by $- 1$ .

$f'' (x) = \frac{- 17 x + 8 x^{2} - 17 x^{3} + 8 x^{4} + (17 x - 4 - 4 x^{2}) (3 x^{2} + 1)}{x^{2} {(1 + x^{2})}^{2}}$

Step 2.11.5.1.5

Expand $(17 x - 4 - 4 x^{2}) (3 x^{2} + 1)$ by multiplying each term in the first expression by each term in the second expression.

$f'' (x) = \frac{- 17 x + 8 x^{2} - 17 x^{3} + 8 x^{4} + 17 x (3 x^{2}) + 17 x \cdot 1 - 4 (3 x^{2}) - 4 \cdot 1 - 4 x^{2} (3 x^{2}) - 4 x^{2} \cdot 1}{x^{2} {(1 + x^{2})}^{2}}$

Step 2.11.5.1.6

Simplify each term.

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Step 2.11.5.1.6.1

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{- 17 x + 8 x^{2} - 17 x^{3} + 8 x^{4} + 17 \cdot (3 x \cdot x^{2}) + 17 x \cdot 1 - 4 (3 x^{2}) - 4 \cdot 1 - 4 x^{2} (3 x^{2}) - 4 x^{2} \cdot 1}{x^{2} {(1 + x^{2})}^{2}}$

Step 2.11.5.1.6.2

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

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Step 2.11.5.1.6.2.1

Move $x^{2}$ .

$f'' (x) = \frac{- 17 x + 8 x^{2} - 17 x^{3} + 8 x^{4} + 17 \cdot (3 (x^{2} x)) + 17 x \cdot 1 - 4 (3 x^{2}) - 4 \cdot 1 - 4 x^{2} (3 x^{2}) - 4 x^{2} \cdot 1}{x^{2} {(1 + x^{2})}^{2}}$

Step 2.11.5.1.6.2.2

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

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Step 2.11.5.1.6.2.2.1

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

$f'' (x) = \frac{- 17 x + 8 x^{2} - 17 x^{3} + 8 x^{4} + 17 \cdot (3 (x^{2} x)) + 17 x \cdot 1 - 4 (3 x^{2}) - 4 \cdot 1 - 4 x^{2} (3 x^{2}) - 4 x^{2} \cdot 1}{x^{2} {(1 + x^{2})}^{2}}$

Step 2.11.5.1.6.2.2.2

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

$f'' (x) = \frac{- 17 x + 8 x^{2} - 17 x^{3} + 8 x^{4} + 17 \cdot (3 x^{2 + 1}) + 17 x \cdot 1 - 4 (3 x^{2}) - 4 \cdot 1 - 4 x^{2} (3 x^{2}) - 4 x^{2} \cdot 1}{x^{2} {(1 + x^{2})}^{2}}$

Step 2.11.5.1.6.2.3

Add $2$ and $1$ .

$f'' (x) = \frac{- 17 x + 8 x^{2} - 17 x^{3} + 8 x^{4} + 17 \cdot (3 x^{3}) + 17 x \cdot 1 - 4 (3 x^{2}) - 4 \cdot 1 - 4 x^{2} (3 x^{2}) - 4 x^{2} \cdot 1}{x^{2} {(1 + x^{2})}^{2}}$

Step 2.11.5.1.6.3

Multiply $17$ by $3$ .

$f'' (x) = \frac{- 17 x + 8 x^{2} - 17 x^{3} + 8 x^{4} + 51 x^{3} + 17 x \cdot 1 - 4 (3 x^{2}) - 4 \cdot 1 - 4 x^{2} (3 x^{2}) - 4 x^{2} \cdot 1}{x^{2} {(1 + x^{2})}^{2}}$

Step 2.11.5.1.6.4

Multiply $17$ by $1$ .

$f'' (x) = \frac{- 17 x + 8 x^{2} - 17 x^{3} + 8 x^{4} + 51 x^{3} + 17 x - 4 (3 x^{2}) - 4 \cdot 1 - 4 x^{2} (3 x^{2}) - 4 x^{2} \cdot 1}{x^{2} {(1 + x^{2})}^{2}}$

Step 2.11.5.1.6.5

Multiply $3$ by $- 4$ .

$f'' (x) = \frac{- 17 x + 8 x^{2} - 17 x^{3} + 8 x^{4} + 51 x^{3} + 17 x - 12 x^{2} - 4 \cdot 1 - 4 x^{2} (3 x^{2}) - 4 x^{2} \cdot 1}{x^{2} {(1 + x^{2})}^{2}}$

Step 2.11.5.1.6.6

Multiply $- 4$ by $1$ .

$f'' (x) = \frac{- 17 x + 8 x^{2} - 17 x^{3} + 8 x^{4} + 51 x^{3} + 17 x - 12 x^{2} - 4 - 4 x^{2} (3 x^{2}) - 4 x^{2} \cdot 1}{x^{2} {(1 + x^{2})}^{2}}$

Step 2.11.5.1.6.7

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{- 17 x + 8 x^{2} - 17 x^{3} + 8 x^{4} + 51 x^{3} + 17 x - 12 x^{2} - 4 - 4 \cdot (3 x^{2} x^{2}) - 4 x^{2} \cdot 1}{x^{2} {(1 + x^{2})}^{2}}$

Step 2.11.5.1.6.8

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

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Step 2.11.5.1.6.8.1

Move $x^{2}$ .

$f'' (x) = \frac{- 17 x + 8 x^{2} - 17 x^{3} + 8 x^{4} + 51 x^{3} + 17 x - 12 x^{2} - 4 - 4 \cdot (3 (x^{2} x^{2})) - 4 x^{2} \cdot 1}{x^{2} {(1 + x^{2})}^{2}}$

Step 2.11.5.1.6.8.2

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

$f'' (x) = \frac{- 17 x + 8 x^{2} - 17 x^{3} + 8 x^{4} + 51 x^{3} + 17 x - 12 x^{2} - 4 - 4 \cdot (3 x^{2 + 2}) - 4 x^{2} \cdot 1}{x^{2} {(1 + x^{2})}^{2}}$

Step 2.11.5.1.6.8.3

Add $2$ and $2$ .

$f'' (x) = \frac{- 17 x + 8 x^{2} - 17 x^{3} + 8 x^{4} + 51 x^{3} + 17 x - 12 x^{2} - 4 - 4 \cdot (3 x^{4}) - 4 x^{2} \cdot 1}{x^{2} {(1 + x^{2})}^{2}}$

Step 2.11.5.1.6.9

Multiply $- 4$ by $3$ .

$f'' (x) = \frac{- 17 x + 8 x^{2} - 17 x^{3} + 8 x^{4} + 51 x^{3} + 17 x - 12 x^{2} - 4 - 12 x^{4} - 4 x^{2} \cdot 1}{x^{2} {(1 + x^{2})}^{2}}$

Step 2.11.5.1.6.10

Multiply $- 4$ by $1$ .

$f'' (x) = \frac{- 17 x + 8 x^{2} - 17 x^{3} + 8 x^{4} + 51 x^{3} + 17 x - 12 x^{2} - 4 - 12 x^{4} - 4 x^{2}}{x^{2} {(1 + x^{2})}^{2}}$

Step 2.11.5.1.7

Subtract $4 x^{2}$ from $- 12 x^{2}$ .

$f'' (x) = \frac{- 17 x + 8 x^{2} - 17 x^{3} + 8 x^{4} + 51 x^{3} + 17 x - 16 x^{2} - 4 - 12 x^{4}}{x^{2} {(1 + x^{2})}^{2}}$

Step 2.11.5.2

Combine the opposite terms in $- 17 x + 8 x^{2} - 17 x^{3} + 8 x^{4} + 51 x^{3} + 17 x - 16 x^{2} - 4 - 12 x^{4}$ .

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Step 2.11.5.2.1

Add $- 17 x$ and $17 x$ .

$f'' (x) = \frac{8 x^{2} - 17 x^{3} + 8 x^{4} + 51 x^{3} + 0 - 16 x^{2} - 4 - 12 x^{4}}{x^{2} {(1 + x^{2})}^{2}}$

Step 2.11.5.2.2

Add $8 x^{2} - 17 x^{3} + 8 x^{4} + 51 x^{3}$ and $0$ .

$f'' (x) = \frac{8 x^{2} - 17 x^{3} + 8 x^{4} + 51 x^{3} - 16 x^{2} - 4 - 12 x^{4}}{x^{2} {(1 + x^{2})}^{2}}$

Step 2.11.5.3

Subtract $16 x^{2}$ from $8 x^{2}$ .

$f'' (x) = \frac{- 17 x^{3} + 8 x^{4} + 51 x^{3} - 8 x^{2} - 4 - 12 x^{4}}{x^{2} {(1 + x^{2})}^{2}}$

Step 2.11.5.4

Add $- 17 x^{3}$ and $51 x^{3}$ .

$f'' (x) = \frac{8 x^{4} + 34 x^{3} - 8 x^{2} - 4 - 12 x^{4}}{x^{2} {(1 + x^{2})}^{2}}$

Step 2.11.5.5

Subtract $12 x^{4}$ from $8 x^{4}$ .

$f'' (x) = \frac{- 4 x^{4} + 34 x^{3} - 8 x^{2} - 4}{x^{2} {(1 + x^{2})}^{2}}$

Step 2.11.6

Factor $2$ out of $- 4 x^{4} + 34 x^{3} - 8 x^{2} - 4$ .

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Step 2.11.6.1

Factor $2$ out of $- 4 x^{4}$ .

$f'' (x) = \frac{2 (- 2 x^{4}) + 34 x^{3} - 8 x^{2} - 4}{x^{2} {(1 + x^{2})}^{2}}$

Step 2.11.6.2

Factor $2$ out of $34 x^{3}$ .

$f'' (x) = \frac{2 (- 2 x^{4}) + 2 (17 x^{3}) - 8 x^{2} - 4}{x^{2} {(1 + x^{2})}^{2}}$

Step 2.11.6.3

Factor $2$ out of $- 8 x^{2}$ .

$f'' (x) = \frac{2 (- 2 x^{4}) + 2 (17 x^{3}) + 2 (- 4 x^{2}) - 4}{x^{2} {(1 + x^{2})}^{2}}$

Step 2.11.6.4

Factor $2$ out of $- 4$ .

$f'' (x) = \frac{2 (- 2 x^{4}) + 2 (17 x^{3}) + 2 (- 4 x^{2}) + 2 (- 2)}{x^{2} {(1 + x^{2})}^{2}}$

Step 2.11.6.5

Factor $2$ out of $2 (- 2 x^{4}) + 2 (17 x^{3})$ .

$f'' (x) = \frac{2 (- 2 x^{4} + 17 x^{3}) + 2 (- 4 x^{2}) + 2 (- 2)}{x^{2} {(1 + x^{2})}^{2}}$

Step 2.11.6.6

Factor $2$ out of $2 (- 2 x^{4} + 17 x^{3}) + 2 (- 4 x^{2})$ .

$f'' (x) = \frac{2 (- 2 x^{4} + 17 x^{3} - 4 x^{2}) + 2 (- 2)}{x^{2} {(1 + x^{2})}^{2}}$

Step 2.11.6.7

Factor $2$ out of $2 (- 2 x^{4} + 17 x^{3} - 4 x^{2}) + 2 (- 2)$ .

$f'' (x) = \frac{2 (- 2 x^{4} + 17 x^{3} - 4 x^{2} - 2)}{x^{2} {(1 + x^{2})}^{2}}$

Step 2.11.7

Factor $- 1$ out of $- 2 x^{4}$ .

$f'' (x) = \frac{2 (- (2 x^{4}) + 17 x^{3} - 4 x^{2} - 2)}{x^{2} {(1 + x^{2})}^{2}}$

Step 2.11.8

Factor $- 1$ out of $17 x^{3}$ .

$f'' (x) = \frac{2 (- (2 x^{4}) - (- 17 x^{3}) - 4 x^{2} - 2)}{x^{2} {(1 + x^{2})}^{2}}$

Step 2.11.9

Factor $- 1$ out of $- (2 x^{4}) - (- 17 x^{3})$ .

$f'' (x) = \frac{2 (- (2 x^{4} - 17 x^{3}) - 4 x^{2} - 2)}{x^{2} {(1 + x^{2})}^{2}}$

Step 2.11.10

Factor $- 1$ out of $- 4 x^{2}$ .

$f'' (x) = \frac{2 (- (2 x^{4} - 17 x^{3}) - (4 x^{2}) - 2)}{x^{2} {(1 + x^{2})}^{2}}$

Step 2.11.11

Factor $- 1$ out of $- (2 x^{4} - 17 x^{3}) - (4 x^{2})$ .

$f'' (x) = \frac{2 (- (2 x^{4} - 17 x^{3} + 4 x^{2}) - 2)}{x^{2} {(1 + x^{2})}^{2}}$

Step 2.11.12

Rewrite $- 2$ as $- 1 (2)$ .

$f'' (x) = \frac{2 (- (2 x^{4} - 17 x^{3} + 4 x^{2}) - 1 \cdot 2)}{x^{2} {(1 + x^{2})}^{2}}$

Step 2.11.13

Factor $- 1$ out of $- (2 x^{4} - 17 x^{3} + 4 x^{2}) - 1 (2)$ .

$f'' (x) = \frac{2 (- (2 x^{4} - 17 x^{3} + 4 x^{2} + 2))}{x^{2} {(1 + x^{2})}^{2}}$

Step 2.11.14

Rewrite $- (2 x^{4} - 17 x^{3} + 4 x^{2} + 2)$ as $- 1 (2 x^{4} - 17 x^{3} + 4 x^{2} + 2)$ .

$f'' (x) = \frac{2 (- 1 (2 x^{4} - 17 x^{3} + 4 x^{2} + 2))}{x^{2} {(1 + x^{2})}^{2}}$

Step 2.11.15

Move the negative in front of the fraction.

$f'' (x) = - \frac{2 (2 x^{4} - 17 x^{3} + 4 x^{2} + 2)}{x^{2} {(1 + x^{2})}^{2}}$

Step 3

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

$\frac{- 17 x + 4 (1 + x^{2})}{x (1 + x^{2})} = 0$

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 $4 \ln (x) - 17 \arctan (x)$ with respect to $x$ is $\frac{d}{d x} [4 \ln (x)] + \frac{d}{d x} [- 17 \arctan (x)]$ .

$\frac{d}{d x} [4 \ln (x)] + \frac{d}{d x} [- 17 \arctan (x)]$

Step 4.1.2

Evaluate $\frac{d}{d x} [4 \ln (x)]$ .

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Step 4.1.2.1

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

$4 \frac{d}{d x} [\ln (x)] + \frac{d}{d x} [- 17 \arctan (x)]$

Step 4.1.2.2

The derivative of $\ln (x)$ with respect to $x$ is $\frac{1}{x}$ .

$4 \frac{1}{x} + \frac{d}{d x} [- 17 \arctan (x)]$

Step 4.1.2.3

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

$\frac{4}{x} + \frac{d}{d x} [- 17 \arctan (x)]$

Step 4.1.3

Evaluate $\frac{d}{d x} [- 17 \arctan (x)]$ .

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Step 4.1.3.1

Since $- 17$ is constant with respect to $x$ , the derivative of $- 17 \arctan (x)$ with respect to $x$ is $- 17 \frac{d}{d x} [\arctan (x)]$ .

$\frac{4}{x} - 17 \frac{d}{d x} [\arctan (x)]$

Step 4.1.3.2

The derivative of $\arctan (x)$ with respect to $x$ is $\frac{1}{1 + x^{2}}$ .

$\frac{4}{x} - 17 \frac{1}{1 + x^{2}}$

Step 4.1.3.3

Combine $- 17$ and $\frac{1}{1 + x^{2}}$ .

$\frac{4}{x} + \frac{- 17}{1 + x^{2}}$

Step 4.1.3.4

Move the negative in front of the fraction.

$\frac{4}{x} - \frac{17}{1 + x^{2}}$

Step 4.1.4

Simplify.

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Step 4.1.4.1

Combine terms.

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Step 4.1.4.1.1

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

$\frac{4}{x} \cdot \frac{1 + x^{2}}{1 + x^{2}} - \frac{17}{1 + x^{2}}$

Step 4.1.4.1.2

To write $- \frac{17}{1 + x^{2}}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{4}{x} \cdot \frac{1 + x^{2}}{1 + x^{2}} - \frac{17}{1 + x^{2}} \cdot \frac{x}{x}$

Step 4.1.4.1.3

Write each expression with a common denominator of $x (1 + x^{2})$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1.4.1.3.1

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

$\frac{4 (1 + x^{2})}{x (1 + x^{2})} - \frac{17}{1 + x^{2}} \cdot \frac{x}{x}$

Step 4.1.4.1.3.2

Multiply $\frac{17}{1 + x^{2}}$ by $\frac{x}{x}$ .

$\frac{4 (1 + x^{2})}{x (1 + x^{2})} - \frac{17 x}{(1 + x^{2}) x}$

Step 4.1.4.1.3.3

Reorder the factors of $(1 + x^{2}) x$ .

$\frac{4 (1 + x^{2})}{x (1 + x^{2})} - \frac{17 x}{x (1 + x^{2})}$

Step 4.1.4.1.4

Combine the numerators over the common denominator.

$\frac{4 (1 + x^{2}) - 17 x}{x (1 + x^{2})}$

Step 4.1.4.2

Reorder terms.

$f' (x) = \frac{- 17 x + 4 (1 + x^{2})}{x (1 + x^{2})}$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{- 17 x + 4 (1 + x^{2})}{x (1 + x^{2})}$ .

$\frac{- 17 x + 4 (1 + x^{2})}{x (1 + x^{2})}$

Step 5

Set the first derivative equal to $0$ then solve the equation $\frac{- 17 x + 4 (1 + x^{2})}{x (1 + x^{2})} = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$\frac{- 17 x + 4 (1 + x^{2})}{x (1 + x^{2})} = 0$

Step 5.2

Set the numerator equal to zero.

$- 17 x + 4 (1 + x^{2}) = 0$

Step 5.3

Solve the equation for $x$ .

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Step 5.3.1

Simplify each term.

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Step 5.3.1.1

Apply the distributive property.

$- 17 x + 4 \cdot 1 + 4 x^{2} = 0$

Step 5.3.1.2

Multiply $4$ by $1$ .

$- 17 x + 4 + 4 x^{2} = 0$

Step 5.3.2

Factor by grouping.

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Step 5.3.2.1

Reorder terms.

$4 x^{2} - 17 x + 4 = 0$

Step 5.3.2.2

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 4 \cdot 4 = 16$ and whose sum is $b = - 17$ .

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Step 5.3.2.2.1

Factor $- 17$ out of $- 17 x$ .

$4 x^{2} - 17 x + 4 = 0$

Step 5.3.2.2.2

Rewrite $- 17$ as $- 1$ plus $- 16$

$4 x^{2} + (- 1 - 16) x + 4 = 0$

Step 5.3.2.2.3

Apply the distributive property.

$4 x^{2} - 1 x - 16 x + 4 = 0$

Step 5.3.2.3

Factor out the greatest common factor from each group.

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Step 5.3.2.3.1

Group the first two terms and the last two terms.

$(4 x^{2} - 1 x) - 16 x + 4 = 0$

Step 5.3.2.3.2

Factor out the greatest common factor (GCF) from each group.

$x (4 x - 1) - 4 (4 x - 1) = 0$

Step 5.3.2.4

Factor the polynomial by factoring out the greatest common factor, $4 x - 1$ .

$(4 x - 1) (x - 4) = 0$

Step 5.3.3

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

$4 x - 1 = 0$

$x - 4 = 0$

Step 5.3.4

Set $4 x - 1$ equal to $0$ and solve for $x$ .

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Step 5.3.4.1

Set $4 x - 1$ equal to $0$ .

$4 x - 1 = 0$

Step 5.3.4.2

Solve $4 x - 1 = 0$ for $x$ .

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Step 5.3.4.2.1

Add $1$ to both sides of the equation.

$4 x = 1$

Step 5.3.4.2.2

Divide each term in $4 x = 1$ by $4$ and simplify.

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Step 5.3.4.2.2.1

Divide each term in $4 x = 1$ by $4$ .

$\frac{4 x}{4} = \frac{1}{4}$

Step 5.3.4.2.2.2

Simplify the left side.

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Step 5.3.4.2.2.2.1

Cancel the common factor of $4$ .

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Step 5.3.4.2.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{1}{4}$

Step 5.3.4.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{4}$

Step 5.3.5

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

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Step 5.3.5.1

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

$x - 4 = 0$

Step 5.3.5.2

Add $4$ to both sides of the equation.

$x = 4$

Step 5.3.6

The final solution is all the values that make $(4 x - 1) (x - 4) = 0$ true.

$x = \frac{1}{4}, 4$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

Set the denominator in $\frac{- 17 x + 4 (1 + x^{2})}{x (1 + x^{2})}$ equal to $0$ to find where the expression is undefined.

$x (1 + x^{2}) = 0$

Step 6.2

Solve for $x$ .

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Step 6.2.1

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

$x = 0$

$1 + x^{2} = 0$

Step 6.2.2

Set $x$ equal to $0$ .

$x = 0$

Step 6.2.3

Set $1 + x^{2}$ equal to $0$ and solve for $x$ .

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Step 6.2.3.1

Set $1 + x^{2}$ equal to $0$ .

$1 + x^{2} = 0$

Step 6.2.3.2

Solve $1 + x^{2} = 0$ for $x$ .

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Step 6.2.3.2.1

Subtract $1$ from both sides of the equation.

$x^{2} = - 1$

Step 6.2.3.2.2

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

$x = \pm \sqrt{- 1}$

Step 6.2.3.2.3

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \pm i$

Step 6.2.3.2.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 6.2.3.2.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x = i$

Step 6.2.3.2.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - i$

Step 6.2.3.2.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = i, - i$

Step 6.2.4

The final solution is all the values that make $x (1 + x^{2}) = 0$ true.

$x = 0, i, - i$

Step 6.3

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 $0$ .

$x = 0$

Step 7

Critical points to evaluate.

$x = \frac{1}{4}, 4$

Step 8

Evaluate the second derivative at $x = \frac{1}{4}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- \frac{2 (2 {(\frac{1}{4})}^{4} - 17 {(\frac{1}{4})}^{3} + 4 {(\frac{1}{4})}^{2} + 2)}{{(\frac{1}{4})}^{2} {(1 + {(\frac{1}{4})}^{2})}^{2}}$

Step 9

Evaluate the second derivative.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Apply the product rule to $\frac{1}{4}$ .

$- \frac{2 (2 \frac{1^{4}}{4^{4}} - 17 {(\frac{1}{4})}^{3} + 4 {(\frac{1}{4})}^{2} + 2)}{{(\frac{1}{4})}^{2} {(1 + {(\frac{1}{4})}^{2})}^{2}}$

Step 9.1.2

One to any power is one.

$- \frac{2 (2 \frac{1}{4^{4}} - 17 {(\frac{1}{4})}^{3} + 4 {(\frac{1}{4})}^{2} + 2)}{{(\frac{1}{4})}^{2} {(1 + {(\frac{1}{4})}^{2})}^{2}}$

Step 9.1.3

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

$- \frac{2 (2 (\frac{1}{256}) - 17 {(\frac{1}{4})}^{3} + 4 {(\frac{1}{4})}^{2} + 2)}{{(\frac{1}{4})}^{2} {(1 + {(\frac{1}{4})}^{2})}^{2}}$

Step 9.1.4

Cancel the common factor of $2$ .

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Step 9.1.4.1

Factor $2$ out of $256$ .

$- \frac{2 (2 \frac{1}{2 (128)} - 17 {(\frac{1}{4})}^{3} + 4 {(\frac{1}{4})}^{2} + 2)}{{(\frac{1}{4})}^{2} {(1 + {(\frac{1}{4})}^{2})}^{2}}$

Step 9.1.4.2

Cancel the common factor.

$- \frac{2 (2 \frac{1}{2 \cdot 128} - 17 {(\frac{1}{4})}^{3} + 4 {(\frac{1}{4})}^{2} + 2)}{{(\frac{1}{4})}^{2} {(1 + {(\frac{1}{4})}^{2})}^{2}}$

Step 9.1.4.3

Rewrite the expression.

$- \frac{2 (\frac{1}{128} - 17 {(\frac{1}{4})}^{3} + 4 {(\frac{1}{4})}^{2} + 2)}{{(\frac{1}{4})}^{2} {(1 + {(\frac{1}{4})}^{2})}^{2}}$

Step 9.1.5

Apply the product rule to $\frac{1}{4}$ .

$- \frac{2 (\frac{1}{128} - 17 \frac{1^{3}}{4^{3}} + 4 {(\frac{1}{4})}^{2} + 2)}{{(\frac{1}{4})}^{2} {(1 + {(\frac{1}{4})}^{2})}^{2}}$

Step 9.1.6

One to any power is one.

$- \frac{2 (\frac{1}{128} - 17 \frac{1}{4^{3}} + 4 {(\frac{1}{4})}^{2} + 2)}{{(\frac{1}{4})}^{2} {(1 + {(\frac{1}{4})}^{2})}^{2}}$

Step 9.1.7

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

$- \frac{2 (\frac{1}{128} - 17 (\frac{1}{64}) + 4 {(\frac{1}{4})}^{2} + 2)}{{(\frac{1}{4})}^{2} {(1 + {(\frac{1}{4})}^{2})}^{2}}$

Step 9.1.8

Combine $- 17$ and $\frac{1}{64}$ .

$- \frac{2 (\frac{1}{128} + \frac{- 17}{64} + 4 {(\frac{1}{4})}^{2} + 2)}{{(\frac{1}{4})}^{2} {(1 + {(\frac{1}{4})}^{2})}^{2}}$

Step 9.1.9

Move the negative in front of the fraction.

$- \frac{2 (\frac{1}{128} - \frac{17}{64} + 4 {(\frac{1}{4})}^{2} + 2)}{{(\frac{1}{4})}^{2} {(1 + {(\frac{1}{4})}^{2})}^{2}}$

Step 9.1.10

Apply the product rule to $\frac{1}{4}$ .

$- \frac{2 (\frac{1}{128} - \frac{17}{64} + 4 \frac{1^{2}}{4^{2}} + 2)}{{(\frac{1}{4})}^{2} {(1 + {(\frac{1}{4})}^{2})}^{2}}$

Step 9.1.11

One to any power is one.

$- \frac{2 (\frac{1}{128} - \frac{17}{64} + 4 \frac{1}{4^{2}} + 2)}{{(\frac{1}{4})}^{2} {(1 + {(\frac{1}{4})}^{2})}^{2}}$

Step 9.1.12

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

$- \frac{2 (\frac{1}{128} - \frac{17}{64} + 4 (\frac{1}{16}) + 2)}{{(\frac{1}{4})}^{2} {(1 + {(\frac{1}{4})}^{2})}^{2}}$

Step 9.1.13

Cancel the common factor of $4$ .

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Step 9.1.13.1

Factor $4$ out of $16$ .

$- \frac{2 (\frac{1}{128} - \frac{17}{64} + 4 \frac{1}{4 (4)} + 2)}{{(\frac{1}{4})}^{2} {(1 + {(\frac{1}{4})}^{2})}^{2}}$

Step 9.1.13.2

Cancel the common factor.

$- \frac{2 (\frac{1}{128} - \frac{17}{64} + 4 \frac{1}{4 \cdot 4} + 2)}{{(\frac{1}{4})}^{2} {(1 + {(\frac{1}{4})}^{2})}^{2}}$

Step 9.1.13.3

Rewrite the expression.

$- \frac{2 (\frac{1}{128} - \frac{17}{64} + \frac{1}{4} + 2)}{{(\frac{1}{4})}^{2} {(1 + {(\frac{1}{4})}^{2})}^{2}}$

Step 9.1.14

To write $- \frac{17}{64}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{2 (\frac{1}{128} - \frac{17}{64} \cdot \frac{2}{2} + \frac{1}{4} + 2)}{{(\frac{1}{4})}^{2} {(1 + {(\frac{1}{4})}^{2})}^{2}}$

Step 9.1.15

Write each expression with a common denominator of $128$ , by multiplying each by an appropriate factor of $1$ .

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Step 9.1.15.1

Multiply $\frac{17}{64}$ by $\frac{2}{2}$ .

$- \frac{2 (\frac{1}{128} - \frac{17 \cdot 2}{64 \cdot 2} + \frac{1}{4} + 2)}{{(\frac{1}{4})}^{2} {(1 + {(\frac{1}{4})}^{2})}^{2}}$

Step 9.1.15.2

Multiply $64$ by $2$ .

$- \frac{2 (\frac{1}{128} - \frac{17 \cdot 2}{128} + \frac{1}{4} + 2)}{{(\frac{1}{4})}^{2} {(1 + {(\frac{1}{4})}^{2})}^{2}}$

Step 9.1.16

Combine the numerators over the common denominator.

$- \frac{2 (\frac{1 - 17 \cdot 2}{128} + \frac{1}{4} + 2)}{{(\frac{1}{4})}^{2} {(1 + {(\frac{1}{4})}^{2})}^{2}}$

Step 9.1.17

Simplify the numerator.

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Step 9.1.17.1

Multiply $- 17$ by $2$ .

$- \frac{2 (\frac{1 - 34}{128} + \frac{1}{4} + 2)}{{(\frac{1}{4})}^{2} {(1 + {(\frac{1}{4})}^{2})}^{2}}$

Step 9.1.17.2

Subtract $34$ from $1$ .

$- \frac{2 (\frac{- 33}{128} + \frac{1}{4} + 2)}{{(\frac{1}{4})}^{2} {(1 + {(\frac{1}{4})}^{2})}^{2}}$

Step 9.1.18

To write $\frac{1}{4}$ as a fraction with a common denominator, multiply by $\frac{32}{32}$ .

$- \frac{2 (\frac{- 33}{128} + \frac{1}{4} \cdot \frac{32}{32} + 2)}{{(\frac{1}{4})}^{2} {(1 + {(\frac{1}{4})}^{2})}^{2}}$

Step 9.1.19

Write each expression with a common denominator of $128$ , by multiplying each by an appropriate factor of $1$ .

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Step 9.1.19.1

Multiply $\frac{1}{4}$ by $\frac{32}{32}$ .

$- \frac{2 (\frac{- 33}{128} + \frac{32}{4 \cdot 32} + 2)}{{(\frac{1}{4})}^{2} {(1 + {(\frac{1}{4})}^{2})}^{2}}$

Step 9.1.19.2

Multiply $4$ by $32$ .

$- \frac{2 (\frac{- 33}{128} + \frac{32}{128} + 2)}{{(\frac{1}{4})}^{2} {(1 + {(\frac{1}{4})}^{2})}^{2}}$

Step 9.1.20

Combine the numerators over the common denominator.

$- \frac{2 (\frac{- 33 + 32}{128} + 2)}{{(\frac{1}{4})}^{2} {(1 + {(\frac{1}{4})}^{2})}^{2}}$

Step 9.1.21

Add $- 33$ and $32$ .

$- \frac{2 (\frac{- 1}{128} + 2)}{{(\frac{1}{4})}^{2} {(1 + {(\frac{1}{4})}^{2})}^{2}}$

Step 9.1.22

To write $2$ as a fraction with a common denominator, multiply by $\frac{128}{128}$ .

$- \frac{2 (\frac{- 1}{128} + 2 \cdot \frac{128}{128})}{{(\frac{1}{4})}^{2} {(1 + {(\frac{1}{4})}^{2})}^{2}}$

Step 9.1.23

Combine $2$ and $\frac{128}{128}$ .

$- \frac{2 (\frac{- 1}{128} + \frac{2 \cdot 128}{128})}{{(\frac{1}{4})}^{2} {(1 + {(\frac{1}{4})}^{2})}^{2}}$

Step 9.1.24

Combine the numerators over the common denominator.

$- \frac{2 \frac{- 1 + 2 \cdot 128}{128}}{{(\frac{1}{4})}^{2} {(1 + {(\frac{1}{4})}^{2})}^{2}}$

Step 9.1.25

Simplify the numerator.

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Step 9.1.25.1

Multiply $2$ by $128$ .

$- \frac{2 \frac{- 1 + 256}{128}}{{(\frac{1}{4})}^{2} {(1 + {(\frac{1}{4})}^{2})}^{2}}$

Step 9.1.25.2

Add $- 1$ and $256$ .

$- \frac{2 (\frac{255}{128})}{{(\frac{1}{4})}^{2} {(1 + {(\frac{1}{4})}^{2})}^{2}}$

Step 9.2

Simplify the denominator.

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Step 9.2.1

Apply the product rule to $\frac{1}{4}$ .

$- \frac{2 (\frac{255}{128})}{\frac{1^{2}}{4^{2}} {(1 + {(\frac{1}{4})}^{2})}^{2}}$

Step 9.2.2

Apply the product rule to $\frac{1}{4}$ .

$- \frac{2 (\frac{255}{128})}{\frac{1^{2}}{4^{2}} {(1 + \frac{1^{2}}{4^{2}})}^{2}}$

Step 9.2.3

One to any power is one.

$- \frac{2 (\frac{255}{128})}{\frac{1^{2}}{4^{2}} {(1 + \frac{1}{4^{2}})}^{2}}$

Step 9.2.4

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

$- \frac{2 (\frac{255}{128})}{\frac{1^{2}}{4^{2}} {(1 + \frac{1}{16})}^{2}}$

Step 9.2.5

Write $1$ as a fraction with a common denominator.

$- \frac{2 (\frac{255}{128})}{\frac{1^{2}}{4^{2}} {(\frac{16}{16} + \frac{1}{16})}^{2}}$

Step 9.2.6

Combine the numerators over the common denominator.

$- \frac{2 (\frac{255}{128})}{\frac{1^{2}}{4^{2}} {(\frac{16 + 1}{16})}^{2}}$

Step 9.2.7

Add $16$ and $1$ .

$- \frac{2 (\frac{255}{128})}{\frac{1^{2}}{4^{2}} {(\frac{17}{16})}^{2}}$

Step 9.2.8

Apply the product rule to $\frac{17}{16}$ .

$- \frac{2 (\frac{255}{128})}{\frac{1^{2}}{4^{2}} \cdot \frac{17^{2}}{16^{2}}}$

Step 9.2.9

One to any power is one.

$- \frac{2 (\frac{255}{128})}{\frac{1}{4^{2}} \cdot \frac{17^{2}}{16^{2}}}$

Step 9.2.10

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

$- \frac{2 (\frac{255}{128})}{\frac{1}{16} \cdot \frac{17^{2}}{16^{2}}}$

Step 9.2.11

Raise $17$ to the power of $2$ .

$- \frac{2 (\frac{255}{128})}{\frac{1}{16} \cdot \frac{289}{16^{2}}}$

Step 9.2.12

Raise $16$ to the power of $2$ .

$- \frac{2 (\frac{255}{128})}{\frac{1}{16} \cdot \frac{289}{256}}$

Step 9.3

Simplify terms.

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Step 9.3.1

Combine $2$ and $\frac{255}{128}$ .

$- \frac{\frac{2 \cdot 255}{128}}{\frac{1}{16} \cdot \frac{289}{256}}$

Step 9.3.2

Multiply $\frac{1}{16}$ by $\frac{289}{256}$ .

$- \frac{\frac{2 \cdot 255}{128}}{\frac{289}{16 \cdot 256}}$

Step 9.3.3

Multiply.

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Step 9.3.3.1

Multiply $2$ by $255$ .

$- \frac{\frac{510}{128}}{\frac{289}{16 \cdot 256}}$

Step 9.3.3.2

Multiply $16$ by $256$ .

$- \frac{\frac{510}{128}}{\frac{289}{4096}}$

Step 9.3.4

Cancel the common factor of $510$ and $128$ .

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Step 9.3.4.1

Factor $2$ out of $510$ .

$- \frac{\frac{2 (255)}{128}}{\frac{289}{4096}}$

Step 9.3.4.2

Cancel the common factors.

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Step 9.3.4.2.1

Factor $2$ out of $128$ .

$- \frac{\frac{2 \cdot 255}{2 \cdot 64}}{\frac{289}{4096}}$

Step 9.3.4.2.2

Cancel the common factor.

$- \frac{\frac{2 \cdot 255}{2 \cdot 64}}{\frac{289}{4096}}$

Step 9.3.4.2.3

Rewrite the expression.

$- \frac{\frac{255}{64}}{\frac{289}{4096}}$

Step 9.4

Multiply the numerator by the reciprocal of the denominator.

$- (\frac{255}{64} \cdot \frac{4096}{289})$

Step 9.5

Cancel the common factor of $17$ .

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Step 9.5.1

Factor $17$ out of $255$ .

$- (\frac{17 (15)}{64} \cdot \frac{4096}{289})$

Step 9.5.2

Factor $17$ out of $289$ .

$- (\frac{17 \cdot 15}{64} \cdot \frac{4096}{17 \cdot 17})$

Step 9.5.3

Cancel the common factor.

$- (\frac{17 \cdot 15}{64} \cdot \frac{4096}{17 \cdot 17})$

Step 9.5.4

Rewrite the expression.

$- (\frac{15}{64} \cdot \frac{4096}{17})$

Step 9.6

Cancel the common factor of $64$ .

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Step 9.6.1

Factor $64$ out of $4096$ .

$- (\frac{15}{64} \cdot \frac{64 (64)}{17})$

Step 9.6.2

Cancel the common factor.

$- (\frac{15}{64} \cdot \frac{64 \cdot 64}{17})$

Step 9.6.3

Rewrite the expression.

$- (15 (\frac{64}{17}))$

Step 9.7

Combine $15$ and $\frac{64}{17}$ .

$- \frac{15 \cdot 64}{17}$

Step 9.8

Multiply $15$ by $64$ .

$- \frac{960}{17}$

Step 10

$x = \frac{1}{4}$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = \frac{1}{4}$ is a local maximum

Step 11

Find the y-value when $x = \frac{1}{4}$ .

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Step 11.1

Replace the variable $x$ with $\frac{1}{4}$ in the expression.

$f (\frac{1}{4}) = 4 \ln (\frac{1}{4}) - 17 \arctan (\frac{1}{4})$

Step 11.2

Simplify the result.

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Step 11.2.1

Simplify each term.

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Step 11.2.1.1

Simplify $4 \ln (\frac{1}{4})$ by moving $4$ inside the logarithm.

$f (\frac{1}{4}) = \ln ({(\frac{1}{4})}^{4}) - 17 \arctan (\frac{1}{4})$

Step 11.2.1.2

Apply the product rule to $\frac{1}{4}$ .

$f (\frac{1}{4}) = \ln (\frac{1^{4}}{4^{4}}) - 17 \arctan (\frac{1}{4})$

Step 11.2.1.3

One to any power is one.

$f (\frac{1}{4}) = \ln (\frac{1}{4^{4}}) - 17 \arctan (\frac{1}{4})$

Step 11.2.1.4

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

$f (\frac{1}{4}) = \ln (\frac{1}{256}) - 17 \arctan (\frac{1}{4})$

Step 11.2.1.5

Evaluate $\arctan (\frac{1}{4})$ .

$f (\frac{1}{4}) = \ln (\frac{1}{256}) - 17 \cdot 0.24497866$

Step 11.2.1.6

Multiply $- 17$ by $0.24497866$ .

$f (\frac{1}{4}) = \ln (\frac{1}{256}) - 4.16463727$

Step 11.2.2

Subtract $4.16463727$ from $\ln (\frac{1}{256})$ .

$f (\frac{1}{4}) = - 9.70981471$

Step 11.2.3

The final answer is $- 9.70981471$ .

$y = - 9.70981471$

Step 12

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

$- \frac{2 (2 {(4)}^{4} - 17 {(4)}^{3} + 4 {(4)}^{2} + 2)}{{(4)}^{2} {(1 + {(4)}^{2})}^{2}}$

Step 13

Evaluate the second derivative.

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Step 13.1

Multiply $4$ by ${(4)}^{2}$ by adding the exponents.

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Step 13.1.1

Multiply $4$ by ${(4)}^{2}$ .

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Step 13.1.1.1

Raise $4$ to the power of $1$ .

$- \frac{2 (2 {(4)}^{4} - 17 {(4)}^{3} + 4^{1} {(4)}^{2} + 2)}{{(4)}^{2} {(1 + {(4)}^{2})}^{2}}$

Step 13.1.1.2

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

$- \frac{2 (2 {(4)}^{4} - 17 {(4)}^{3} + 4^{1 + 2} + 2)}{{(4)}^{2} {(1 + {(4)}^{2})}^{2}}$

Step 13.1.2

Add $1$ and $2$ .

$- \frac{2 (2 {(4)}^{4} - 17 {(4)}^{3} + 4^{3} + 2)}{{(4)}^{2} {(1 + {(4)}^{2})}^{2}}$

Step 13.2

Simplify the numerator.

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Step 13.2.1

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

$- \frac{2 (2 \cdot 256 - 17 \cdot 4^{3} + 4^{3} + 2)}{4^{2} {(1 + 4^{2})}^{2}}$

Step 13.2.2

Multiply $2$ by $256$ .

$- \frac{2 (512 - 17 \cdot 4^{3} + 4^{3} + 2)}{4^{2} {(1 + 4^{2})}^{2}}$

Step 13.2.3

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

$- \frac{2 (512 - 17 \cdot 64 + 4^{3} + 2)}{4^{2} {(1 + 4^{2})}^{2}}$

Step 13.2.4

Multiply $- 17$ by $64$ .

$- \frac{2 (512 - 1088 + 4^{3} + 2)}{4^{2} {(1 + 4^{2})}^{2}}$

Step 13.2.5

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

$- \frac{2 (512 - 1088 + 64 + 2)}{4^{2} {(1 + 4^{2})}^{2}}$

Step 13.2.6

Subtract $1088$ from $512$ .

$- \frac{2 (- 576 + 64 + 2)}{4^{2} {(1 + 4^{2})}^{2}}$

Step 13.2.7

Add $- 576$ and $64$ .

$- \frac{2 (- 512 + 2)}{4^{2} {(1 + 4^{2})}^{2}}$

Step 13.2.8

Add $- 512$ and $2$ .

$- \frac{2 \cdot - 510}{4^{2} {(1 + 4^{2})}^{2}}$

Step 13.3

Simplify the denominator.

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Step 13.3.1

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

$- \frac{2 \cdot - 510}{4^{2} {(1 + 16)}^{2}}$

Step 13.3.2

Add $1$ and $16$ .

$- \frac{2 \cdot - 510}{4^{2} \cdot 17^{2}}$

Step 13.3.3

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

$- \frac{2 \cdot - 510}{16 \cdot 17^{2}}$

Step 13.3.4

Raise $17$ to the power of $2$ .

$- \frac{2 \cdot - 510}{16 \cdot 289}$

Step 13.4

Reduce the expression by cancelling the common factors.

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Step 13.4.1

Multiply $2$ by $- 510$ .

$- \frac{- 1020}{16 \cdot 289}$

Step 13.4.2

Multiply $16$ by $289$ .

$- \frac{- 1020}{4624}$

Step 13.4.3

Cancel the common factor of $- 1020$ and $4624$ .

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Step 13.4.3.1

Factor $68$ out of $- 1020$ .

$- \frac{68 (- 15)}{4624}$

Step 13.4.3.2

Cancel the common factors.

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Step 13.4.3.2.1

Factor $68$ out of $4624$ .

$- \frac{68 \cdot - 15}{68 \cdot 68}$

Step 13.4.3.2.2

Cancel the common factor.

$- \frac{68 \cdot - 15}{68 \cdot 68}$

Step 13.4.3.2.3

Rewrite the expression.

$- \frac{- 15}{68}$

Step 13.4.4

Move the negative in front of the fraction.

$- - \frac{15}{68}$

Step 13.5

Multiply $- - \frac{15}{68}$ .

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Step 13.5.1

Multiply $- 1$ by $- 1$ .

$1 (\frac{15}{68})$

Step 13.5.2

Multiply $\frac{15}{68}$ by $1$ .

$\frac{15}{68}$

Step 14

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

$x = 4$ is a local minimum

Step 15

Find the y-value when $x = 4$ .

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Step 15.1

Replace the variable $x$ with $4$ in the expression.

$f (4) = 4 \ln (4) - 17 \arctan (4)$

Step 15.2

Simplify the result.

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Step 15.2.1

Simplify each term.

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Step 15.2.1.1

Simplify $4 \ln (4)$ by moving $4$ inside the logarithm.

$f (4) = \ln (4^{4}) - 17 \arctan (4)$

Step 15.2.1.2

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

$f (4) = \ln (256) - 17 \arctan (4)$

Step 15.2.1.3

Evaluate $\arctan (4)$ .

$f (4) = \ln (256) - 17 \cdot 1.32581766$

Step 15.2.1.4

Multiply $- 17$ by $1.32581766$ .

$f (4) = \ln (256) - 22.53890028$

Step 15.2.2

Subtract $22.53890028$ from $\ln (256)$ .

$f (4) = - 16.99372283$

Step 15.2.3

The final answer is $- 16.99372283$ .

$y = - 16.99372283$

Step 16

These are the local extrema for $f (x) = 4 \ln (x) - 17 \arctan (x)$ .

$(\frac{1}{4}, - 9.70981471)$ is a local maxima

$(4, - 16.99372283)$ is a local minima

Step 17