Calculus Examples

$f (x) = x - 6 \ln (x^{2} + 1)$

Step 1

Find the first derivative of the function.

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Step 1.1

Differentiate.

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Step 1.1.1

By the Sum Rule, the derivative of $x - 6 \ln (x^{2} + 1)$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 6 \ln (x^{2} + 1)]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [- 6 \ln (x^{2} + 1)]$

Step 1.1.2

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

$1 + \frac{d}{d x} [- 6 \ln (x^{2} + 1)]$

Step 1.2

Evaluate $\frac{d}{d x} [- 6 \ln (x^{2} + 1)]$ .

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Step 1.2.1

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 \ln (x^{2} + 1)$ with respect to $x$ is $- 6 \frac{d}{d x} [\ln (x^{2} + 1)]$ .

$1 - 6 \frac{d}{d x} [\ln (x^{2} + 1)]$

Step 1.2.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = x^{2} + 1$ .

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Step 1.2.2.1

To apply the Chain Rule, set $u$ as $x^{2} + 1$ .

$1 - 6 (\frac{d}{d u} [\ln (u)] \frac{d}{d x} [x^{2} + 1])$

Step 1.2.2.2

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

$1 - 6 (\frac{1}{u} \frac{d}{d x} [x^{2} + 1])$

Step 1.2.2.3

Replace all occurrences of $u$ with $x^{2} + 1$ .

$1 - 6 (\frac{1}{x^{2} + 1} \frac{d}{d x} [x^{2} + 1])$

Step 1.2.3

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

$1 - 6 (\frac{1}{x^{2} + 1} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]))$

Step 1.2.4

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

$1 - 6 (\frac{1}{x^{2} + 1} (2 x + \frac{d}{d x} [1]))$

Step 1.2.5

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

$1 - 6 (\frac{1}{x^{2} + 1} (2 x + 0))$

Step 1.2.6

Add $2 x$ and $0$ .

$1 - 6 (\frac{1}{x^{2} + 1} (2 x))$

Step 1.2.7

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

$1 - 6 (\frac{2}{x^{2} + 1} x)$

Step 1.2.8

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

$1 - 6 \frac{2 x}{x^{2} + 1}$

Step 1.2.9

Combine $- 6$ and $\frac{2 x}{x^{2} + 1}$ .

$1 + \frac{- 6 (2 x)}{x^{2} + 1}$

Step 1.2.10

Multiply $2$ by $- 6$ .

$1 + \frac{- 12 x}{x^{2} + 1}$

Step 1.2.11

Move the negative in front of the fraction.

$1 - \frac{12 x}{x^{2} + 1}$

Step 1.3

Simplify.

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Step 1.3.1

Combine terms.

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Step 1.3.1.1

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

$\frac{x^{2} + 1}{x^{2} + 1} - \frac{12 x}{x^{2} + 1}$

Step 1.3.1.2

Combine the numerators over the common denominator.

$\frac{x^{2} + 1 - 12 x}{x^{2} + 1}$

Step 1.3.2

Reorder terms.

$\frac{x^{2} - 12 x + 1}{x^{2} + 1}$

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 $f (x) = x^{2} - 12 x + 1$ and $g (x) = x^{2} + 1$ .

$f'' (x) = \frac{(x^{2} + 1) \frac{d}{d x} (x^{2} - 12 x + 1) - (x^{2} - 12 x + 1) \frac{d}{d x} (x^{2} + 1)}{{(x^{2} + 1)}^{2}}$

Step 2.2

Differentiate.

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Step 2.2.1

By the Sum Rule, the derivative of $x^{2} - 12 x + 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 12 x] + \frac{d}{d x} [1]$ .

$f'' (x) = \frac{(x^{2} + 1) (\frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 12 x) + \frac{d}{d x} (1)) - (x^{2} - 12 x + 1) \frac{d}{d x} (x^{2} + 1)}{{(x^{2} + 1)}^{2}}$

Step 2.2.2

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

$f'' (x) = \frac{(x^{2} + 1) (2 x + \frac{d}{d x} (- 12 x) + \frac{d}{d x} (1)) - (x^{2} - 12 x + 1) \frac{d}{d x} (x^{2} + 1)}{{(x^{2} + 1)}^{2}}$

Step 2.2.3

Since $- 12$ is constant with respect to $x$ , the derivative of $- 12 x$ with respect to $x$ is $- 12 \frac{d}{d x} [x]$ .

$f'' (x) = \frac{(x^{2} + 1) (2 x - 12 \frac{d}{d x} x + \frac{d}{d x} (1)) - (x^{2} - 12 x + 1) \frac{d}{d x} (x^{2} + 1)}{{(x^{2} + 1)}^{2}}$

Step 2.2.4

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

$f'' (x) = \frac{(x^{2} + 1) (2 x - 12 \cdot 1 + \frac{d}{d x} (1)) - (x^{2} - 12 x + 1) \frac{d}{d x} (x^{2} + 1)}{{(x^{2} + 1)}^{2}}$

Step 2.2.5

Multiply $- 12$ by $1$ .

$f'' (x) = \frac{(x^{2} + 1) (2 x - 12 + \frac{d}{d x} (1)) - (x^{2} - 12 x + 1) \frac{d}{d x} (x^{2} + 1)}{{(x^{2} + 1)}^{2}}$

Step 2.2.6

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

$f'' (x) = \frac{(x^{2} + 1) (2 x - 12 + 0) - (x^{2} - 12 x + 1) \frac{d}{d x} (x^{2} + 1)}{{(x^{2} + 1)}^{2}}$

Step 2.2.7

Add $2 x - 12$ and $0$ .

$f'' (x) = \frac{(x^{2} + 1) (2 x - 12) - (x^{2} - 12 x + 1) \frac{d}{d x} (x^{2} + 1)}{{(x^{2} + 1)}^{2}}$

Step 2.2.8

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

$f'' (x) = \frac{(x^{2} + 1) (2 x - 12) - (x^{2} - 12 x + 1) (\frac{d}{d x} (x^{2}) + \frac{d}{d x} (1))}{{(x^{2} + 1)}^{2}}$

Step 2.2.9

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

$f'' (x) = \frac{(x^{2} + 1) (2 x - 12) - (x^{2} - 12 x + 1) (2 x + \frac{d}{d x} (1))}{{(x^{2} + 1)}^{2}}$

Step 2.2.10

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

$f'' (x) = \frac{(x^{2} + 1) (2 x - 12) - (x^{2} - 12 x + 1) (2 x + 0)}{{(x^{2} + 1)}^{2}}$

Step 2.2.11

Simplify the expression.

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Step 2.2.11.1

Add $2 x$ and $0$ .

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

Step 2.2.11.2

Multiply $2$ by $- 1$ .

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

Step 2.3

Simplify.

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Step 2.3.1

Apply the distributive property.

$f'' (x) = \frac{(x^{2} + 1) (2 x - 12) + (- 2 x^{2} - 2 (- 12 x) - 2 \cdot 1) x}{{(x^{2} + 1)}^{2}}$

Step 2.3.2

Apply the distributive property.

$f'' (x) = \frac{(x^{2} + 1) (2 x - 12) - 2 x^{2} x - 2 (- 12 x) x - 2 \cdot (1 x)}{{(x^{2} + 1)}^{2}}$

Step 2.3.3

Simplify the numerator.

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Step 2.3.3.1

Simplify each term.

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Step 2.3.3.1.1

Expand $(x^{2} + 1) (2 x - 12)$ using the FOIL Method.

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Step 2.3.3.1.1.1

Apply the distributive property.

$f'' (x) = \frac{x^{2} (2 x - 12) + 1 (2 x - 12) - 2 x^{2} x - 2 (- 12 x) x - 2 \cdot (1 x)}{{(x^{2} + 1)}^{2}}$

Step 2.3.3.1.1.2

Apply the distributive property.

$f'' (x) = \frac{x^{2} (2 x) + x^{2} \cdot - 12 + 1 (2 x - 12) - 2 x^{2} x - 2 (- 12 x) x - 2 \cdot (1 x)}{{(x^{2} + 1)}^{2}}$

Step 2.3.3.1.1.3

Apply the distributive property.

$f'' (x) = \frac{x^{2} (2 x) + x^{2} \cdot - 12 + 1 (2 x) + 1 \cdot - 12 - 2 x^{2} x - 2 (- 12 x) x - 2 \cdot (1 x)}{{(x^{2} + 1)}^{2}}$

Step 2.3.3.1.2

Simplify each term.

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Step 2.3.3.1.2.1

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{2 x^{2} x + x^{2} \cdot - 12 + 1 (2 x) + 1 \cdot - 12 - 2 x^{2} x - 2 (- 12 x) x - 2 \cdot (1 x)}{{(x^{2} + 1)}^{2}}$

Step 2.3.3.1.2.2

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

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Step 2.3.3.1.2.2.1

Move $x$ .

$f'' (x) = \frac{2 (x \cdot x^{2}) + x^{2} \cdot - 12 + 1 (2 x) + 1 \cdot - 12 - 2 x^{2} x - 2 (- 12 x) x - 2 \cdot (1 x)}{{(x^{2} + 1)}^{2}}$

Step 2.3.3.1.2.2.2

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

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Step 2.3.3.1.2.2.2.1

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

$f'' (x) = \frac{2 (x \cdot x^{2}) + x^{2} \cdot - 12 + 1 (2 x) + 1 \cdot - 12 - 2 x^{2} x - 2 (- 12 x) x - 2 \cdot (1 x)}{{(x^{2} + 1)}^{2}}$

Step 2.3.3.1.2.2.2.2

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

$f'' (x) = \frac{2 x^{1 + 2} + x^{2} \cdot - 12 + 1 (2 x) + 1 \cdot - 12 - 2 x^{2} x - 2 (- 12 x) x - 2 \cdot (1 x)}{{(x^{2} + 1)}^{2}}$

Step 2.3.3.1.2.2.3

Add $1$ and $2$ .

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

Step 2.3.3.1.2.3

Move $- 12$ to the left of $x^{2}$ .

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

Step 2.3.3.1.2.4

Multiply $2 x$ by $1$ .

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

Step 2.3.3.1.2.5

Multiply $- 12$ by $1$ .

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

Step 2.3.3.1.3

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

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Step 2.3.3.1.3.1

Move $x$ .

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

Step 2.3.3.1.3.2

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

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Step 2.3.3.1.3.2.1

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

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

Step 2.3.3.1.3.2.2

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

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

Step 2.3.3.1.3.3

Add $1$ and $2$ .

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

Step 2.3.3.1.4

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

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Step 2.3.3.1.4.1

Move $x$ .

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

Step 2.3.3.1.4.2

Multiply $x$ by $x$ .

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

Step 2.3.3.1.5

Multiply $- 12$ by $- 2$ .

$f'' (x) = \frac{2 x^{3} - 12 x^{2} + 2 x - 12 - 2 x^{3} + 24 x^{2} - 2 \cdot (1 x)}{{(x^{2} + 1)}^{2}}$

Step 2.3.3.1.6

Multiply $- 2$ by $1$ .

$f'' (x) = \frac{2 x^{3} - 12 x^{2} + 2 x - 12 - 2 x^{3} + 24 x^{2} - 2 x}{{(x^{2} + 1)}^{2}}$

Step 2.3.3.2

Combine the opposite terms in $2 x^{3} - 12 x^{2} + 2 x - 12 - 2 x^{3} + 24 x^{2} - 2 x$ .

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Step 2.3.3.2.1

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

$f'' (x) = \frac{- 12 x^{2} + 2 x - 12 + 0 + 24 x^{2} - 2 x}{{(x^{2} + 1)}^{2}}$

Step 2.3.3.2.2

Add $- 12 x^{2} + 2 x - 12$ and $0$ .

$f'' (x) = \frac{- 12 x^{2} + 2 x - 12 + 24 x^{2} - 2 x}{{(x^{2} + 1)}^{2}}$

Step 2.3.3.2.3

Subtract $2 x$ from $2 x$ .

$f'' (x) = \frac{- 12 x^{2} - 12 + 24 x^{2} + 0}{{(x^{2} + 1)}^{2}}$

Step 2.3.3.2.4

Add $- 12 x^{2} - 12 + 24 x^{2}$ and $0$ .

$f'' (x) = \frac{- 12 x^{2} - 12 + 24 x^{2}}{{(x^{2} + 1)}^{2}}$

Step 2.3.3.3

Add $- 12 x^{2}$ and $24 x^{2}$ .

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

Step 2.3.4

Simplify the numerator.

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Step 2.3.4.1

Factor $12$ out of $12 x^{2} - 12$ .

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Step 2.3.4.1.1

Factor $12$ out of $12 x^{2}$ .

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

Step 2.3.4.1.2

Factor $12$ out of $- 12$ .

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

Step 2.3.4.1.3

Factor $12$ out of $12 (x^{2}) + 12 (- 1)$ .

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

Step 2.3.4.2

Rewrite $1$ as $1^{2}$ .

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

Step 2.3.4.3

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 $b = 1$ .

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

Step 3

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

$\frac{x^{2} - 12 x + 1}{x^{2} + 1} = 0$

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

Differentiate.

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Step 4.1.1.1

By the Sum Rule, the derivative of $x - 6 \ln (x^{2} + 1)$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 6 \ln (x^{2} + 1)]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [- 6 \ln (x^{2} + 1)]$

Step 4.1.1.2

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

$1 + \frac{d}{d x} [- 6 \ln (x^{2} + 1)]$

Step 4.1.2

Evaluate $\frac{d}{d x} [- 6 \ln (x^{2} + 1)]$ .

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Step 4.1.2.1

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 \ln (x^{2} + 1)$ with respect to $x$ is $- 6 \frac{d}{d x} [\ln (x^{2} + 1)]$ .

$1 - 6 \frac{d}{d x} [\ln (x^{2} + 1)]$

Step 4.1.2.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = x^{2} + 1$ .

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Step 4.1.2.2.1

To apply the Chain Rule, set $u$ as $x^{2} + 1$ .

$1 - 6 (\frac{d}{d u} [\ln (u)] \frac{d}{d x} [x^{2} + 1])$

Step 4.1.2.2.2

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

$1 - 6 (\frac{1}{u} \frac{d}{d x} [x^{2} + 1])$

Step 4.1.2.2.3

Replace all occurrences of $u$ with $x^{2} + 1$ .

$1 - 6 (\frac{1}{x^{2} + 1} \frac{d}{d x} [x^{2} + 1])$

Step 4.1.2.3

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

$1 - 6 (\frac{1}{x^{2} + 1} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]))$

Step 4.1.2.4

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

$1 - 6 (\frac{1}{x^{2} + 1} (2 x + \frac{d}{d x} [1]))$

Step 4.1.2.5

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

$1 - 6 (\frac{1}{x^{2} + 1} (2 x + 0))$

Step 4.1.2.6

Add $2 x$ and $0$ .

$1 - 6 (\frac{1}{x^{2} + 1} (2 x))$

Step 4.1.2.7

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

$1 - 6 (\frac{2}{x^{2} + 1} x)$

Step 4.1.2.8

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

$1 - 6 \frac{2 x}{x^{2} + 1}$

Step 4.1.2.9

Combine $- 6$ and $\frac{2 x}{x^{2} + 1}$ .

$1 + \frac{- 6 (2 x)}{x^{2} + 1}$

Step 4.1.2.10

Multiply $2$ by $- 6$ .

$1 + \frac{- 12 x}{x^{2} + 1}$

Step 4.1.2.11

Move the negative in front of the fraction.

$1 - \frac{12 x}{x^{2} + 1}$

Step 4.1.3

Simplify.

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Step 4.1.3.1

Combine terms.

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Step 4.1.3.1.1

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

$\frac{x^{2} + 1}{x^{2} + 1} - \frac{12 x}{x^{2} + 1}$

Step 4.1.3.1.2

Combine the numerators over the common denominator.

$\frac{x^{2} + 1 - 12 x}{x^{2} + 1}$

Step 4.1.3.2

Reorder terms.

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

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{x^{2} - 12 x + 1}{x^{2} + 1}$ .

$\frac{x^{2} - 12 x + 1}{x^{2} + 1}$

Step 5

Set the first derivative equal to $0$ then solve the equation $\frac{x^{2} - 12 x + 1}{x^{2} + 1} = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$\frac{x^{2} - 12 x + 1}{x^{2} + 1} = 0$

Step 5.2

Set the numerator equal to zero.

$x^{2} - 12 x + 1 = 0$

Step 5.3

Solve the equation for $x$ .

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Step 5.3.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 5.3.2

Substitute the values $a = 1$ , $b = - 12$ , and $c = 1$ into the quadratic formula and solve for $x$ .

$\frac{12 \pm \sqrt{{(- 12)}^{2} - 4 \cdot (1 \cdot 1)}}{2 \cdot 1}$

Step 5.3.3

Simplify.

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Step 5.3.3.1

Simplify the numerator.

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Step 5.3.3.1.1

Raise $- 12$ to the power of $2$ .

$x = \frac{12 \pm \sqrt{144 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Step 5.3.3.1.2

Multiply $- 4 \cdot 1 \cdot 1$ .

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Step 5.3.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{12 \pm \sqrt{144 - 4 \cdot 1}}{2 \cdot 1}$

Step 5.3.3.1.2.2

Multiply $- 4$ by $1$ .

$x = \frac{12 \pm \sqrt{144 - 4}}{2 \cdot 1}$

Step 5.3.3.1.3

Subtract $4$ from $144$ .

$x = \frac{12 \pm \sqrt{140}}{2 \cdot 1}$

Step 5.3.3.1.4

Rewrite $140$ as $2^{2} \cdot 35$ .

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Step 5.3.3.1.4.1

Factor $4$ out of $140$ .

$x = \frac{12 \pm \sqrt{4 (35)}}{2 \cdot 1}$

Step 5.3.3.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{12 \pm \sqrt{2^{2} \cdot 35}}{2 \cdot 1}$

Step 5.3.3.1.5

Pull terms out from under the radical.

$x = \frac{12 \pm 2 \sqrt{35}}{2 \cdot 1}$

Step 5.3.3.2

Multiply $2$ by $1$ .

$x = \frac{12 \pm 2 \sqrt{35}}{2}$

Step 5.3.3.3

Simplify $\frac{12 \pm 2 \sqrt{35}}{2}$ .

$x = 6 \pm \sqrt{35}$

Step 5.3.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 5.3.4.1

Simplify the numerator.

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Step 5.3.4.1.1

Raise $- 12$ to the power of $2$ .

$x = \frac{12 \pm \sqrt{144 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Step 5.3.4.1.2

Multiply $- 4 \cdot 1 \cdot 1$ .

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Step 5.3.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{12 \pm \sqrt{144 - 4 \cdot 1}}{2 \cdot 1}$

Step 5.3.4.1.2.2

Multiply $- 4$ by $1$ .

$x = \frac{12 \pm \sqrt{144 - 4}}{2 \cdot 1}$

Step 5.3.4.1.3

Subtract $4$ from $144$ .

$x = \frac{12 \pm \sqrt{140}}{2 \cdot 1}$

Step 5.3.4.1.4

Rewrite $140$ as $2^{2} \cdot 35$ .

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Step 5.3.4.1.4.1

Factor $4$ out of $140$ .

$x = \frac{12 \pm \sqrt{4 (35)}}{2 \cdot 1}$

Step 5.3.4.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{12 \pm \sqrt{2^{2} \cdot 35}}{2 \cdot 1}$

Step 5.3.4.1.5

Pull terms out from under the radical.

$x = \frac{12 \pm 2 \sqrt{35}}{2 \cdot 1}$

Step 5.3.4.2

Multiply $2$ by $1$ .

$x = \frac{12 \pm 2 \sqrt{35}}{2}$

Step 5.3.4.3

Simplify $\frac{12 \pm 2 \sqrt{35}}{2}$ .

$x = 6 \pm \sqrt{35}$

Step 5.3.4.4

Change the $\pm$ to $+$ .

$x = 6 + \sqrt{35}$

Step 5.3.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 5.3.5.1

Simplify the numerator.

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Step 5.3.5.1.1

Raise $- 12$ to the power of $2$ .

$x = \frac{12 \pm \sqrt{144 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Step 5.3.5.1.2

Multiply $- 4 \cdot 1 \cdot 1$ .

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Step 5.3.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{12 \pm \sqrt{144 - 4 \cdot 1}}{2 \cdot 1}$

Step 5.3.5.1.2.2

Multiply $- 4$ by $1$ .

$x = \frac{12 \pm \sqrt{144 - 4}}{2 \cdot 1}$

Step 5.3.5.1.3

Subtract $4$ from $144$ .

$x = \frac{12 \pm \sqrt{140}}{2 \cdot 1}$

Step 5.3.5.1.4

Rewrite $140$ as $2^{2} \cdot 35$ .

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Step 5.3.5.1.4.1

Factor $4$ out of $140$ .

$x = \frac{12 \pm \sqrt{4 (35)}}{2 \cdot 1}$

Step 5.3.5.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{12 \pm \sqrt{2^{2} \cdot 35}}{2 \cdot 1}$

Step 5.3.5.1.5

Pull terms out from under the radical.

$x = \frac{12 \pm 2 \sqrt{35}}{2 \cdot 1}$

Step 5.3.5.2

Multiply $2$ by $1$ .

$x = \frac{12 \pm 2 \sqrt{35}}{2}$

Step 5.3.5.3

Simplify $\frac{12 \pm 2 \sqrt{35}}{2}$ .

$x = 6 \pm \sqrt{35}$

Step 5.3.5.4

Change the $\pm$ to $-$ .

$x = 6 - \sqrt{35}$

Step 5.3.6

The final answer is the combination of both solutions.

$x = 6 + \sqrt{35}, 6 - \sqrt{35}$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 7

Critical points to evaluate.

$x = 6 + \sqrt{35}, 6 - \sqrt{35}$

Step 8

Evaluate the second derivative at $x = 6 + \sqrt{35}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{12 ((6 + \sqrt{35}) + 1) ((6 + \sqrt{35}) - 1)}{{({(6 + \sqrt{35})}^{2} + 1)}^{2}}$

Step 9

Evaluate the second derivative.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Add $6$ and $1$ .

$\frac{12 (7 + \sqrt{35}) (6 + \sqrt{35} - 1)}{{({(6 + \sqrt{35})}^{2} + 1)}^{2}}$

Step 9.1.2

Subtract $1$ from $6$ .

$\frac{12 (7 + \sqrt{35}) (5 + \sqrt{35})}{{({(6 + \sqrt{35})}^{2} + 1)}^{2}}$

Step 9.2

Simplify the denominator.

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Step 9.2.1

Rewrite ${(6 + \sqrt{35})}^{2}$ as $(6 + \sqrt{35}) (6 + \sqrt{35})$ .

$\frac{12 (7 + \sqrt{35}) (5 + \sqrt{35})}{{((6 + \sqrt{35}) (6 + \sqrt{35}) + 1)}^{2}}$

Step 9.2.2

Expand $(6 + \sqrt{35}) (6 + \sqrt{35})$ using the FOIL Method.

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Step 9.2.2.1

Apply the distributive property.

$\frac{12 (7 + \sqrt{35}) (5 + \sqrt{35})}{{(6 (6 + \sqrt{35}) + \sqrt{35} (6 + \sqrt{35}) + 1)}^{2}}$

Step 9.2.2.2

Apply the distributive property.

$\frac{12 (7 + \sqrt{35}) (5 + \sqrt{35})}{{(6 \cdot 6 + 6 \sqrt{35} + \sqrt{35} (6 + \sqrt{35}) + 1)}^{2}}$

Step 9.2.2.3

Apply the distributive property.

$\frac{12 (7 + \sqrt{35}) (5 + \sqrt{35})}{{(6 \cdot 6 + 6 \sqrt{35} + \sqrt{35} \cdot 6 + \sqrt{35} \sqrt{35} + 1)}^{2}}$

Step 9.2.3

Simplify and combine like terms.

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Step 9.2.3.1

Simplify each term.

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Step 9.2.3.1.1

Multiply $6$ by $6$ .

$\frac{12 (7 + \sqrt{35}) (5 + \sqrt{35})}{{(36 + 6 \sqrt{35} + \sqrt{35} \cdot 6 + \sqrt{35} \sqrt{35} + 1)}^{2}}$

Step 9.2.3.1.2

Move $6$ to the left of $\sqrt{35}$ .

$\frac{12 (7 + \sqrt{35}) (5 + \sqrt{35})}{{(36 + 6 \sqrt{35} + 6 \cdot \sqrt{35} + \sqrt{35} \sqrt{35} + 1)}^{2}}$

Step 9.2.3.1.3

Combine using the product rule for radicals.

$\frac{12 (7 + \sqrt{35}) (5 + \sqrt{35})}{{(36 + 6 \sqrt{35} + 6 \sqrt{35} + \sqrt{35 \cdot 35} + 1)}^{2}}$

Step 9.2.3.1.4

Multiply $35$ by $35$ .

$\frac{12 (7 + \sqrt{35}) (5 + \sqrt{35})}{{(36 + 6 \sqrt{35} + 6 \sqrt{35} + \sqrt{1225} + 1)}^{2}}$

Step 9.2.3.1.5

Rewrite $1225$ as $35^{2}$ .

$\frac{12 (7 + \sqrt{35}) (5 + \sqrt{35})}{{(36 + 6 \sqrt{35} + 6 \sqrt{35} + \sqrt{35^{2}} + 1)}^{2}}$

Step 9.2.3.1.6

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

$\frac{12 (7 + \sqrt{35}) (5 + \sqrt{35})}{{(36 + 6 \sqrt{35} + 6 \sqrt{35} + 35 + 1)}^{2}}$

Step 9.2.3.2

Add $36$ and $35$ .

$\frac{12 (7 + \sqrt{35}) (5 + \sqrt{35})}{{(71 + 6 \sqrt{35} + 6 \sqrt{35} + 1)}^{2}}$

Step 9.2.3.3

Add $6 \sqrt{35}$ and $6 \sqrt{35}$ .

$\frac{12 (7 + \sqrt{35}) (5 + \sqrt{35})}{{(71 + 12 \sqrt{35} + 1)}^{2}}$

Step 9.2.4

Add $71$ and $1$ .

$\frac{12 (7 + \sqrt{35}) (5 + \sqrt{35})}{{(72 + 12 \sqrt{35})}^{2}}$

Step 9.3

Group $5 + \sqrt{35}$ and $7 + \sqrt{35}$ together.

$\frac{12 ((5 + \sqrt{35}) (7 + \sqrt{35}))}{{(72 + 12 \sqrt{35})}^{2}}$

Step 9.4

Expand $(5 + \sqrt{35}) (7 + \sqrt{35})$ using the FOIL Method.

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Step 9.4.1

Apply the distributive property.

$\frac{12 (5 (7 + \sqrt{35}) + \sqrt{35} (7 + \sqrt{35}))}{{(72 + 12 \sqrt{35})}^{2}}$

Step 9.4.2

Apply the distributive property.

$\frac{12 (5 \cdot 7 + 5 \sqrt{35} + \sqrt{35} (7 + \sqrt{35}))}{{(72 + 12 \sqrt{35})}^{2}}$

Step 9.4.3

Apply the distributive property.

$\frac{12 (5 \cdot 7 + 5 \sqrt{35} + \sqrt{35} \cdot 7 + \sqrt{35} \sqrt{35})}{{(72 + 12 \sqrt{35})}^{2}}$

Step 9.5

Simplify and combine like terms.

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Step 9.5.1

Simplify each term.

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Step 9.5.1.1

Multiply $5$ by $7$ .

$\frac{12 (35 + 5 \sqrt{35} + \sqrt{35} \cdot 7 + \sqrt{35} \sqrt{35})}{{(72 + 12 \sqrt{35})}^{2}}$

Step 9.5.1.2

Move $7$ to the left of $\sqrt{35}$ .

$\frac{12 (35 + 5 \sqrt{35} + 7 \cdot \sqrt{35} + \sqrt{35} \sqrt{35})}{{(72 + 12 \sqrt{35})}^{2}}$

Step 9.5.1.3

Combine using the product rule for radicals.

$\frac{12 (35 + 5 \sqrt{35} + 7 \sqrt{35} + \sqrt{35 \cdot 35})}{{(72 + 12 \sqrt{35})}^{2}}$

Step 9.5.1.4

Multiply $35$ by $35$ .

$\frac{12 (35 + 5 \sqrt{35} + 7 \sqrt{35} + \sqrt{1225})}{{(72 + 12 \sqrt{35})}^{2}}$

Step 9.5.1.5

Rewrite $1225$ as $35^{2}$ .

$\frac{12 (35 + 5 \sqrt{35} + 7 \sqrt{35} + \sqrt{35^{2}})}{{(72 + 12 \sqrt{35})}^{2}}$

Step 9.5.1.6

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

$\frac{12 (35 + 5 \sqrt{35} + 7 \sqrt{35} + 35)}{{(72 + 12 \sqrt{35})}^{2}}$

Step 9.5.2

Add $35$ and $35$ .

$\frac{12 (70 + 5 \sqrt{35} + 7 \sqrt{35})}{{(72 + 12 \sqrt{35})}^{2}}$

Step 9.5.3

Add $5 \sqrt{35}$ and $7 \sqrt{35}$ .

$\frac{12 (70 + 12 \sqrt{35})}{{(72 + 12 \sqrt{35})}^{2}}$

Step 9.6

Rewrite ${(72 + 12 \sqrt{35})}^{2}$ as $(72 + 12 \sqrt{35}) (72 + 12 \sqrt{35})$ .

$\frac{12 (70 + 12 \sqrt{35})}{(72 + 12 \sqrt{35}) (72 + 12 \sqrt{35})}$

Step 9.7

Expand $(72 + 12 \sqrt{35}) (72 + 12 \sqrt{35})$ using the FOIL Method.

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Step 9.7.1

Apply the distributive property.

$\frac{12 (70 + 12 \sqrt{35})}{72 (72 + 12 \sqrt{35}) + 12 \sqrt{35} (72 + 12 \sqrt{35})}$

Step 9.7.2

Apply the distributive property.

$\frac{12 (70 + 12 \sqrt{35})}{72 \cdot 72 + 72 (12 \sqrt{35}) + 12 \sqrt{35} (72 + 12 \sqrt{35})}$

Step 9.7.3

Apply the distributive property.

$\frac{12 (70 + 12 \sqrt{35})}{72 \cdot 72 + 72 (12 \sqrt{35}) + 12 \sqrt{35} \cdot 72 + 12 \sqrt{35} (12 \sqrt{35})}$

Step 9.8

Simplify and combine like terms.

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Step 9.8.1

Simplify each term.

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Step 9.8.1.1

Multiply $72$ by $72$ .

$\frac{12 (70 + 12 \sqrt{35})}{5184 + 72 (12 \sqrt{35}) + 12 \sqrt{35} \cdot 72 + 12 \sqrt{35} (12 \sqrt{35})}$

Step 9.8.1.2

Multiply $12$ by $72$ .

$\frac{12 (70 + 12 \sqrt{35})}{5184 + 864 \sqrt{35} + 12 \sqrt{35} \cdot 72 + 12 \sqrt{35} (12 \sqrt{35})}$

Step 9.8.1.3

Multiply $72$ by $12$ .

$\frac{12 (70 + 12 \sqrt{35})}{5184 + 864 \sqrt{35} + 864 \sqrt{35} + 12 \sqrt{35} (12 \sqrt{35})}$

Step 9.8.1.4

Multiply $12 \sqrt{35} (12 \sqrt{35})$ .

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Step 9.8.1.4.1

Multiply $12$ by $12$ .

$\frac{12 (70 + 12 \sqrt{35})}{5184 + 864 \sqrt{35} + 864 \sqrt{35} + 144 \sqrt{35} \sqrt{35}}$

Step 9.8.1.4.2

Raise $\sqrt{35}$ to the power of $1$ .

$\frac{12 (70 + 12 \sqrt{35})}{5184 + 864 \sqrt{35} + 864 \sqrt{35} + 144 ({\sqrt{35}}^{1} \sqrt{35})}$

Step 9.8.1.4.3

Raise $\sqrt{35}$ to the power of $1$ .

$\frac{12 (70 + 12 \sqrt{35})}{5184 + 864 \sqrt{35} + 864 \sqrt{35} + 144 ({\sqrt{35}}^{1} {\sqrt{35}}^{1})}$

Step 9.8.1.4.4

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

$\frac{12 (70 + 12 \sqrt{35})}{5184 + 864 \sqrt{35} + 864 \sqrt{35} + 144 {\sqrt{35}}^{1 + 1}}$

Step 9.8.1.4.5

Add $1$ and $1$ .

$\frac{12 (70 + 12 \sqrt{35})}{5184 + 864 \sqrt{35} + 864 \sqrt{35} + 144 {\sqrt{35}}^{2}}$

Step 9.8.1.5

Rewrite ${\sqrt{35}}^{2}$ as $35$ .

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Step 9.8.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{35}$ as $35^{\frac{1}{2}}$ .

$\frac{12 (70 + 12 \sqrt{35})}{5184 + 864 \sqrt{35} + 864 \sqrt{35} + 144 {(35^{\frac{1}{2}})}^{2}}$

Step 9.8.1.5.2

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

$\frac{12 (70 + 12 \sqrt{35})}{5184 + 864 \sqrt{35} + 864 \sqrt{35} + 144 \cdot 35^{\frac{1}{2} \cdot 2}}$

Step 9.8.1.5.3

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

$\frac{12 (70 + 12 \sqrt{35})}{5184 + 864 \sqrt{35} + 864 \sqrt{35} + 144 \cdot 35^{\frac{2}{2}}}$

Step 9.8.1.5.4

Cancel the common factor of $2$ .

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Step 9.8.1.5.4.1

Cancel the common factor.

$\frac{12 (70 + 12 \sqrt{35})}{5184 + 864 \sqrt{35} + 864 \sqrt{35} + 144 \cdot 35^{\frac{2}{2}}}$

Step 9.8.1.5.4.2

Rewrite the expression.

$\frac{12 (70 + 12 \sqrt{35})}{5184 + 864 \sqrt{35} + 864 \sqrt{35} + 144 \cdot 35^{1}}$

Step 9.8.1.5.5

Evaluate the exponent.

$\frac{12 (70 + 12 \sqrt{35})}{5184 + 864 \sqrt{35} + 864 \sqrt{35} + 144 \cdot 35}$

Step 9.8.1.6

Multiply $144$ by $35$ .

$\frac{12 (70 + 12 \sqrt{35})}{5184 + 864 \sqrt{35} + 864 \sqrt{35} + 5040}$

Step 9.8.2

Add $5184$ and $5040$ .

$\frac{12 (70 + 12 \sqrt{35})}{10224 + 864 \sqrt{35} + 864 \sqrt{35}}$

Step 9.8.3

Add $864 \sqrt{35}$ and $864 \sqrt{35}$ .

$\frac{12 (70 + 12 \sqrt{35})}{10224 + 1728 \sqrt{35}}$

Step 9.9

Cancel the common factors.

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Step 9.9.1

Factor $12$ out of $10224$ .

$\frac{12 (70 + 12 \sqrt{35})}{12 \cdot 852 + 1728 \sqrt{35}}$

Step 9.9.2

Factor $12$ out of $1728 \sqrt{35}$ .

$\frac{12 (70 + 12 \sqrt{35})}{12 \cdot 852 + 12 (144 \sqrt{35})}$

Step 9.9.3

Factor $12$ out of $12 (852) + 12 (144 \sqrt{35})$ .

$\frac{12 (70 + 12 \sqrt{35})}{12 (852 + 144 \sqrt{35})}$

Step 9.9.4

Cancel the common factor.

$\frac{12 (70 + 12 \sqrt{35})}{12 (852 + 144 \sqrt{35})}$

Step 9.9.5

Rewrite the expression.

$\frac{70 + 12 \sqrt{35}}{852 + 144 \sqrt{35}}$

Step 9.10

Cancel the common factor of $70 + 12 \sqrt{35}$ and $852 + 144 \sqrt{35}$ .

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Step 9.10.1

Factor $2$ out of $70$ .

$\frac{2 (35) + 12 \sqrt{35}}{852 + 144 \sqrt{35}}$

Step 9.10.2

Factor $2$ out of $12 \sqrt{35}$ .

$\frac{2 (35) + 2 (6 \sqrt{35})}{852 + 144 \sqrt{35}}$

Step 9.10.3

Factor $2$ out of $2 (35) + 2 (6 \sqrt{35})$ .

$\frac{2 (35 + 6 \sqrt{35})}{852 + 144 \sqrt{35}}$

Step 9.10.4

Cancel the common factors.

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Step 9.10.4.1

Factor $2$ out of $852$ .

$\frac{2 (35 + 6 \sqrt{35})}{2 \cdot 426 + 144 \sqrt{35}}$

Step 9.10.4.2

Factor $2$ out of $144 \sqrt{35}$ .

$\frac{2 (35 + 6 \sqrt{35})}{2 \cdot 426 + 2 (72 \sqrt{35})}$

Step 9.10.4.3

Factor $2$ out of $2 (426) + 2 (72 \sqrt{35})$ .

$\frac{2 (35 + 6 \sqrt{35})}{2 (426 + 72 \sqrt{35})}$

Step 9.10.4.4

Cancel the common factor.

$\frac{2 (35 + 6 \sqrt{35})}{2 (426 + 72 \sqrt{35})}$

Step 9.10.4.5

Rewrite the expression.

$\frac{35 + 6 \sqrt{35}}{426 + 72 \sqrt{35}}$

Step 9.11

Multiply $\frac{35 + 6 \sqrt{35}}{426 + 72 \sqrt{35}}$ by $\frac{426 - 72 \sqrt{35}}{426 - 72 \sqrt{35}}$ .

$\frac{35 + 6 \sqrt{35}}{426 + 72 \sqrt{35}} \cdot \frac{426 - 72 \sqrt{35}}{426 - 72 \sqrt{35}}$

Step 9.12

Simplify terms.

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Step 9.12.1

Multiply $\frac{35 + 6 \sqrt{35}}{426 + 72 \sqrt{35}}$ by $\frac{426 - 72 \sqrt{35}}{426 - 72 \sqrt{35}}$ .

$\frac{(35 + 6 \sqrt{35}) (426 - 72 \sqrt{35})}{(426 + 72 \sqrt{35}) (426 - 72 \sqrt{35})}$

Step 9.12.2

Expand the denominator using the FOIL method.

$\frac{(35 + 6 \sqrt{35}) (426 - 72 \sqrt{35})}{181476 - 30672 \sqrt{35} + 30672 \sqrt{35} - 5184 {\sqrt{35}}^{2}}$

Step 9.12.3

Simplify.

$\frac{(35 + 6 \sqrt{35}) (426 - 72 \sqrt{35})}{36}$

Step 9.12.4

Cancel the common factor of $426 - 72 \sqrt{35}$ and $36$ .

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Step 9.12.4.1

Factor $6$ out of $(35 + 6 \sqrt{35}) (426 - 72 \sqrt{35})$ .

$\frac{6 ((35 + 6 \sqrt{35}) (71 - 12 \sqrt{35}))}{36}$

Step 9.12.4.2

Cancel the common factors.

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Step 9.12.4.2.1

Factor $6$ out of $36$ .

$\frac{6 ((35 + 6 \sqrt{35}) (71 - 12 \sqrt{35}))}{6 (6)}$

Step 9.12.4.2.2

Cancel the common factor.

$\frac{6 ((35 + 6 \sqrt{35}) (71 - 12 \sqrt{35}))}{6 \cdot 6}$

Step 9.12.4.2.3

Rewrite the expression.

$\frac{(35 + 6 \sqrt{35}) (71 - 12 \sqrt{35})}{6}$

Step 9.13

Expand $(35 + 6 \sqrt{35}) (71 - 12 \sqrt{35})$ using the FOIL Method.

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Step 9.13.1

Apply the distributive property.

$\frac{35 (71 - 12 \sqrt{35}) + 6 \sqrt{35} (71 - 12 \sqrt{35})}{6}$

Step 9.13.2

Apply the distributive property.

$\frac{35 \cdot 71 + 35 (- 12 \sqrt{35}) + 6 \sqrt{35} (71 - 12 \sqrt{35})}{6}$

Step 9.13.3

Apply the distributive property.

$\frac{35 \cdot 71 + 35 (- 12 \sqrt{35}) + 6 \sqrt{35} \cdot 71 + 6 \sqrt{35} (- 12 \sqrt{35})}{6}$

Step 9.14

Simplify and combine like terms.

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Step 9.14.1

Simplify each term.

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Step 9.14.1.1

Multiply $35$ by $71$ .

$\frac{2485 + 35 (- 12 \sqrt{35}) + 6 \sqrt{35} \cdot 71 + 6 \sqrt{35} (- 12 \sqrt{35})}{6}$

Step 9.14.1.2

Multiply $- 12$ by $35$ .

$\frac{2485 - 420 \sqrt{35} + 6 \sqrt{35} \cdot 71 + 6 \sqrt{35} (- 12 \sqrt{35})}{6}$

Step 9.14.1.3

Multiply $71$ by $6$ .

$\frac{2485 - 420 \sqrt{35} + 426 \sqrt{35} + 6 \sqrt{35} (- 12 \sqrt{35})}{6}$

Step 9.14.1.4

Multiply $6 \sqrt{35} (- 12 \sqrt{35})$ .

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Step 9.14.1.4.1

Multiply $- 12$ by $6$ .

$\frac{2485 - 420 \sqrt{35} + 426 \sqrt{35} - 72 \sqrt{35} \sqrt{35}}{6}$

Step 9.14.1.4.2

Raise $\sqrt{35}$ to the power of $1$ .

$\frac{2485 - 420 \sqrt{35} + 426 \sqrt{35} - 72 ({\sqrt{35}}^{1} \sqrt{35})}{6}$

Step 9.14.1.4.3

Raise $\sqrt{35}$ to the power of $1$ .

$\frac{2485 - 420 \sqrt{35} + 426 \sqrt{35} - 72 ({\sqrt{35}}^{1} {\sqrt{35}}^{1})}{6}$

Step 9.14.1.4.4

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

$\frac{2485 - 420 \sqrt{35} + 426 \sqrt{35} - 72 {\sqrt{35}}^{1 + 1}}{6}$

Step 9.14.1.4.5

Add $1$ and $1$ .

$\frac{2485 - 420 \sqrt{35} + 426 \sqrt{35} - 72 {\sqrt{35}}^{2}}{6}$

Step 9.14.1.5

Rewrite ${\sqrt{35}}^{2}$ as $35$ .

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Step 9.14.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{35}$ as $35^{\frac{1}{2}}$ .

$\frac{2485 - 420 \sqrt{35} + 426 \sqrt{35} - 72 {(35^{\frac{1}{2}})}^{2}}{6}$

Step 9.14.1.5.2

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

$\frac{2485 - 420 \sqrt{35} + 426 \sqrt{35} - 72 \cdot 35^{\frac{1}{2} \cdot 2}}{6}$

Step 9.14.1.5.3

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

$\frac{2485 - 420 \sqrt{35} + 426 \sqrt{35} - 72 \cdot 35^{\frac{2}{2}}}{6}$

Step 9.14.1.5.4

Cancel the common factor of $2$ .

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Step 9.14.1.5.4.1

Cancel the common factor.

$\frac{2485 - 420 \sqrt{35} + 426 \sqrt{35} - 72 \cdot 35^{\frac{2}{2}}}{6}$

Step 9.14.1.5.4.2

Rewrite the expression.

$\frac{2485 - 420 \sqrt{35} + 426 \sqrt{35} - 72 \cdot 35^{1}}{6}$

Step 9.14.1.5.5

Evaluate the exponent.

$\frac{2485 - 420 \sqrt{35} + 426 \sqrt{35} - 72 \cdot 35}{6}$

Step 9.14.1.6

Multiply $- 72$ by $35$ .

$\frac{2485 - 420 \sqrt{35} + 426 \sqrt{35} - 2520}{6}$

Step 9.14.2

Subtract $2520$ from $2485$ .

$\frac{- 35 - 420 \sqrt{35} + 426 \sqrt{35}}{6}$

Step 9.14.3

Add $- 420 \sqrt{35}$ and $426 \sqrt{35}$ .

$\frac{- 35 + 6 \sqrt{35}}{6}$

Step 9.15

Rewrite $- 35$ as $- 1 (35)$ .

$\frac{- 1 (35) + 6 \sqrt{35}}{6}$

Step 9.16

Factor $- 1$ out of $6 \sqrt{35}$ .

$\frac{- 1 (35) - (- 6 \sqrt{35})}{6}$

Step 9.17

Factor $- 1$ out of $- 1 (35) - (- 6 \sqrt{35})$ .

$\frac{- 1 (35 - 6 \sqrt{35})}{6}$

Step 9.18

Move the negative in front of the fraction.

$- \frac{35 - 6 \sqrt{35}}{6}$

Step 10

$x = 6 + \sqrt{35}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = 6 + \sqrt{35}$ is a local minimum

Step 11

Find the y-value when $x = 6 + \sqrt{35}$ .

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Step 11.1

Replace the variable $x$ with $6 + \sqrt{35}$ in the expression.

$f (6 + \sqrt{35}) = (6 + \sqrt{35}) - 6 \ln ({(6 + \sqrt{35})}^{2} + 1)$

Step 11.2

Simplify the result.

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Step 11.2.1

Remove parentheses.

$f (6 + \sqrt{35}) = 6 + \sqrt{35} - 6 \ln ({(6 + \sqrt{35})}^{2} + 1)$

Step 11.2.2

The final answer is $6 + \sqrt{35} - 6 \ln ({(6 + \sqrt{35})}^{2} + 1)$ .

$y = 6 + \sqrt{35} - 6 \ln ({(6 + \sqrt{35})}^{2} + 1)$

Step 12

Evaluate the second derivative at $x = 6 - \sqrt{35}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{12 ((6 - \sqrt{35}) + 1) ((6 - \sqrt{35}) - 1)}{{({(6 - \sqrt{35})}^{2} + 1)}^{2}}$

Step 13

Evaluate the second derivative.

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Step 13.1

Simplify the numerator.

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Step 13.1.1

Add $6$ and $1$ .

$\frac{12 (7 - \sqrt{35}) (6 - \sqrt{35} - 1)}{{({(6 - \sqrt{35})}^{2} + 1)}^{2}}$

Step 13.1.2

Subtract $1$ from $6$ .

$\frac{12 (7 - \sqrt{35}) (5 - \sqrt{35})}{{({(6 - \sqrt{35})}^{2} + 1)}^{2}}$

Step 13.2

Simplify the denominator.

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Step 13.2.1

Rewrite ${(6 - \sqrt{35})}^{2}$ as $(6 - \sqrt{35}) (6 - \sqrt{35})$ .

$\frac{12 (7 - \sqrt{35}) (5 - \sqrt{35})}{{((6 - \sqrt{35}) (6 - \sqrt{35}) + 1)}^{2}}$

Step 13.2.2

Expand $(6 - \sqrt{35}) (6 - \sqrt{35})$ using the FOIL Method.

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Step 13.2.2.1

Apply the distributive property.

$\frac{12 (7 - \sqrt{35}) (5 - \sqrt{35})}{{(6 (6 - \sqrt{35}) - \sqrt{35} (6 - \sqrt{35}) + 1)}^{2}}$

Step 13.2.2.2

Apply the distributive property.

$\frac{12 (7 - \sqrt{35}) (5 - \sqrt{35})}{{(6 \cdot 6 + 6 (- \sqrt{35}) - \sqrt{35} (6 - \sqrt{35}) + 1)}^{2}}$

Step 13.2.2.3

Apply the distributive property.

$\frac{12 (7 - \sqrt{35}) (5 - \sqrt{35})}{{(6 \cdot 6 + 6 (- \sqrt{35}) - \sqrt{35} \cdot 6 - \sqrt{35} (- \sqrt{35}) + 1)}^{2}}$

Step 13.2.3

Simplify and combine like terms.

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Step 13.2.3.1

Simplify each term.

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Step 13.2.3.1.1

Multiply $6$ by $6$ .

$\frac{12 (7 - \sqrt{35}) (5 - \sqrt{35})}{{(36 + 6 (- \sqrt{35}) - \sqrt{35} \cdot 6 - \sqrt{35} (- \sqrt{35}) + 1)}^{2}}$

Step 13.2.3.1.2

Multiply $- 1$ by $6$ .

$\frac{12 (7 - \sqrt{35}) (5 - \sqrt{35})}{{(36 - 6 \sqrt{35} - \sqrt{35} \cdot 6 - \sqrt{35} (- \sqrt{35}) + 1)}^{2}}$

Step 13.2.3.1.3

Multiply $6$ by $- 1$ .

$\frac{12 (7 - \sqrt{35}) (5 - \sqrt{35})}{{(36 - 6 \sqrt{35} - 6 \sqrt{35} - \sqrt{35} (- \sqrt{35}) + 1)}^{2}}$

Step 13.2.3.1.4

Multiply $- \sqrt{35} (- \sqrt{35})$ .

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Step 13.2.3.1.4.1

Multiply $- 1$ by $- 1$ .

$\frac{12 (7 - \sqrt{35}) (5 - \sqrt{35})}{{(36 - 6 \sqrt{35} - 6 \sqrt{35} + 1 \sqrt{35} \sqrt{35} + 1)}^{2}}$

Step 13.2.3.1.4.2

Multiply $\sqrt{35}$ by $1$ .

$\frac{12 (7 - \sqrt{35}) (5 - \sqrt{35})}{{(36 - 6 \sqrt{35} - 6 \sqrt{35} + \sqrt{35} \sqrt{35} + 1)}^{2}}$

Step 13.2.3.1.4.3

Raise $\sqrt{35}$ to the power of $1$ .

$\frac{12 (7 - \sqrt{35}) (5 - \sqrt{35})}{{(36 - 6 \sqrt{35} - 6 \sqrt{35} + {\sqrt{35}}^{1} \sqrt{35} + 1)}^{2}}$

Step 13.2.3.1.4.4

Raise $\sqrt{35}$ to the power of $1$ .

$\frac{12 (7 - \sqrt{35}) (5 - \sqrt{35})}{{(36 - 6 \sqrt{35} - 6 \sqrt{35} + {\sqrt{35}}^{1} {\sqrt{35}}^{1} + 1)}^{2}}$

Step 13.2.3.1.4.5

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

$\frac{12 (7 - \sqrt{35}) (5 - \sqrt{35})}{{(36 - 6 \sqrt{35} - 6 \sqrt{35} + {\sqrt{35}}^{1 + 1} + 1)}^{2}}$

Step 13.2.3.1.4.6

Add $1$ and $1$ .

$\frac{12 (7 - \sqrt{35}) (5 - \sqrt{35})}{{(36 - 6 \sqrt{35} - 6 \sqrt{35} + {\sqrt{35}}^{2} + 1)}^{2}}$

Step 13.2.3.1.5

Rewrite ${\sqrt{35}}^{2}$ as $35$ .

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Step 13.2.3.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{35}$ as $35^{\frac{1}{2}}$ .

$\frac{12 (7 - \sqrt{35}) (5 - \sqrt{35})}{{(36 - 6 \sqrt{35} - 6 \sqrt{35} + {(35^{\frac{1}{2}})}^{2} + 1)}^{2}}$

Step 13.2.3.1.5.2

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

$\frac{12 (7 - \sqrt{35}) (5 - \sqrt{35})}{{(36 - 6 \sqrt{35} - 6 \sqrt{35} + 35^{\frac{1}{2} \cdot 2} + 1)}^{2}}$

Step 13.2.3.1.5.3

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

$\frac{12 (7 - \sqrt{35}) (5 - \sqrt{35})}{{(36 - 6 \sqrt{35} - 6 \sqrt{35} + 35^{\frac{2}{2}} + 1)}^{2}}$

Step 13.2.3.1.5.4

Cancel the common factor of $2$ .

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Step 13.2.3.1.5.4.1

Cancel the common factor.

$\frac{12 (7 - \sqrt{35}) (5 - \sqrt{35})}{{(36 - 6 \sqrt{35} - 6 \sqrt{35} + 35^{\frac{2}{2}} + 1)}^{2}}$

Step 13.2.3.1.5.4.2

Rewrite the expression.

$\frac{12 (7 - \sqrt{35}) (5 - \sqrt{35})}{{(36 - 6 \sqrt{35} - 6 \sqrt{35} + 35^{1} + 1)}^{2}}$

Step 13.2.3.1.5.5

Evaluate the exponent.

$\frac{12 (7 - \sqrt{35}) (5 - \sqrt{35})}{{(36 - 6 \sqrt{35} - 6 \sqrt{35} + 35 + 1)}^{2}}$

Step 13.2.3.2

Add $36$ and $35$ .

$\frac{12 (7 - \sqrt{35}) (5 - \sqrt{35})}{{(71 - 6 \sqrt{35} - 6 \sqrt{35} + 1)}^{2}}$

Step 13.2.3.3

Subtract $6 \sqrt{35}$ from $- 6 \sqrt{35}$ .

$\frac{12 (7 - \sqrt{35}) (5 - \sqrt{35})}{{(71 - 12 \sqrt{35} + 1)}^{2}}$

Step 13.2.4

Add $71$ and $1$ .

$\frac{12 (7 - \sqrt{35}) (5 - \sqrt{35})}{{(72 - 12 \sqrt{35})}^{2}}$

Step 13.3

Group $5 - \sqrt{35}$ and $7 - \sqrt{35}$ together.

$\frac{12 ((5 - \sqrt{35}) (7 - \sqrt{35}))}{{(72 - 12 \sqrt{35})}^{2}}$

Step 13.4

Expand $(5 - \sqrt{35}) (7 - \sqrt{35})$ using the FOIL Method.

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Step 13.4.1

Apply the distributive property.

$\frac{12 (5 (7 - \sqrt{35}) - \sqrt{35} (7 - \sqrt{35}))}{{(72 - 12 \sqrt{35})}^{2}}$

Step 13.4.2

Apply the distributive property.

$\frac{12 (5 \cdot 7 + 5 (- \sqrt{35}) - \sqrt{35} (7 - \sqrt{35}))}{{(72 - 12 \sqrt{35})}^{2}}$

Step 13.4.3

Apply the distributive property.

$\frac{12 (5 \cdot 7 + 5 (- \sqrt{35}) - \sqrt{35} \cdot 7 - \sqrt{35} (- \sqrt{35}))}{{(72 - 12 \sqrt{35})}^{2}}$

Step 13.5

Simplify and combine like terms.

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Step 13.5.1

Simplify each term.

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Step 13.5.1.1

Multiply $5$ by $7$ .

$\frac{12 (35 + 5 (- \sqrt{35}) - \sqrt{35} \cdot 7 - \sqrt{35} (- \sqrt{35}))}{{(72 - 12 \sqrt{35})}^{2}}$

Step 13.5.1.2

Multiply $- 1$ by $5$ .

$\frac{12 (35 - 5 \sqrt{35} - \sqrt{35} \cdot 7 - \sqrt{35} (- \sqrt{35}))}{{(72 - 12 \sqrt{35})}^{2}}$

Step 13.5.1.3

Multiply $7$ by $- 1$ .

$\frac{12 (35 - 5 \sqrt{35} - 7 \sqrt{35} - \sqrt{35} (- \sqrt{35}))}{{(72 - 12 \sqrt{35})}^{2}}$

Step 13.5.1.4

Multiply $- \sqrt{35} (- \sqrt{35})$ .

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Step 13.5.1.4.1

Multiply $- 1$ by $- 1$ .

$\frac{12 (35 - 5 \sqrt{35} - 7 \sqrt{35} + 1 \sqrt{35} \sqrt{35})}{{(72 - 12 \sqrt{35})}^{2}}$

Step 13.5.1.4.2

Multiply $\sqrt{35}$ by $1$ .

$\frac{12 (35 - 5 \sqrt{35} - 7 \sqrt{35} + \sqrt{35} \sqrt{35})}{{(72 - 12 \sqrt{35})}^{2}}$

Step 13.5.1.4.3

Raise $\sqrt{35}$ to the power of $1$ .

$\frac{12 (35 - 5 \sqrt{35} - 7 \sqrt{35} + {\sqrt{35}}^{1} \sqrt{35})}{{(72 - 12 \sqrt{35})}^{2}}$

Step 13.5.1.4.4

Raise $\sqrt{35}$ to the power of $1$ .

$\frac{12 (35 - 5 \sqrt{35} - 7 \sqrt{35} + {\sqrt{35}}^{1} {\sqrt{35}}^{1})}{{(72 - 12 \sqrt{35})}^{2}}$

Step 13.5.1.4.5

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

$\frac{12 (35 - 5 \sqrt{35} - 7 \sqrt{35} + {\sqrt{35}}^{1 + 1})}{{(72 - 12 \sqrt{35})}^{2}}$

Step 13.5.1.4.6

Add $1$ and $1$ .

$\frac{12 (35 - 5 \sqrt{35} - 7 \sqrt{35} + {\sqrt{35}}^{2})}{{(72 - 12 \sqrt{35})}^{2}}$

Step 13.5.1.5

Rewrite ${\sqrt{35}}^{2}$ as $35$ .

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Step 13.5.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{35}$ as $35^{\frac{1}{2}}$ .

$\frac{12 (35 - 5 \sqrt{35} - 7 \sqrt{35} + {(35^{\frac{1}{2}})}^{2})}{{(72 - 12 \sqrt{35})}^{2}}$

Step 13.5.1.5.2

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

$\frac{12 (35 - 5 \sqrt{35} - 7 \sqrt{35} + 35^{\frac{1}{2} \cdot 2})}{{(72 - 12 \sqrt{35})}^{2}}$

Step 13.5.1.5.3

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

$\frac{12 (35 - 5 \sqrt{35} - 7 \sqrt{35} + 35^{\frac{2}{2}})}{{(72 - 12 \sqrt{35})}^{2}}$

Step 13.5.1.5.4

Cancel the common factor of $2$ .

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Step 13.5.1.5.4.1

Cancel the common factor.

$\frac{12 (35 - 5 \sqrt{35} - 7 \sqrt{35} + 35^{\frac{2}{2}})}{{(72 - 12 \sqrt{35})}^{2}}$

Step 13.5.1.5.4.2

Rewrite the expression.

$\frac{12 (35 - 5 \sqrt{35} - 7 \sqrt{35} + 35^{1})}{{(72 - 12 \sqrt{35})}^{2}}$

Step 13.5.1.5.5

Evaluate the exponent.

$\frac{12 (35 - 5 \sqrt{35} - 7 \sqrt{35} + 35)}{{(72 - 12 \sqrt{35})}^{2}}$

Step 13.5.2

Add $35$ and $35$ .

$\frac{12 (70 - 5 \sqrt{35} - 7 \sqrt{35})}{{(72 - 12 \sqrt{35})}^{2}}$

Step 13.5.3

Subtract $7 \sqrt{35}$ from $- 5 \sqrt{35}$ .

$\frac{12 (70 - 12 \sqrt{35})}{{(72 - 12 \sqrt{35})}^{2}}$

Step 13.6

Rewrite ${(72 - 12 \sqrt{35})}^{2}$ as $(72 - 12 \sqrt{35}) (72 - 12 \sqrt{35})$ .

$\frac{12 (70 - 12 \sqrt{35})}{(72 - 12 \sqrt{35}) (72 - 12 \sqrt{35})}$

Step 13.7

Expand $(72 - 12 \sqrt{35}) (72 - 12 \sqrt{35})$ using the FOIL Method.

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Step 13.7.1

Apply the distributive property.

$\frac{12 (70 - 12 \sqrt{35})}{72 (72 - 12 \sqrt{35}) - 12 \sqrt{35} (72 - 12 \sqrt{35})}$

Step 13.7.2

Apply the distributive property.

$\frac{12 (70 - 12 \sqrt{35})}{72 \cdot 72 + 72 (- 12 \sqrt{35}) - 12 \sqrt{35} (72 - 12 \sqrt{35})}$

Step 13.7.3

Apply the distributive property.

$\frac{12 (70 - 12 \sqrt{35})}{72 \cdot 72 + 72 (- 12 \sqrt{35}) - 12 \sqrt{35} \cdot 72 - 12 \sqrt{35} (- 12 \sqrt{35})}$

Step 13.8

Simplify and combine like terms.

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Step 13.8.1

Simplify each term.

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Step 13.8.1.1

Multiply $72$ by $72$ .

$\frac{12 (70 - 12 \sqrt{35})}{5184 + 72 (- 12 \sqrt{35}) - 12 \sqrt{35} \cdot 72 - 12 \sqrt{35} (- 12 \sqrt{35})}$

Step 13.8.1.2

Multiply $- 12$ by $72$ .

$\frac{12 (70 - 12 \sqrt{35})}{5184 - 864 \sqrt{35} - 12 \sqrt{35} \cdot 72 - 12 \sqrt{35} (- 12 \sqrt{35})}$

Step 13.8.1.3

Multiply $72$ by $- 12$ .

$\frac{12 (70 - 12 \sqrt{35})}{5184 - 864 \sqrt{35} - 864 \sqrt{35} - 12 \sqrt{35} (- 12 \sqrt{35})}$

Step 13.8.1.4

Multiply $- 12 \sqrt{35} (- 12 \sqrt{35})$ .

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Step 13.8.1.4.1

Multiply $- 12$ by $- 12$ .

$\frac{12 (70 - 12 \sqrt{35})}{5184 - 864 \sqrt{35} - 864 \sqrt{35} + 144 \sqrt{35} \sqrt{35}}$

Step 13.8.1.4.2

Raise $\sqrt{35}$ to the power of $1$ .

$\frac{12 (70 - 12 \sqrt{35})}{5184 - 864 \sqrt{35} - 864 \sqrt{35} + 144 ({\sqrt{35}}^{1} \sqrt{35})}$

Step 13.8.1.4.3

Raise $\sqrt{35}$ to the power of $1$ .

$\frac{12 (70 - 12 \sqrt{35})}{5184 - 864 \sqrt{35} - 864 \sqrt{35} + 144 ({\sqrt{35}}^{1} {\sqrt{35}}^{1})}$

Step 13.8.1.4.4

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

$\frac{12 (70 - 12 \sqrt{35})}{5184 - 864 \sqrt{35} - 864 \sqrt{35} + 144 {\sqrt{35}}^{1 + 1}}$

Step 13.8.1.4.5

Add $1$ and $1$ .

$\frac{12 (70 - 12 \sqrt{35})}{5184 - 864 \sqrt{35} - 864 \sqrt{35} + 144 {\sqrt{35}}^{2}}$

Step 13.8.1.5

Rewrite ${\sqrt{35}}^{2}$ as $35$ .

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Step 13.8.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{35}$ as $35^{\frac{1}{2}}$ .

$\frac{12 (70 - 12 \sqrt{35})}{5184 - 864 \sqrt{35} - 864 \sqrt{35} + 144 {(35^{\frac{1}{2}})}^{2}}$

Step 13.8.1.5.2

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

$\frac{12 (70 - 12 \sqrt{35})}{5184 - 864 \sqrt{35} - 864 \sqrt{35} + 144 \cdot 35^{\frac{1}{2} \cdot 2}}$

Step 13.8.1.5.3

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

$\frac{12 (70 - 12 \sqrt{35})}{5184 - 864 \sqrt{35} - 864 \sqrt{35} + 144 \cdot 35^{\frac{2}{2}}}$

Step 13.8.1.5.4

Cancel the common factor of $2$ .

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Step 13.8.1.5.4.1

Cancel the common factor.

$\frac{12 (70 - 12 \sqrt{35})}{5184 - 864 \sqrt{35} - 864 \sqrt{35} + 144 \cdot 35^{\frac{2}{2}}}$

Step 13.8.1.5.4.2

Rewrite the expression.

$\frac{12 (70 - 12 \sqrt{35})}{5184 - 864 \sqrt{35} - 864 \sqrt{35} + 144 \cdot 35^{1}}$

Step 13.8.1.5.5

Evaluate the exponent.

$\frac{12 (70 - 12 \sqrt{35})}{5184 - 864 \sqrt{35} - 864 \sqrt{35} + 144 \cdot 35}$

Step 13.8.1.6

Multiply $144$ by $35$ .

$\frac{12 (70 - 12 \sqrt{35})}{5184 - 864 \sqrt{35} - 864 \sqrt{35} + 5040}$

Step 13.8.2

Add $5184$ and $5040$ .

$\frac{12 (70 - 12 \sqrt{35})}{10224 - 864 \sqrt{35} - 864 \sqrt{35}}$

Step 13.8.3

Subtract $864 \sqrt{35}$ from $- 864 \sqrt{35}$ .

$\frac{12 (70 - 12 \sqrt{35})}{10224 - 1728 \sqrt{35}}$

Step 13.9

Cancel the common factors.

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Step 13.9.1

Factor $12$ out of $10224$ .

$\frac{12 (70 - 12 \sqrt{35})}{12 \cdot 852 - 1728 \sqrt{35}}$

Step 13.9.2

Factor $12$ out of $- 1728 \sqrt{35}$ .

$\frac{12 (70 - 12 \sqrt{35})}{12 \cdot 852 + 12 (- 144 \sqrt{35})}$

Step 13.9.3

Factor $12$ out of $12 (852) + 12 (- 144 \sqrt{35})$ .

$\frac{12 (70 - 12 \sqrt{35})}{12 (852 - 144 \sqrt{35})}$

Step 13.9.4

Cancel the common factor.

$\frac{12 (70 - 12 \sqrt{35})}{12 (852 - 144 \sqrt{35})}$

Step 13.9.5

Rewrite the expression.

$\frac{70 - 12 \sqrt{35}}{852 - 144 \sqrt{35}}$

Step 13.10

Cancel the common factor of $70 - 12 \sqrt{35}$ and $852 - 144 \sqrt{35}$ .

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Step 13.10.1

Factor $2$ out of $70$ .

$\frac{2 (35) - 12 \sqrt{35}}{852 - 144 \sqrt{35}}$

Step 13.10.2

Factor $2$ out of $- 12 \sqrt{35}$ .

$\frac{2 (35) + 2 (- 6 \sqrt{35})}{852 - 144 \sqrt{35}}$

Step 13.10.3

Factor $2$ out of $2 (35) + 2 (- 6 \sqrt{35})$ .

$\frac{2 (35 - 6 \sqrt{35})}{852 - 144 \sqrt{35}}$

Step 13.10.4

Cancel the common factors.

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Step 13.10.4.1

Factor $2$ out of $852$ .

$\frac{2 (35 - 6 \sqrt{35})}{2 \cdot 426 - 144 \sqrt{35}}$

Step 13.10.4.2

Factor $2$ out of $- 144 \sqrt{35}$ .

$\frac{2 (35 - 6 \sqrt{35})}{2 \cdot 426 + 2 (- 72 \sqrt{35})}$

Step 13.10.4.3

Factor $2$ out of $2 (426) + 2 (- 72 \sqrt{35})$ .

$\frac{2 (35 - 6 \sqrt{35})}{2 (426 - 72 \sqrt{35})}$

Step 13.10.4.4

Cancel the common factor.

$\frac{2 (35 - 6 \sqrt{35})}{2 (426 - 72 \sqrt{35})}$

Step 13.10.4.5

Rewrite the expression.

$\frac{35 - 6 \sqrt{35}}{426 - 72 \sqrt{35}}$

Step 13.11

Multiply $\frac{35 - 6 \sqrt{35}}{426 - 72 \sqrt{35}}$ by $\frac{426 + 72 \sqrt{35}}{426 + 72 \sqrt{35}}$ .

$\frac{35 - 6 \sqrt{35}}{426 - 72 \sqrt{35}} \cdot \frac{426 + 72 \sqrt{35}}{426 + 72 \sqrt{35}}$

Step 13.12

Simplify terms.

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Step 13.12.1

Multiply $\frac{35 - 6 \sqrt{35}}{426 - 72 \sqrt{35}}$ by $\frac{426 + 72 \sqrt{35}}{426 + 72 \sqrt{35}}$ .

$\frac{(35 - 6 \sqrt{35}) (426 + 72 \sqrt{35})}{(426 - 72 \sqrt{35}) (426 + 72 \sqrt{35})}$

Step 13.12.2

Expand the denominator using the FOIL method.

$\frac{(35 - 6 \sqrt{35}) (426 + 72 \sqrt{35})}{181476 + 30672 \sqrt{35} - 30672 \sqrt{35} - 5184 {\sqrt{35}}^{2}}$

Step 13.12.3

Simplify.

$\frac{(35 - 6 \sqrt{35}) (426 + 72 \sqrt{35})}{36}$

Step 13.12.4

Cancel the common factor of $426 + 72 \sqrt{35}$ and $36$ .

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Step 13.12.4.1

Factor $6$ out of $(35 - 6 \sqrt{35}) (426 + 72 \sqrt{35})$ .

$\frac{6 ((35 - 6 \sqrt{35}) (71 + 12 \sqrt{35}))}{36}$

Step 13.12.4.2

Cancel the common factors.

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Step 13.12.4.2.1

Factor $6$ out of $36$ .

$\frac{6 ((35 - 6 \sqrt{35}) (71 + 12 \sqrt{35}))}{6 (6)}$

Step 13.12.4.2.2

Cancel the common factor.

$\frac{6 ((35 - 6 \sqrt{35}) (71 + 12 \sqrt{35}))}{6 \cdot 6}$

Step 13.12.4.2.3

Rewrite the expression.

$\frac{(35 - 6 \sqrt{35}) (71 + 12 \sqrt{35})}{6}$

Step 13.13

Expand $(35 - 6 \sqrt{35}) (71 + 12 \sqrt{35})$ using the FOIL Method.

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Step 13.13.1

Apply the distributive property.

$\frac{35 (71 + 12 \sqrt{35}) - 6 \sqrt{35} (71 + 12 \sqrt{35})}{6}$

Step 13.13.2

Apply the distributive property.

$\frac{35 \cdot 71 + 35 (12 \sqrt{35}) - 6 \sqrt{35} (71 + 12 \sqrt{35})}{6}$

Step 13.13.3

Apply the distributive property.

$\frac{35 \cdot 71 + 35 (12 \sqrt{35}) - 6 \sqrt{35} \cdot 71 - 6 \sqrt{35} (12 \sqrt{35})}{6}$

Step 13.14

Simplify and combine like terms.

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Step 13.14.1

Simplify each term.

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Step 13.14.1.1

Multiply $35$ by $71$ .

$\frac{2485 + 35 (12 \sqrt{35}) - 6 \sqrt{35} \cdot 71 - 6 \sqrt{35} (12 \sqrt{35})}{6}$

Step 13.14.1.2

Multiply $12$ by $35$ .

$\frac{2485 + 420 \sqrt{35} - 6 \sqrt{35} \cdot 71 - 6 \sqrt{35} (12 \sqrt{35})}{6}$

Step 13.14.1.3

Multiply $71$ by $- 6$ .

$\frac{2485 + 420 \sqrt{35} - 426 \sqrt{35} - 6 \sqrt{35} (12 \sqrt{35})}{6}$

Step 13.14.1.4

Multiply $- 6 \sqrt{35} (12 \sqrt{35})$ .

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Step 13.14.1.4.1

Multiply $12$ by $- 6$ .

$\frac{2485 + 420 \sqrt{35} - 426 \sqrt{35} - 72 \sqrt{35} \sqrt{35}}{6}$

Step 13.14.1.4.2

Raise $\sqrt{35}$ to the power of $1$ .

$\frac{2485 + 420 \sqrt{35} - 426 \sqrt{35} - 72 ({\sqrt{35}}^{1} \sqrt{35})}{6}$

Step 13.14.1.4.3

Raise $\sqrt{35}$ to the power of $1$ .

$\frac{2485 + 420 \sqrt{35} - 426 \sqrt{35} - 72 ({\sqrt{35}}^{1} {\sqrt{35}}^{1})}{6}$

Step 13.14.1.4.4

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

$\frac{2485 + 420 \sqrt{35} - 426 \sqrt{35} - 72 {\sqrt{35}}^{1 + 1}}{6}$

Step 13.14.1.4.5

Add $1$ and $1$ .

$\frac{2485 + 420 \sqrt{35} - 426 \sqrt{35} - 72 {\sqrt{35}}^{2}}{6}$

Step 13.14.1.5

Rewrite ${\sqrt{35}}^{2}$ as $35$ .

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Step 13.14.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{35}$ as $35^{\frac{1}{2}}$ .

$\frac{2485 + 420 \sqrt{35} - 426 \sqrt{35} - 72 {(35^{\frac{1}{2}})}^{2}}{6}$

Step 13.14.1.5.2

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

$\frac{2485 + 420 \sqrt{35} - 426 \sqrt{35} - 72 \cdot 35^{\frac{1}{2} \cdot 2}}{6}$

Step 13.14.1.5.3

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

$\frac{2485 + 420 \sqrt{35} - 426 \sqrt{35} - 72 \cdot 35^{\frac{2}{2}}}{6}$

Step 13.14.1.5.4

Cancel the common factor of $2$ .

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Step 13.14.1.5.4.1

Cancel the common factor.

$\frac{2485 + 420 \sqrt{35} - 426 \sqrt{35} - 72 \cdot 35^{\frac{2}{2}}}{6}$

Step 13.14.1.5.4.2

Rewrite the expression.

$\frac{2485 + 420 \sqrt{35} - 426 \sqrt{35} - 72 \cdot 35^{1}}{6}$

Step 13.14.1.5.5

Evaluate the exponent.

$\frac{2485 + 420 \sqrt{35} - 426 \sqrt{35} - 72 \cdot 35}{6}$

Step 13.14.1.6

Multiply $- 72$ by $35$ .

$\frac{2485 + 420 \sqrt{35} - 426 \sqrt{35} - 2520}{6}$

Step 13.14.2

Subtract $2520$ from $2485$ .

$\frac{- 35 + 420 \sqrt{35} - 426 \sqrt{35}}{6}$

Step 13.14.3

Subtract $426 \sqrt{35}$ from $420 \sqrt{35}$ .

$\frac{- 35 - 6 \sqrt{35}}{6}$

Step 13.15

Rewrite $- 35$ as $- 1 (35)$ .

$\frac{- 1 (35) - 6 \sqrt{35}}{6}$

Step 13.16

Factor $- 1$ out of $- 6 \sqrt{35}$ .

$\frac{- 1 (35) - (6 \sqrt{35})}{6}$

Step 13.17

Factor $- 1$ out of $- 1 (35) - (6 \sqrt{35})$ .

$\frac{- 1 (35 + 6 \sqrt{35})}{6}$

Step 13.18

Move the negative in front of the fraction.

$- \frac{35 + 6 \sqrt{35}}{6}$

Step 14

$x = 6 - \sqrt{35}$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = 6 - \sqrt{35}$ is a local maximum

Step 15

Find the y-value when $x = 6 - \sqrt{35}$ .

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Step 15.1

Replace the variable $x$ with $6 - \sqrt{35}$ in the expression.

$f (6 - \sqrt{35}) = (6 - \sqrt{35}) - 6 \ln ({(6 - \sqrt{35})}^{2} + 1)$

Step 15.2

Simplify the result.

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Step 15.2.1

Simplify $- 6 \ln ({(6 - \sqrt{35})}^{2} + 1)$ by moving $6$ inside the logarithm.

$f (6 - \sqrt{35}) = 6 - \sqrt{35} - \ln ({({(6 - \sqrt{35})}^{2} + 1)}^{6})$

Step 15.2.2

The final answer is $6 - \sqrt{35} - \ln ({({(6 - \sqrt{35})}^{2} + 1)}^{6})$ .

$y = 6 - \sqrt{35} - \ln ({({(6 - \sqrt{35})}^{2} + 1)}^{6})$

Step 16

These are the local extrema for $f (x) = x - 6 \ln (x^{2} + 1)$ .

$(6 + \sqrt{35}, 6 + \sqrt{35} - 6 \ln ({(6 + \sqrt{35})}^{2} + 1))$ is a local minima

$(6 - \sqrt{35}, 6 - \sqrt{35} - \ln ({({(6 - \sqrt{35})}^{2} + 1)}^{6}))$ is a local maxima

Step 17