Calculus Examples

$y = \frac{(1 + x)}{(1 + x^{2})}$

Step 1

Write $y = \frac{(1 + x)}{(1 + x^{2})}$ as a function.

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

Step 2

Find the second derivative.

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Step 2.1

Find the first derivative.

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Step 2.1.1

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

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

Step 2.1.2

Differentiate.

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Step 2.1.2.1

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

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

Step 2.1.2.2

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

$\frac{(1 + x^{2}) (0 + \frac{d}{d x} [x]) - (1 + x) \frac{d}{d x} [1 + x^{2}]}{{(1 + x^{2})}^{2}}$

Step 2.1.2.3

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

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

Step 2.1.2.4

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

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

Step 2.1.2.5

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

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

Step 2.1.2.6

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

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

Step 2.1.2.7

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

$\frac{1 + x^{2} - (1 + x) (0 + \frac{d}{d x} [x^{2}])}{{(1 + x^{2})}^{2}}$

Step 2.1.2.8

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

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

Step 2.1.2.9

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

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

Step 2.1.2.10

Multiply $2$ by $- 1$ .

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

Step 2.1.3

Simplify.

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Step 2.1.3.1

Apply the distributive property.

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

Step 2.1.3.2

Apply the distributive property.

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

Step 2.1.3.3

Simplify the numerator.

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Step 2.1.3.3.1

Simplify each term.

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Step 2.1.3.3.1.1

Multiply $- 2$ by $1$ .

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

Step 2.1.3.3.1.2

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

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Step 2.1.3.3.1.2.1

Move $x$ .

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

Step 2.1.3.3.1.2.2

Multiply $x$ by $x$ .

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

Step 2.1.3.3.2

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

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

Step 2.1.3.4

Reorder terms.

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

Step 2.1.3.5

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

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

Step 2.1.3.6

Factor $- 1$ out of $- 2 x$ .

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

Step 2.1.3.7

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

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

Step 2.1.3.8

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

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

Step 2.1.3.9

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

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

Step 2.1.3.10

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

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

Step 2.1.3.11

Move the negative in front of the fraction.

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

Step 2.2

Find the second derivative.

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Step 2.2.1

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

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

Step 2.2.2

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

Step 2.2.3

Differentiate.

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Step 2.2.3.1

Multiply the exponents in ${({(x^{2} + 1)}^{2})}^{2}$ .

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Step 2.2.3.1.1

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

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

Step 2.2.3.1.2

Multiply $2$ by $2$ .

$- \frac{{(x^{2} + 1)}^{2} \frac{d}{d x} [x^{2} + 2 x - 1] - (x^{2} + 2 x - 1) \frac{d}{d x} [{(x^{2} + 1)}^{2}]}{{(x^{2} + 1)}^{4}} + \frac{x^{2} + 2 x - 1}{{(x^{2} + 1)}^{2}} \frac{d}{d x} [- 1]$

Step 2.2.3.2

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

$- \frac{{(x^{2} + 1)}^{2} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [2 x] + \frac{d}{d x} [- 1]) - (x^{2} + 2 x - 1) \frac{d}{d x} [{(x^{2} + 1)}^{2}]}{{(x^{2} + 1)}^{4}} + \frac{x^{2} + 2 x - 1}{{(x^{2} + 1)}^{2}} \frac{d}{d x} [- 1]$

Step 2.2.3.3

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

$- \frac{{(x^{2} + 1)}^{2} (2 x + \frac{d}{d x} [2 x] + \frac{d}{d x} [- 1]) - (x^{2} + 2 x - 1) \frac{d}{d x} [{(x^{2} + 1)}^{2}]}{{(x^{2} + 1)}^{4}} + \frac{x^{2} + 2 x - 1}{{(x^{2} + 1)}^{2}} \frac{d}{d x} [- 1]$

Step 2.2.3.4

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

$- \frac{{(x^{2} + 1)}^{2} (2 x + 2 \frac{d}{d x} [x] + \frac{d}{d x} [- 1]) - (x^{2} + 2 x - 1) \frac{d}{d x} [{(x^{2} + 1)}^{2}]}{{(x^{2} + 1)}^{4}} + \frac{x^{2} + 2 x - 1}{{(x^{2} + 1)}^{2}} \frac{d}{d x} [- 1]$

Step 2.2.3.5

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

$- \frac{{(x^{2} + 1)}^{2} (2 x + 2 \cdot 1 + \frac{d}{d x} [- 1]) - (x^{2} + 2 x - 1) \frac{d}{d x} [{(x^{2} + 1)}^{2}]}{{(x^{2} + 1)}^{4}} + \frac{x^{2} + 2 x - 1}{{(x^{2} + 1)}^{2}} \frac{d}{d x} [- 1]$

Step 2.2.3.6

Multiply $2$ by $1$ .

$- \frac{{(x^{2} + 1)}^{2} (2 x + 2 + \frac{d}{d x} [- 1]) - (x^{2} + 2 x - 1) \frac{d}{d x} [{(x^{2} + 1)}^{2}]}{{(x^{2} + 1)}^{4}} + \frac{x^{2} + 2 x - 1}{{(x^{2} + 1)}^{2}} \frac{d}{d x} [- 1]$

Step 2.2.3.7

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

$- \frac{{(x^{2} + 1)}^{2} (2 x + 2 + 0) - (x^{2} + 2 x - 1) \frac{d}{d x} [{(x^{2} + 1)}^{2}]}{{(x^{2} + 1)}^{4}} + \frac{x^{2} + 2 x - 1}{{(x^{2} + 1)}^{2}} \frac{d}{d x} [- 1]$

Step 2.2.3.8

Add $2 x + 2$ and $0$ .

$- \frac{{(x^{2} + 1)}^{2} (2 x + 2) - (x^{2} + 2 x - 1) \frac{d}{d x} [{(x^{2} + 1)}^{2}]}{{(x^{2} + 1)}^{4}} + \frac{x^{2} + 2 x - 1}{{(x^{2} + 1)}^{2}} \frac{d}{d x} [- 1]$

Step 2.2.4

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

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Step 2.2.4.1

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

$- \frac{{(x^{2} + 1)}^{2} (2 x + 2) - (x^{2} + 2 x - 1) (\frac{d}{d u} [u^{2}] \frac{d}{d x} [x^{2} + 1])}{{(x^{2} + 1)}^{4}} + \frac{x^{2} + 2 x - 1}{{(x^{2} + 1)}^{2}} \frac{d}{d x} [- 1]$

Step 2.2.4.2

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

$- \frac{{(x^{2} + 1)}^{2} (2 x + 2) - (x^{2} + 2 x - 1) (2 u \frac{d}{d x} [x^{2} + 1])}{{(x^{2} + 1)}^{4}} + \frac{x^{2} + 2 x - 1}{{(x^{2} + 1)}^{2}} \frac{d}{d x} [- 1]$

Step 2.2.4.3

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

$- \frac{{(x^{2} + 1)}^{2} (2 x + 2) - (x^{2} + 2 x - 1) (2 (x^{2} + 1) \frac{d}{d x} [x^{2} + 1])}{{(x^{2} + 1)}^{4}} + \frac{x^{2} + 2 x - 1}{{(x^{2} + 1)}^{2}} \frac{d}{d x} [- 1]$

Step 2.2.5

Simplify with factoring out.

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Step 2.2.5.1

Multiply $2$ by $- 1$ .

$- \frac{{(x^{2} + 1)}^{2} (2 x + 2) - 2 (x^{2} + 2 x - 1) ((x^{2} + 1) \frac{d}{d x} [x^{2} + 1])}{{(x^{2} + 1)}^{4}} + \frac{x^{2} + 2 x - 1}{{(x^{2} + 1)}^{2}} \frac{d}{d x} [- 1]$

Step 2.2.5.2

Factor $x^{2} + 1$ out of ${(x^{2} + 1)}^{2} (2 x + 2) - 2 (x^{2} + 2 x - 1) ((x^{2} + 1) \frac{d}{d x} [x^{2} + 1])$ .

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Step 2.2.5.2.1

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

$- \frac{(x^{2} + 1) ((x^{2} + 1) (2 x + 2)) - 2 (x^{2} + 2 x - 1) ((x^{2} + 1) \frac{d}{d x} [x^{2} + 1])}{{(x^{2} + 1)}^{4}} + \frac{x^{2} + 2 x - 1}{{(x^{2} + 1)}^{2}} \frac{d}{d x} [- 1]$

Step 2.2.5.2.2

Factor $x^{2} + 1$ out of $- 2 (x^{2} + 2 x - 1) ((x^{2} + 1) \frac{d}{d x} [x^{2} + 1])$ .

$- \frac{(x^{2} + 1) ((x^{2} + 1) (2 x + 2)) + (x^{2} + 1) (- 2 (x^{2} + 2 x - 1) (\frac{d}{d x} [x^{2} + 1]))}{{(x^{2} + 1)}^{4}} + \frac{x^{2} + 2 x - 1}{{(x^{2} + 1)}^{2}} \frac{d}{d x} [- 1]$

Step 2.2.5.2.3

Factor $x^{2} + 1$ out of $(x^{2} + 1) ((x^{2} + 1) (2 x + 2)) + (x^{2} + 1) (- 2 (x^{2} + 2 x - 1) (\frac{d}{d x} [x^{2} + 1]))$ .

$- \frac{(x^{2} + 1) ((x^{2} + 1) (2 x + 2) - 2 (x^{2} + 2 x - 1) (\frac{d}{d x} [x^{2} + 1]))}{{(x^{2} + 1)}^{4}} + \frac{x^{2} + 2 x - 1}{{(x^{2} + 1)}^{2}} \frac{d}{d x} [- 1]$

Step 2.2.6

Cancel the common factors.

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Step 2.2.6.1

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

$- \frac{(x^{2} + 1) ((x^{2} + 1) (2 x + 2) - 2 (x^{2} + 2 x - 1) (\frac{d}{d x} [x^{2} + 1]))}{(x^{2} + 1) {(x^{2} + 1)}^{3}} + \frac{x^{2} + 2 x - 1}{{(x^{2} + 1)}^{2}} \frac{d}{d x} [- 1]$

Step 2.2.6.2

Cancel the common factor.

$- \frac{(x^{2} + 1) ((x^{2} + 1) (2 x + 2) - 2 (x^{2} + 2 x - 1) (\frac{d}{d x} [x^{2} + 1]))}{(x^{2} + 1) {(x^{2} + 1)}^{3}} + \frac{x^{2} + 2 x - 1}{{(x^{2} + 1)}^{2}} \frac{d}{d x} [- 1]$

Step 2.2.6.3

Rewrite the expression.

$- \frac{(x^{2} + 1) (2 x + 2) - 2 (x^{2} + 2 x - 1) (\frac{d}{d x} [x^{2} + 1])}{{(x^{2} + 1)}^{3}} + \frac{x^{2} + 2 x - 1}{{(x^{2} + 1)}^{2}} \frac{d}{d x} [- 1]$

Step 2.2.7

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

$- \frac{(x^{2} + 1) (2 x + 2) - 2 (x^{2} + 2 x - 1) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [1])}{{(x^{2} + 1)}^{3}} + \frac{x^{2} + 2 x - 1}{{(x^{2} + 1)}^{2}} \frac{d}{d x} [- 1]$

Step 2.2.8

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

$- \frac{(x^{2} + 1) (2 x + 2) - 2 (x^{2} + 2 x - 1) (2 x + \frac{d}{d x} [1])}{{(x^{2} + 1)}^{3}} + \frac{x^{2} + 2 x - 1}{{(x^{2} + 1)}^{2}} \frac{d}{d x} [- 1]$

Step 2.2.9

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

$- \frac{(x^{2} + 1) (2 x + 2) - 2 (x^{2} + 2 x - 1) (2 x + 0)}{{(x^{2} + 1)}^{3}} + \frac{x^{2} + 2 x - 1}{{(x^{2} + 1)}^{2}} \frac{d}{d x} [- 1]$

Step 2.2.10

Simplify the expression.

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Step 2.2.10.1

Add $2 x$ and $0$ .

$- \frac{(x^{2} + 1) (2 x + 2) - 2 (x^{2} + 2 x - 1) (2 x)}{{(x^{2} + 1)}^{3}} + \frac{x^{2} + 2 x - 1}{{(x^{2} + 1)}^{2}} \frac{d}{d x} [- 1]$

Step 2.2.10.2

Multiply $2$ by $- 2$ .

$- \frac{(x^{2} + 1) (2 x + 2) - 4 (x^{2} + 2 x - 1) x}{{(x^{2} + 1)}^{3}} + \frac{x^{2} + 2 x - 1}{{(x^{2} + 1)}^{2}} \frac{d}{d x} [- 1]$

Step 2.2.11

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

$- \frac{(x^{2} + 1) (2 x + 2) - 4 (x^{2} + 2 x - 1) x}{{(x^{2} + 1)}^{3}} + \frac{x^{2} + 2 x - 1}{{(x^{2} + 1)}^{2}} \cdot 0$

Step 2.2.12

Simplify the expression.

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Step 2.2.12.1

Multiply $\frac{x^{2} + 2 x - 1}{{(x^{2} + 1)}^{2}}$ by $0$ .

$- \frac{(x^{2} + 1) (2 x + 2) - 4 (x^{2} + 2 x - 1) x}{{(x^{2} + 1)}^{3}} + 0$

Step 2.2.12.2

Add $- \frac{(x^{2} + 1) (2 x + 2) - 4 (x^{2} + 2 x - 1) x}{{(x^{2} + 1)}^{3}}$ and $0$ .

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

Step 2.2.13

Simplify.

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Step 2.2.13.1

Apply the distributive property.

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

Step 2.2.13.2

Apply the distributive property.

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

Step 2.2.13.3

Simplify the numerator.

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Step 2.2.13.3.1

Simplify each term.

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Step 2.2.13.3.1.1

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

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Step 2.2.13.3.1.1.1

Apply the distributive property.

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

Step 2.2.13.3.1.1.2

Apply the distributive property.

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

Step 2.2.13.3.1.1.3

Apply the distributive property.

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

Step 2.2.13.3.1.2

Simplify each term.

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Step 2.2.13.3.1.2.1

Rewrite using the commutative property of multiplication.

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

Step 2.2.13.3.1.2.2

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

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Step 2.2.13.3.1.2.2.1

Move $x$ .

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

Step 2.2.13.3.1.2.2.2

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

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Step 2.2.13.3.1.2.2.2.1

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

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

Step 2.2.13.3.1.2.2.2.2

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

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

Step 2.2.13.3.1.2.2.3

Add $1$ and $2$ .

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

Step 2.2.13.3.1.2.3

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

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

Step 2.2.13.3.1.2.4

Multiply $2 x$ by $1$ .

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

Step 2.2.13.3.1.2.5

Multiply $2$ by $1$ .

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

Step 2.2.13.3.1.3

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

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Step 2.2.13.3.1.3.1

Move $x$ .

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

Step 2.2.13.3.1.3.2

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

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Step 2.2.13.3.1.3.2.1

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

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

Step 2.2.13.3.1.3.2.2

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

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

Step 2.2.13.3.1.3.3

Add $1$ and $2$ .

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

Step 2.2.13.3.1.4

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

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Step 2.2.13.3.1.4.1

Move $x$ .

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

Step 2.2.13.3.1.4.2

Multiply $x$ by $x$ .

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

Step 2.2.13.3.1.5

Multiply $2$ by $- 4$ .

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

Step 2.2.13.3.1.6

Multiply $- 4$ by $- 1$ .

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

Step 2.2.13.3.2

Subtract $4 x^{3}$ from $2 x^{3}$ .

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

Step 2.2.13.3.3

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

$- \frac{- 2 x^{3} - 6 x^{2} + 2 x + 2 + 4 x}{{(x^{2} + 1)}^{3}}$

Step 2.2.13.3.4

Add $2 x$ and $4 x$ .

$- \frac{- 2 x^{3} - 6 x^{2} + 6 x + 2}{{(x^{2} + 1)}^{3}}$

Step 2.2.13.4

Factor $2$ out of $- 2 x^{3} - 6 x^{2} + 6 x + 2$ .

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Step 2.2.13.4.1

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

$- \frac{2 (- x^{3}) - 6 x^{2} + 6 x + 2}{{(x^{2} + 1)}^{3}}$

Step 2.2.13.4.2

Factor $2$ out of $- 6 x^{2}$ .

$- \frac{2 (- x^{3}) + 2 (- 3 x^{2}) + 6 x + 2}{{(x^{2} + 1)}^{3}}$

Step 2.2.13.4.3

Factor $2$ out of $6 x$ .

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

Step 2.2.13.4.4

Factor $2$ out of $2$ .

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

Step 2.2.13.4.5

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

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

Step 2.2.13.4.6

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

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

Step 2.2.13.4.7

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

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

Step 2.2.13.5

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

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

Step 2.2.13.6

Factor $- 1$ out of $- 3 x^{2}$ .

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

Step 2.2.13.7

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

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

Step 2.2.13.8

Factor $- 1$ out of $3 x$ .

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

Step 2.2.13.9

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

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

Step 2.2.13.10

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

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

Step 2.2.13.11

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

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

Step 2.2.13.12

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

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

Step 2.2.13.13

Move the negative in front of the fraction.

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

Step 2.2.13.14

Multiply $- 1$ by $- 1$ .

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

Step 2.2.13.15

Multiply $\frac{2 (x^{3} + 3 x^{2} - 3 x - 1)}{{(x^{2} + 1)}^{3}}$ by $1$ .

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

Step 2.3

The second derivative of $f (x)$ with respect to $x$ is $\frac{2 (x^{3} + 3 x^{2} - 3 x - 1)}{{(x^{2} + 1)}^{3}}$ .

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

Step 3

Set the second derivative equal to $0$ then solve the equation $\frac{2 (x^{3} + 3 x^{2} - 3 x - 1)}{{(x^{2} + 1)}^{3}} = 0$ .

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Step 3.1

Set the second derivative equal to $0$ .

$\frac{2 (x^{3} + 3 x^{2} - 3 x - 1)}{{(x^{2} + 1)}^{3}} = 0$

Step 3.2

Set the numerator equal to zero.

$2 (x^{3} + 3 x^{2} - 3 x - 1) = 0$

Step 3.3

Solve the equation for $x$ .

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Step 3.3.1

Divide each term in $2 (x^{3} + 3 x^{2} - 3 x - 1) = 0$ by $2$ and simplify.

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Step 3.3.1.1

Divide each term in $2 (x^{3} + 3 x^{2} - 3 x - 1) = 0$ by $2$ .

$\frac{2 (x^{3} + 3 x^{2} - 3 x - 1)}{2} = \frac{0}{2}$

Step 3.3.1.2

Simplify the left side.

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Step 3.3.1.2.1

Cancel the common factor of $2$ .

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Step 3.3.1.2.1.1

Cancel the common factor.

$\frac{2 (x^{3} + 3 x^{2} - 3 x - 1)}{2} = \frac{0}{2}$

Step 3.3.1.2.1.2

Divide $x^{3} + 3 x^{2} - 3 x - 1$ by $1$ .

$x^{3} + 3 x^{2} - 3 x - 1 = \frac{0}{2}$

Step 3.3.1.3

Simplify the right side.

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Step 3.3.1.3.1

Divide $0$ by $2$ .

$x^{3} + 3 x^{2} - 3 x - 1 = 0$

Step 3.3.2

Factor the left side of the equation.

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Step 3.3.2.1

Regroup terms.

$x^{3} - 1 + 3 x^{2} - 3 x = 0$

Step 3.3.2.2

Rewrite $1$ as $1^{3}$ .

$x^{3} - 1^{3} + 3 x^{2} - 3 x = 0$

Step 3.3.2.3

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = x$ and $b = 1$ .

$(x - 1) (x^{2} + x \cdot 1 + 1^{2}) + 3 x^{2} - 3 x = 0$

Step 3.3.2.4

Simplify.

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Step 3.3.2.4.1

Multiply $x$ by $1$ .

$(x - 1) (x^{2} + x + 1^{2}) + 3 x^{2} - 3 x = 0$

Step 3.3.2.4.2

One to any power is one.

$(x - 1) (x^{2} + x + 1) + 3 x^{2} - 3 x = 0$

Step 3.3.2.5

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

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Step 3.3.2.5.1

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

$(x - 1) (x^{2} + x + 1) + 3 x (x) - 3 x = 0$

Step 3.3.2.5.2

Factor $3 x$ out of $- 3 x$ .

$(x - 1) (x^{2} + x + 1) + 3 x (x) + 3 x (- 1) = 0$

Step 3.3.2.5.3

Factor $3 x$ out of $3 x (x) + 3 x (- 1)$ .

$(x - 1) (x^{2} + x + 1) + 3 x (x - 1) = 0$

Step 3.3.2.6

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

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Step 3.3.2.6.1

Factor $x - 1$ out of $3 x (x - 1)$ .

$(x - 1) (x^{2} + x + 1) + (x - 1) (3 x) = 0$

Step 3.3.2.6.2

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

$(x - 1) (x^{2} + x + 1 + 3 x) = 0$

Step 3.3.2.7

Add $x$ and $3 x$ .

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

Step 3.3.3

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

$x - 1 = 0$

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

Step 3.3.4

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

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Step 3.3.4.1

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

$x - 1 = 0$

Step 3.3.4.2

Add $1$ to both sides of the equation.

$x = 1$

Step 3.3.5

Set $x^{2} + 4 x + 1$ equal to $0$ and solve for $x$ .

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Step 3.3.5.1

Set $x^{2} + 4 x + 1$ equal to $0$ .

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

Step 3.3.5.2

Solve $x^{2} + 4 x + 1 = 0$ for $x$ .

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Step 3.3.5.2.1

Use the quadratic formula to find the solutions.

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

Step 3.3.5.2.2

Substitute the values $a = 1$ , $b = 4$ , and $c = 1$ into the quadratic formula and solve for $x$ .

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

Step 3.3.5.2.3

Simplify.

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Step 3.3.5.2.3.1

Simplify the numerator.

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Step 3.3.5.2.3.1.1

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

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Step 3.3.5.2.3.1.2

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

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Step 3.3.5.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1}}{2 \cdot 1}$

Step 3.3.5.2.3.1.2.2

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4}}{2 \cdot 1}$

Step 3.3.5.2.3.1.3

Subtract $4$ from $16$ .

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

Step 3.3.5.2.3.1.4

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 3.3.5.2.3.1.4.1

Factor $4$ out of $12$ .

$x = \frac{- 4 \pm \sqrt{4 (3)}}{2 \cdot 1}$

Step 3.3.5.2.3.1.4.2

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

$x = \frac{- 4 \pm \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

Step 3.3.5.2.3.1.5

Pull terms out from under the radical.

$x = \frac{- 4 \pm 2 \sqrt{3}}{2 \cdot 1}$

Step 3.3.5.2.3.2

Multiply $2$ by $1$ .

$x = \frac{- 4 \pm 2 \sqrt{3}}{2}$

Step 3.3.5.2.3.3

Simplify $\frac{- 4 \pm 2 \sqrt{3}}{2}$ .

$x = - 2 \pm \sqrt{3}$

Step 3.3.5.2.4

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

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Step 3.3.5.2.4.1

Simplify the numerator.

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Step 3.3.5.2.4.1.1

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

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Step 3.3.5.2.4.1.2

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

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Step 3.3.5.2.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1}}{2 \cdot 1}$

Step 3.3.5.2.4.1.2.2

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4}}{2 \cdot 1}$

Step 3.3.5.2.4.1.3

Subtract $4$ from $16$ .

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

Step 3.3.5.2.4.1.4

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 3.3.5.2.4.1.4.1

Factor $4$ out of $12$ .

$x = \frac{- 4 \pm \sqrt{4 (3)}}{2 \cdot 1}$

Step 3.3.5.2.4.1.4.2

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

$x = \frac{- 4 \pm \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

Step 3.3.5.2.4.1.5

Pull terms out from under the radical.

$x = \frac{- 4 \pm 2 \sqrt{3}}{2 \cdot 1}$

Step 3.3.5.2.4.2

Multiply $2$ by $1$ .

$x = \frac{- 4 \pm 2 \sqrt{3}}{2}$

Step 3.3.5.2.4.3

Simplify $\frac{- 4 \pm 2 \sqrt{3}}{2}$ .

$x = - 2 \pm \sqrt{3}$

Step 3.3.5.2.4.4

Change the $\pm$ to $+$ .

$x = - 2 + \sqrt{3}$

Step 3.3.5.2.5

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

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Step 3.3.5.2.5.1

Simplify the numerator.

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Step 3.3.5.2.5.1.1

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

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Step 3.3.5.2.5.1.2

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

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Step 3.3.5.2.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1}}{2 \cdot 1}$

Step 3.3.5.2.5.1.2.2

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4}}{2 \cdot 1}$

Step 3.3.5.2.5.1.3

Subtract $4$ from $16$ .

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

Step 3.3.5.2.5.1.4

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 3.3.5.2.5.1.4.1

Factor $4$ out of $12$ .

$x = \frac{- 4 \pm \sqrt{4 (3)}}{2 \cdot 1}$

Step 3.3.5.2.5.1.4.2

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

$x = \frac{- 4 \pm \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

Step 3.3.5.2.5.1.5

Pull terms out from under the radical.

$x = \frac{- 4 \pm 2 \sqrt{3}}{2 \cdot 1}$

Step 3.3.5.2.5.2

Multiply $2$ by $1$ .

$x = \frac{- 4 \pm 2 \sqrt{3}}{2}$

Step 3.3.5.2.5.3

Simplify $\frac{- 4 \pm 2 \sqrt{3}}{2}$ .

$x = - 2 \pm \sqrt{3}$

Step 3.3.5.2.5.4

Change the $\pm$ to $-$ .

$x = - 2 - \sqrt{3}$

Step 3.3.5.2.6

The final answer is the combination of both solutions.

$x = - 2 + \sqrt{3}, - 2 - \sqrt{3}$

Step 3.3.6

The final solution is all the values that make $(x - 1) (x^{2} + 4 x + 1) = 0$ true.

$x = 1, - 2 + \sqrt{3}, - 2 - \sqrt{3}$

Step 4

Find the points where the second derivative is $0$ .

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Step 4.1

Substitute $1$ in $f (x) = \frac{1 + x}{1 + x^{2}}$ to find the value of $y$ .

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Step 4.1.1

Replace the variable $x$ with $1$ in the expression.

$f (1) = \frac{1 + 1}{1 + {(1)}^{2}}$

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

Simplify the expression.

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Step 4.1.2.1.1

Remove parentheses.

$f (1) = \frac{1 + 1}{1 + {(1)}^{2}}$

Step 4.1.2.1.2

Add $1$ and $1$ .

$f (1) = \frac{2}{1 + 1^{2}}$

Step 4.1.2.2

Simplify the denominator.

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Step 4.1.2.2.1

One to any power is one.

$f (1) = \frac{2}{1 + 1}$

Step 4.1.2.2.2

Add $1$ and $1$ .

$f (1) = \frac{2}{2}$

Step 4.1.2.3

Divide $2$ by $2$ .

$f (1) = 1$

Step 4.1.2.4

The final answer is $1$ .

$1$

Step 4.2

The point found by substituting $1$ in $f (x) = \frac{1 + x}{1 + x^{2}}$ is $(1, 1)$ . This point can be an inflection point.

$(1, 1)$

Step 4.3

Substitute $- 2 + \sqrt{3}$ in $f (x) = \frac{1 + x}{1 + x^{2}}$ to find the value of $y$ .

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Step 4.3.1

Replace the variable $x$ with $- 2 + \sqrt{3}$ in the expression.

$f (- 2 + \sqrt{3}) = \frac{1 - 2 + \sqrt{3}}{1 + {(- 2 + \sqrt{3})}^{2}}$

Step 4.3.2

Simplify the result.

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Step 4.3.2.1

Simplify the expression.

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Step 4.3.2.1.1

Remove parentheses.

$f (- 2 + \sqrt{3}) = \frac{1 - 2 + \sqrt{3}}{1 + {(- 2 + \sqrt{3})}^{2}}$

Step 4.3.2.1.2

Subtract $2$ from $1$ .

$f (- 2 + \sqrt{3}) = \frac{- 1 + \sqrt{3}}{1 + {(- 2 + \sqrt{3})}^{2}}$

Step 4.3.2.2

Simplify the denominator.

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Step 4.3.2.2.1

Rewrite ${(- 2 + \sqrt{3})}^{2}$ as $(- 2 + \sqrt{3}) (- 2 + \sqrt{3})$ .

$f (- 2 + \sqrt{3}) = \frac{- 1 + \sqrt{3}}{1 + (- 2 + \sqrt{3}) (- 2 + \sqrt{3})}$

Step 4.3.2.2.2

Expand $(- 2 + \sqrt{3}) (- 2 + \sqrt{3})$ using the FOIL Method.

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Step 4.3.2.2.2.1

Apply the distributive property.

$f (- 2 + \sqrt{3}) = \frac{- 1 + \sqrt{3}}{1 - 2 (- 2 + \sqrt{3}) + \sqrt{3} (- 2 + \sqrt{3})}$

Step 4.3.2.2.2.2

Apply the distributive property.

$f (- 2 + \sqrt{3}) = \frac{- 1 + \sqrt{3}}{1 - 2 \cdot - 2 - 2 \sqrt{3} + \sqrt{3} (- 2 + \sqrt{3})}$

Step 4.3.2.2.2.3

Apply the distributive property.

$f (- 2 + \sqrt{3}) = \frac{- 1 + \sqrt{3}}{1 - 2 \cdot - 2 - 2 \sqrt{3} + \sqrt{3} \cdot - 2 + \sqrt{3} \sqrt{3}}$

Step 4.3.2.2.3

Simplify and combine like terms.

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Step 4.3.2.2.3.1

Simplify each term.

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Step 4.3.2.2.3.1.1

Multiply $- 2$ by $- 2$ .

$f (- 2 + \sqrt{3}) = \frac{- 1 + \sqrt{3}}{1 + 4 - 2 \sqrt{3} + \sqrt{3} \cdot - 2 + \sqrt{3} \sqrt{3}}$

Step 4.3.2.2.3.1.2

Move $- 2$ to the left of $\sqrt{3}$ .

$f (- 2 + \sqrt{3}) = \frac{- 1 + \sqrt{3}}{1 + 4 - 2 \sqrt{3} - 2 \cdot \sqrt{3} + \sqrt{3} \sqrt{3}}$

Step 4.3.2.2.3.1.3

Combine using the product rule for radicals.

$f (- 2 + \sqrt{3}) = \frac{- 1 + \sqrt{3}}{1 + 4 - 2 \sqrt{3} - 2 \sqrt{3} + \sqrt{3 \cdot 3}}$

Step 4.3.2.2.3.1.4

Multiply $3$ by $3$ .

$f (- 2 + \sqrt{3}) = \frac{- 1 + \sqrt{3}}{1 + 4 - 2 \sqrt{3} - 2 \sqrt{3} + \sqrt{9}}$

Step 4.3.2.2.3.1.5

Rewrite $9$ as $3^{2}$ .

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

Step 4.3.2.2.3.1.6

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

$f (- 2 + \sqrt{3}) = \frac{- 1 + \sqrt{3}}{1 + 4 - 2 \sqrt{3} - 2 \sqrt{3} + 3}$

Step 4.3.2.2.3.2

Add $4$ and $3$ .

$f (- 2 + \sqrt{3}) = \frac{- 1 + \sqrt{3}}{1 + 7 - 2 \sqrt{3} - 2 \sqrt{3}}$

Step 4.3.2.2.3.3

Subtract $2 \sqrt{3}$ from $- 2 \sqrt{3}$ .

$f (- 2 + \sqrt{3}) = \frac{- 1 + \sqrt{3}}{1 + 7 - 4 \sqrt{3}}$

Step 4.3.2.2.4

Add $1$ and $7$ .

$f (- 2 + \sqrt{3}) = \frac{- 1 + \sqrt{3}}{8 - 4 \sqrt{3}}$

Step 4.3.2.3

Multiply $\frac{- 1 + \sqrt{3}}{8 - 4 \sqrt{3}}$ by $\frac{8 + 4 \sqrt{3}}{8 + 4 \sqrt{3}}$ .

$f (- 2 + \sqrt{3}) = \frac{- 1 + \sqrt{3}}{8 - 4 \sqrt{3}} \cdot \frac{8 + 4 \sqrt{3}}{8 + 4 \sqrt{3}}$

Step 4.3.2.4

Simplify terms.

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Step 4.3.2.4.1

Multiply $\frac{- 1 + \sqrt{3}}{8 - 4 \sqrt{3}}$ by $\frac{8 + 4 \sqrt{3}}{8 + 4 \sqrt{3}}$ .

$f (- 2 + \sqrt{3}) = \frac{(- 1 + \sqrt{3}) (8 + 4 \sqrt{3})}{(8 - 4 \sqrt{3}) (8 + 4 \sqrt{3})}$

Step 4.3.2.4.2

Expand the denominator using the FOIL method.

$f (- 2 + \sqrt{3}) = \frac{(- 1 + \sqrt{3}) (8 + 4 \sqrt{3})}{64 + 32 \sqrt{3} - 32 \sqrt{3} - 16 {\sqrt{3}}^{2}}$

Step 4.3.2.4.3

Simplify.

$f (- 2 + \sqrt{3}) = \frac{(- 1 + \sqrt{3}) (8 + 4 \sqrt{3})}{16}$

Step 4.3.2.4.4

Cancel the common factor of $8 + 4 \sqrt{3}$ and $16$ .

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Step 4.3.2.4.4.1

Factor $4$ out of $(- 1 + \sqrt{3}) (8 + 4 \sqrt{3})$ .

$f (- 2 + \sqrt{3}) = \frac{4 ((- 1 + \sqrt{3}) (2 + \sqrt{3}))}{16}$

Step 4.3.2.4.4.2

Cancel the common factors.

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Step 4.3.2.4.4.2.1

Factor $4$ out of $16$ .

$f (- 2 + \sqrt{3}) = \frac{4 ((- 1 + \sqrt{3}) (2 + \sqrt{3}))}{4 (4)}$

Step 4.3.2.4.4.2.2

Cancel the common factor.

$f (- 2 + \sqrt{3}) = \frac{4 ((- 1 + \sqrt{3}) (2 + \sqrt{3}))}{4 \cdot 4}$

Step 4.3.2.4.4.2.3

Rewrite the expression.

$f (- 2 + \sqrt{3}) = \frac{(- 1 + \sqrt{3}) (2 + \sqrt{3})}{4}$

Step 4.3.2.5

Expand $(- 1 + \sqrt{3}) (2 + \sqrt{3})$ using the FOIL Method.

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Step 4.3.2.5.1

Apply the distributive property.

$f (- 2 + \sqrt{3}) = \frac{- 1 (2 + \sqrt{3}) + \sqrt{3} (2 + \sqrt{3})}{4}$

Step 4.3.2.5.2

Apply the distributive property.

$f (- 2 + \sqrt{3}) = \frac{- 1 \cdot 2 - 1 \sqrt{3} + \sqrt{3} (2 + \sqrt{3})}{4}$

Step 4.3.2.5.3

Apply the distributive property.

$f (- 2 + \sqrt{3}) = \frac{- 1 \cdot 2 - 1 \sqrt{3} + \sqrt{3} \cdot 2 + \sqrt{3} \sqrt{3}}{4}$

Step 4.3.2.6

Simplify and combine like terms.

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Step 4.3.2.6.1

Simplify each term.

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Step 4.3.2.6.1.1

Multiply $- 1$ by $2$ .

$f (- 2 + \sqrt{3}) = \frac{- 2 - 1 \sqrt{3} + \sqrt{3} \cdot 2 + \sqrt{3} \sqrt{3}}{4}$

Step 4.3.2.6.1.2

Rewrite $- 1 \sqrt{3}$ as $- \sqrt{3}$ .

$f (- 2 + \sqrt{3}) = \frac{- 2 - \sqrt{3} + \sqrt{3} \cdot 2 + \sqrt{3} \sqrt{3}}{4}$

Step 4.3.2.6.1.3

Move $2$ to the left of $\sqrt{3}$ .

$f (- 2 + \sqrt{3}) = \frac{- 2 - \sqrt{3} + 2 \cdot \sqrt{3} + \sqrt{3} \sqrt{3}}{4}$

Step 4.3.2.6.1.4

Combine using the product rule for radicals.

$f (- 2 + \sqrt{3}) = \frac{- 2 - \sqrt{3} + 2 \sqrt{3} + \sqrt{3 \cdot 3}}{4}$

Step 4.3.2.6.1.5

Multiply $3$ by $3$ .

$f (- 2 + \sqrt{3}) = \frac{- 2 - \sqrt{3} + 2 \sqrt{3} + \sqrt{9}}{4}$

Step 4.3.2.6.1.6

Rewrite $9$ as $3^{2}$ .

$f (- 2 + \sqrt{3}) = \frac{- 2 - \sqrt{3} + 2 \sqrt{3} + \sqrt{3^{2}}}{4}$

Step 4.3.2.6.1.7

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

$f (- 2 + \sqrt{3}) = \frac{- 2 - \sqrt{3} + 2 \sqrt{3} + 3}{4}$

Step 4.3.2.6.2

Add $- 2$ and $3$ .

$f (- 2 + \sqrt{3}) = \frac{1 - \sqrt{3} + 2 \sqrt{3}}{4}$

Step 4.3.2.6.3

Add $- \sqrt{3}$ and $2 \sqrt{3}$ .

$f (- 2 + \sqrt{3}) = \frac{1 + \sqrt{3}}{4}$

Step 4.3.2.7

The final answer is $\frac{1 + \sqrt{3}}{4}$ .

$\frac{1 + \sqrt{3}}{4}$

Step 4.4

The point found by substituting $- 2 + \sqrt{3}$ in $f (x) = \frac{1 + x}{1 + x^{2}}$ is $(- 2 + \sqrt{3}, \frac{1 + \sqrt{3}}{4})$ . This point can be an inflection point.

$(- 2 + \sqrt{3}, \frac{1 + \sqrt{3}}{4})$

Step 4.5

Substitute $- 2 - \sqrt{3}$ in $f (x) = \frac{1 + x}{1 + x^{2}}$ to find the value of $y$ .

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Step 4.5.1

Replace the variable $x$ with $- 2 - \sqrt{3}$ in the expression.

$f (- 2 - \sqrt{3}) = \frac{1 - 2 - \sqrt{3}}{1 + {(- 2 - \sqrt{3})}^{2}}$

Step 4.5.2

Simplify the result.

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Step 4.5.2.1

Simplify the expression.

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Step 4.5.2.1.1

Remove parentheses.

$f (- 2 - \sqrt{3}) = \frac{1 - 2 - \sqrt{3}}{1 + {(- 2 - \sqrt{3})}^{2}}$

Step 4.5.2.1.2

Subtract $2$ from $1$ .

$f (- 2 - \sqrt{3}) = \frac{- 1 - \sqrt{3}}{1 + {(- 2 - \sqrt{3})}^{2}}$

Step 4.5.2.2

Simplify the denominator.

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Step 4.5.2.2.1

Rewrite ${(- 2 - \sqrt{3})}^{2}$ as $(- 2 - \sqrt{3}) (- 2 - \sqrt{3})$ .

$f (- 2 - \sqrt{3}) = \frac{- 1 - \sqrt{3}}{1 + (- 2 - \sqrt{3}) (- 2 - \sqrt{3})}$

Step 4.5.2.2.2

Expand $(- 2 - \sqrt{3}) (- 2 - \sqrt{3})$ using the FOIL Method.

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Step 4.5.2.2.2.1

Apply the distributive property.

$f (- 2 - \sqrt{3}) = \frac{- 1 - \sqrt{3}}{1 - 2 (- 2 - \sqrt{3}) - \sqrt{3} (- 2 - \sqrt{3})}$

Step 4.5.2.2.2.2

Apply the distributive property.

$f (- 2 - \sqrt{3}) = \frac{- 1 - \sqrt{3}}{1 - 2 \cdot - 2 - 2 (- \sqrt{3}) - \sqrt{3} (- 2 - \sqrt{3})}$

Step 4.5.2.2.2.3

Apply the distributive property.

$f (- 2 - \sqrt{3}) = \frac{- 1 - \sqrt{3}}{1 - 2 \cdot - 2 - 2 (- \sqrt{3}) - \sqrt{3} \cdot - 2 - \sqrt{3} (- \sqrt{3})}$

Step 4.5.2.2.3

Simplify and combine like terms.

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Step 4.5.2.2.3.1

Simplify each term.

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Step 4.5.2.2.3.1.1

Multiply $- 2$ by $- 2$ .

$f (- 2 - \sqrt{3}) = \frac{- 1 - \sqrt{3}}{1 + 4 - 2 (- \sqrt{3}) - \sqrt{3} \cdot - 2 - \sqrt{3} (- \sqrt{3})}$

Step 4.5.2.2.3.1.2

Multiply $- 1$ by $- 2$ .

$f (- 2 - \sqrt{3}) = \frac{- 1 - \sqrt{3}}{1 + 4 + 2 \sqrt{3} - \sqrt{3} \cdot - 2 - \sqrt{3} (- \sqrt{3})}$

Step 4.5.2.2.3.1.3

Multiply $- 2$ by $- 1$ .

$f (- 2 - \sqrt{3}) = \frac{- 1 - \sqrt{3}}{1 + 4 + 2 \sqrt{3} + 2 \sqrt{3} - \sqrt{3} (- \sqrt{3})}$

Step 4.5.2.2.3.1.4

Multiply $- \sqrt{3} (- \sqrt{3})$ .

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Step 4.5.2.2.3.1.4.1

Multiply $- 1$ by $- 1$ .

$f (- 2 - \sqrt{3}) = \frac{- 1 - \sqrt{3}}{1 + 4 + 2 \sqrt{3} + 2 \sqrt{3} + 1 \sqrt{3} \sqrt{3}}$

Step 4.5.2.2.3.1.4.2

Multiply $\sqrt{3}$ by $1$ .

$f (- 2 - \sqrt{3}) = \frac{- 1 - \sqrt{3}}{1 + 4 + 2 \sqrt{3} + 2 \sqrt{3} + \sqrt{3} \sqrt{3}}$

Step 4.5.2.2.3.1.4.3

Raise $\sqrt{3}$ to the power of $1$ .

$f (- 2 - \sqrt{3}) = \frac{- 1 - \sqrt{3}}{1 + 4 + 2 \sqrt{3} + 2 \sqrt{3} + \sqrt{3} \sqrt{3}}$

Step 4.5.2.2.3.1.4.4

Raise $\sqrt{3}$ to the power of $1$ .

$f (- 2 - \sqrt{3}) = \frac{- 1 - \sqrt{3}}{1 + 4 + 2 \sqrt{3} + 2 \sqrt{3} + \sqrt{3} \sqrt{3}}$

Step 4.5.2.2.3.1.4.5

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

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

Step 4.5.2.2.3.1.4.6

Add $1$ and $1$ .

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

Step 4.5.2.2.3.1.5

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 4.5.2.2.3.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

Step 4.5.2.2.3.1.5.2

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

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

Step 4.5.2.2.3.1.5.3

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

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

Step 4.5.2.2.3.1.5.4

Cancel the common factor of $2$ .

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Step 4.5.2.2.3.1.5.4.1

Cancel the common factor.

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

Step 4.5.2.2.3.1.5.4.2

Rewrite the expression.

$f (- 2 - \sqrt{3}) = \frac{- 1 - \sqrt{3}}{1 + 4 + 2 \sqrt{3} + 2 \sqrt{3} + 3}$

Step 4.5.2.2.3.1.5.5

Evaluate the exponent.

$f (- 2 - \sqrt{3}) = \frac{- 1 - \sqrt{3}}{1 + 4 + 2 \sqrt{3} + 2 \sqrt{3} + 3}$

Step 4.5.2.2.3.2

Add $4$ and $3$ .

$f (- 2 - \sqrt{3}) = \frac{- 1 - \sqrt{3}}{1 + 7 + 2 \sqrt{3} + 2 \sqrt{3}}$

Step 4.5.2.2.3.3

Add $2 \sqrt{3}$ and $2 \sqrt{3}$ .

$f (- 2 - \sqrt{3}) = \frac{- 1 - \sqrt{3}}{1 + 7 + 4 \sqrt{3}}$

Step 4.5.2.2.4

Add $1$ and $7$ .

$f (- 2 - \sqrt{3}) = \frac{- 1 - \sqrt{3}}{8 + 4 \sqrt{3}}$

Step 4.5.2.3

Multiply $\frac{- 1 - \sqrt{3}}{8 + 4 \sqrt{3}}$ by $\frac{8 - 4 \sqrt{3}}{8 - 4 \sqrt{3}}$ .

$f (- 2 - \sqrt{3}) = \frac{- 1 - \sqrt{3}}{8 + 4 \sqrt{3}} \cdot \frac{8 - 4 \sqrt{3}}{8 - 4 \sqrt{3}}$

Step 4.5.2.4

Simplify terms.

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Step 4.5.2.4.1

Multiply $\frac{- 1 - \sqrt{3}}{8 + 4 \sqrt{3}}$ by $\frac{8 - 4 \sqrt{3}}{8 - 4 \sqrt{3}}$ .

$f (- 2 - \sqrt{3}) = \frac{(- 1 - \sqrt{3}) (8 - 4 \sqrt{3})}{(8 + 4 \sqrt{3}) (8 - 4 \sqrt{3})}$

Step 4.5.2.4.2

Expand the denominator using the FOIL method.

$f (- 2 - \sqrt{3}) = \frac{(- 1 - \sqrt{3}) (8 - 4 \sqrt{3})}{64 - 32 \sqrt{3} + 32 \sqrt{3} - 16 {\sqrt{3}}^{2}}$

Step 4.5.2.4.3

Simplify.

$f (- 2 - \sqrt{3}) = \frac{(- 1 - \sqrt{3}) (8 - 4 \sqrt{3})}{16}$

Step 4.5.2.4.4

Cancel the common factor of $8 - 4 \sqrt{3}$ and $16$ .

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Step 4.5.2.4.4.1

Factor $4$ out of $(- 1 - \sqrt{3}) (8 - 4 \sqrt{3})$ .

$f (- 2 - \sqrt{3}) = \frac{4 ((- 1 - \sqrt{3}) (2 - \sqrt{3}))}{16}$

Step 4.5.2.4.4.2

Cancel the common factors.

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Step 4.5.2.4.4.2.1

Factor $4$ out of $16$ .

$f (- 2 - \sqrt{3}) = \frac{4 ((- 1 - \sqrt{3}) (2 - \sqrt{3}))}{4 (4)}$

Step 4.5.2.4.4.2.2

Cancel the common factor.

$f (- 2 - \sqrt{3}) = \frac{4 ((- 1 - \sqrt{3}) (2 - \sqrt{3}))}{4 \cdot 4}$

Step 4.5.2.4.4.2.3

Rewrite the expression.

$f (- 2 - \sqrt{3}) = \frac{(- 1 - \sqrt{3}) (2 - \sqrt{3})}{4}$

Step 4.5.2.5

Expand $(- 1 - \sqrt{3}) (2 - \sqrt{3})$ using the FOIL Method.

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Step 4.5.2.5.1

Apply the distributive property.

$f (- 2 - \sqrt{3}) = \frac{- 1 (2 - \sqrt{3}) - \sqrt{3} (2 - \sqrt{3})}{4}$

Step 4.5.2.5.2

Apply the distributive property.

$f (- 2 - \sqrt{3}) = \frac{- 1 \cdot 2 - 1 (- \sqrt{3}) - \sqrt{3} (2 - \sqrt{3})}{4}$

Step 4.5.2.5.3

Apply the distributive property.

$f (- 2 - \sqrt{3}) = \frac{- 1 \cdot 2 - 1 (- \sqrt{3}) - \sqrt{3} \cdot 2 - \sqrt{3} (- \sqrt{3})}{4}$

Step 4.5.2.6

Simplify and combine like terms.

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Step 4.5.2.6.1

Simplify each term.

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Step 4.5.2.6.1.1

Multiply $- 1$ by $2$ .

$f (- 2 - \sqrt{3}) = \frac{- 2 - 1 (- \sqrt{3}) - \sqrt{3} \cdot 2 - \sqrt{3} (- \sqrt{3})}{4}$

Step 4.5.2.6.1.2

Multiply $- 1 (- \sqrt{3})$ .

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Step 4.5.2.6.1.2.1

Multiply $- 1$ by $- 1$ .

$f (- 2 - \sqrt{3}) = \frac{- 2 + 1 \sqrt{3} - \sqrt{3} \cdot 2 - \sqrt{3} (- \sqrt{3})}{4}$

Step 4.5.2.6.1.2.2

Multiply $\sqrt{3}$ by $1$ .

$f (- 2 - \sqrt{3}) = \frac{- 2 + \sqrt{3} - \sqrt{3} \cdot 2 - \sqrt{3} (- \sqrt{3})}{4}$

Step 4.5.2.6.1.3

Multiply $2$ by $- 1$ .

$f (- 2 - \sqrt{3}) = \frac{- 2 + \sqrt{3} - 2 \sqrt{3} - \sqrt{3} (- \sqrt{3})}{4}$

Step 4.5.2.6.1.4

Multiply $- \sqrt{3} (- \sqrt{3})$ .

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Step 4.5.2.6.1.4.1

Multiply $- 1$ by $- 1$ .

$f (- 2 - \sqrt{3}) = \frac{- 2 + \sqrt{3} - 2 \sqrt{3} + 1 \sqrt{3} \sqrt{3}}{4}$

Step 4.5.2.6.1.4.2

Multiply $\sqrt{3}$ by $1$ .

$f (- 2 - \sqrt{3}) = \frac{- 2 + \sqrt{3} - 2 \sqrt{3} + \sqrt{3} \sqrt{3}}{4}$

Step 4.5.2.6.1.4.3

Raise $\sqrt{3}$ to the power of $1$ .

$f (- 2 - \sqrt{3}) = \frac{- 2 + \sqrt{3} - 2 \sqrt{3} + \sqrt{3} \sqrt{3}}{4}$

Step 4.5.2.6.1.4.4

Raise $\sqrt{3}$ to the power of $1$ .

$f (- 2 - \sqrt{3}) = \frac{- 2 + \sqrt{3} - 2 \sqrt{3} + \sqrt{3} \sqrt{3}}{4}$

Step 4.5.2.6.1.4.5

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

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

Step 4.5.2.6.1.4.6

Add $1$ and $1$ .

$f (- 2 - \sqrt{3}) = \frac{- 2 + \sqrt{3} - 2 \sqrt{3} + {\sqrt{3}}^{2}}{4}$

Step 4.5.2.6.1.5

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 4.5.2.6.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

Step 4.5.2.6.1.5.2

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

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

Step 4.5.2.6.1.5.3

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

$f (- 2 - \sqrt{3}) = \frac{- 2 + \sqrt{3} - 2 \sqrt{3} + 3^{\frac{2}{2}}}{4}$

Step 4.5.2.6.1.5.4

Cancel the common factor of $2$ .

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Step 4.5.2.6.1.5.4.1

Cancel the common factor.

$f (- 2 - \sqrt{3}) = \frac{- 2 + \sqrt{3} - 2 \sqrt{3} + 3^{\frac{2}{2}}}{4}$

Step 4.5.2.6.1.5.4.2

Rewrite the expression.

$f (- 2 - \sqrt{3}) = \frac{- 2 + \sqrt{3} - 2 \sqrt{3} + 3}{4}$

Step 4.5.2.6.1.5.5

Evaluate the exponent.

$f (- 2 - \sqrt{3}) = \frac{- 2 + \sqrt{3} - 2 \sqrt{3} + 3}{4}$

Step 4.5.2.6.2

Add $- 2$ and $3$ .

$f (- 2 - \sqrt{3}) = \frac{1 + \sqrt{3} - 2 \sqrt{3}}{4}$

Step 4.5.2.6.3

Subtract $2 \sqrt{3}$ from $\sqrt{3}$ .

$f (- 2 - \sqrt{3}) = \frac{1 - \sqrt{3}}{4}$

Step 4.5.2.7

The final answer is $\frac{1 - \sqrt{3}}{4}$ .

$\frac{1 - \sqrt{3}}{4}$

Step 4.6

The point found by substituting $- 2 - \sqrt{3}$ in $f (x) = \frac{1 + x}{1 + x^{2}}$ is $(- 2 - \sqrt{3}, \frac{1 - \sqrt{3}}{4})$ . This point can be an inflection point.

$(- 2 - \sqrt{3}, \frac{1 - \sqrt{3}}{4})$

Step 4.7

Determine the points that could be inflection points.

$(1, 1), (- 2 + \sqrt{3}, \frac{1 + \sqrt{3}}{4}), (- 2 - \sqrt{3}, \frac{1 - \sqrt{3}}{4})$

Step 5

Split $(- \infty, \infty)$ into intervals around the points that could potentially be inflection points.

$(- \infty, - 2 - \sqrt{3}) \cup (- 2 - \sqrt{3}, - 2 + \sqrt{3}) \cup (- 2 + \sqrt{3}, 1) \cup (1, \infty)$

Step 6

Substitute a value from the interval $(- \infty, - 2 - \sqrt{3})$ into the second derivative to determine if it is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $- 3.8320508$ in the expression.

$f'' (- 3.8320508) = \frac{2 ({(- 3.8320508)}^{3} + 3 {(- 3.8320508)}^{2} - 3 \cdot - 3.8320508 - 1)}{{({(- 3.8320508)}^{2} + 1)}^{3}}$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify the numerator.

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Step 6.2.1.1

Raise $- 3.8320508$ to the power of $3$ .

$f'' (- 3.8320508) = \frac{2 (- 56.2721846 + 3 {(- 3.8320508)}^{2} - 3 \cdot - 3.8320508 - 1)}{{({(- 3.8320508)}^{2} + 1)}^{3}}$

Step 6.2.1.2

Raise $- 3.8320508$ to the power of $2$ .

$f'' (- 3.8320508) = \frac{2 (- 56.2721846 + 3 \cdot 14.68461339 - 3 \cdot - 3.8320508 - 1)}{{({(- 3.8320508)}^{2} + 1)}^{3}}$

Step 6.2.1.3

Multiply $3$ by $14.68461339$ .

$f'' (- 3.8320508) = \frac{2 (- 56.2721846 + 44.05384017 - 3 \cdot - 3.8320508 - 1)}{{({(- 3.8320508)}^{2} + 1)}^{3}}$

Step 6.2.1.4

Multiply $- 3$ by $- 3.8320508$ .

$f'' (- 3.8320508) = \frac{2 (- 56.2721846 + 44.05384017 + 11.49615242 - 1)}{{({(- 3.8320508)}^{2} + 1)}^{3}}$

Step 6.2.1.5

Add $- 56.2721846$ and $44.05384017$ .

$f'' (- 3.8320508) = \frac{2 (- 12.21834443 + 11.49615242 - 1)}{{({(- 3.8320508)}^{2} + 1)}^{3}}$

Step 6.2.1.6

Add $- 12.21834443$ and $11.49615242$ .

$f'' (- 3.8320508) = \frac{2 (- 0.722192 - 1)}{{({(- 3.8320508)}^{2} + 1)}^{3}}$

Step 6.2.1.7

Subtract $1$ from $- 0.722192$ .

$f'' (- 3.8320508) = \frac{2 \cdot - 1.722192}{{({(- 3.8320508)}^{2} + 1)}^{3}}$

Step 6.2.2

Simplify the denominator.

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Step 6.2.2.1

Raise $- 3.8320508$ to the power of $2$ .

$f'' (- 3.8320508) = \frac{2 \cdot - 1.722192}{{(14.68461339 + 1)}^{3}}$

Step 6.2.2.2

Add $14.68461339$ and $1$ .

$f'' (- 3.8320508) = \frac{2 \cdot - 1.722192}{{15.68461339}^{3}}$

Step 6.2.2.3

Raise $15.68461339$ to the power of $3$ .

$f'' (- 3.8320508) = \frac{2 \cdot - 1.722192}{3858.526212}$

Step 6.2.3

Simplify the expression.

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Step 6.2.3.1

Multiply $2$ by $- 1.722192$ .

$f'' (- 3.8320508) = \frac{- 3.44438401}{3858.526212}$

Step 6.2.3.2

Divide $- 3.44438401$ by $3858.526212$ .

$f'' (- 3.8320508) = - 0.00089266$

Step 6.2.4

The final answer is $- 0.00089266$ .

$- 0.00089266$

Step 6.3

At $- 3.8320508$ , the second derivative is $- 0.00089266$ . Since this is negative, the second derivative is decreasing on the interval $(- \infty, - 2 - \sqrt{3})$

Decreasing on $(- \infty, - 2 - \sqrt{3})$ since $f'' (x) < 0$

Step 7

Substitute a value from the interval $(- 2 - \sqrt{3}, - 2 + \sqrt{3})$ into the second derivative to determine if it is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $- 2$ in the expression.

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

Step 7.2

Simplify the result.

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Step 7.2.1

Simplify the numerator.

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Step 7.2.1.1

Raise $- 2$ to the power of $3$ .

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

Step 7.2.1.2

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

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

Step 7.2.1.3

Multiply $3$ by $4$ .

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

Step 7.2.1.4

Multiply $- 3$ by $- 2$ .

$f'' (- 2) = \frac{2 (- 8 + 12 + 6 - 1)}{{({(- 2)}^{2} + 1)}^{3}}$

Step 7.2.1.5

Add $- 8$ and $12$ .

$f'' (- 2) = \frac{2 (4 + 6 - 1)}{{({(- 2)}^{2} + 1)}^{3}}$

Step 7.2.1.6

Add $4$ and $6$ .

$f'' (- 2) = \frac{2 (10 - 1)}{{({(- 2)}^{2} + 1)}^{3}}$

Step 7.2.1.7

Subtract $1$ from $10$ .

$f'' (- 2) = \frac{2 \cdot 9}{{({(- 2)}^{2} + 1)}^{3}}$

Step 7.2.2

Simplify the denominator.

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Step 7.2.2.1

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

$f'' (- 2) = \frac{2 \cdot 9}{{(4 + 1)}^{3}}$

Step 7.2.2.2

Add $4$ and $1$ .

$f'' (- 2) = \frac{2 \cdot 9}{5^{3}}$

Step 7.2.2.3

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

$f'' (- 2) = \frac{2 \cdot 9}{125}$

Step 7.2.3

Simplify the expression.

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Step 7.2.3.1

Multiply $2$ by $9$ .

$f'' (- 2) = \frac{18}{125}$

Step 7.2.3.2

Divide $18$ by $125$ .

$f'' (- 2) = 0.144$

Step 7.2.4

The final answer is $0.144$ .

$0.144$

Step 7.3

At $- 2$ , the second derivative is $0.144$ . Since this is positive, the second derivative is increasing on the interval $(- 2 - \sqrt{3}, - 2 + \sqrt{3})$ .

Increasing on $(- 2 - \sqrt{3}, - 2 + \sqrt{3})$ since $f'' (x) > 0$

Step 8

Substitute a value from the interval $(- 2 + \sqrt{3}, 1)$ into the second derivative to determine if it is increasing or decreasing.

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Step 8.1

Replace the variable $x$ with $0.3660254$ in the expression.

$f'' (0.3660254) = \frac{2 ({(0.3660254)}^{3} + 3 {(0.3660254)}^{2} - 3 \cdot 0.3660254 - 1)}{{({(0.3660254)}^{2} + 1)}^{3}}$

Step 8.2

Simplify the result.

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Step 8.2.1

Simplify the numerator.

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Step 8.2.1.1

Raise $0.3660254$ to the power of $3$ .

$f'' (0.3660254) = \frac{2 (0.0490381 + 3 \cdot {0.3660254}^{2} - 3 \cdot 0.3660254 - 1)}{{({0.3660254}^{2} + 1)}^{3}}$

Step 8.2.1.2

Raise $0.3660254$ to the power of $2$ .

$f'' (0.3660254) = \frac{2 (0.0490381 + 3 \cdot 0.13397459 - 3 \cdot 0.3660254 - 1)}{{({0.3660254}^{2} + 1)}^{3}}$

Step 8.2.1.3

Multiply $3$ by $0.13397459$ .

$f'' (0.3660254) = \frac{2 (0.0490381 + 0.40192378 - 3 \cdot 0.3660254 - 1)}{{({0.3660254}^{2} + 1)}^{3}}$

Step 8.2.1.4

Multiply $- 3$ by $0.3660254$ .

$f'' (0.3660254) = \frac{2 (0.0490381 + 0.40192378 - 1.09807621 - 1)}{{({0.3660254}^{2} + 1)}^{3}}$

Step 8.2.1.5

Add $0.0490381$ and $0.40192378$ .

$f'' (0.3660254) = \frac{2 (0.45096189 - 1.09807621 - 1)}{{({0.3660254}^{2} + 1)}^{3}}$

Step 8.2.1.6

Subtract $1.09807621$ from $0.45096189$ .

$f'' (0.3660254) = \frac{2 (- 0.64711431 - 1)}{{({0.3660254}^{2} + 1)}^{3}}$

Step 8.2.1.7

Subtract $1$ from $- 0.64711431$ .

$f'' (0.3660254) = \frac{2 \cdot - 1.64711431}{{({0.3660254}^{2} + 1)}^{3}}$

Step 8.2.2

Simplify the denominator.

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Step 8.2.2.1

Raise $0.3660254$ to the power of $2$ .

$f'' (0.3660254) = \frac{2 \cdot - 1.64711431}{{(0.13397459 + 1)}^{3}}$

Step 8.2.2.2

Add $0.13397459$ and $1$ .

$f'' (0.3660254) = \frac{2 \cdot - 1.64711431}{{1.13397459}^{3}}$

Step 8.2.2.3

Raise $1.13397459$ to the power of $3$ .

$f'' (0.3660254) = \frac{2 \cdot - 1.64711431}{1.4581761}$

Step 8.2.3

Simplify the expression.

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Step 8.2.3.1

Multiply $2$ by $- 1.64711431$ .

$f'' (0.3660254) = \frac{- 3.29422863}{1.4581761}$

Step 8.2.3.2

Divide $- 3.29422863$ by $1.4581761$ .

$f'' (0.3660254) = - 2.2591432$

Step 8.2.4

The final answer is $- 2.2591432$ .

$- 2.2591432$

Step 8.3

At $0.3660254$ , the second derivative is $- 2.2591432$ . Since this is negative, the second derivative is decreasing on the interval $(- 2 + \sqrt{3}, 1)$

Decreasing on $(- 2 + \sqrt{3}, 1)$ since $f'' (x) < 0$

Step 9

Substitute a value from the interval $(1, \infty)$ into the second derivative to determine if it is increasing or decreasing.

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Step 9.1

Replace the variable $x$ with $1.1$ in the expression.

$f'' (1.1) = \frac{2 ({(1.1)}^{3} + 3 {(1.1)}^{2} - 3 \cdot 1.1 - 1)}{{({(1.1)}^{2} + 1)}^{3}}$

Step 9.2

Simplify the result.

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Step 9.2.1

Simplify the numerator.

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Step 9.2.1.1

Raise $1.1$ to the power of $3$ .

$f'' (1.1) = \frac{2 (1.331 + 3 \cdot {1.1}^{2} - 3 \cdot 1.1 - 1)}{{({1.1}^{2} + 1)}^{3}}$

Step 9.2.1.2

Raise $1.1$ to the power of $2$ .

$f'' (1.1) = \frac{2 (1.331 + 3 \cdot 1.21 - 3 \cdot 1.1 - 1)}{{({1.1}^{2} + 1)}^{3}}$

Step 9.2.1.3

Multiply $3$ by $1.21$ .

$f'' (1.1) = \frac{2 (1.331 + 3.63 - 3 \cdot 1.1 - 1)}{{({1.1}^{2} + 1)}^{3}}$

Step 9.2.1.4

Multiply $- 3$ by $1.1$ .

$f'' (1.1) = \frac{2 (1.331 + 3.63 - 3.3 - 1)}{{({1.1}^{2} + 1)}^{3}}$

Step 9.2.1.5

Add $1.331$ and $3.63$ .

$f'' (1.1) = \frac{2 (4.961 - 3.3 - 1)}{{({1.1}^{2} + 1)}^{3}}$

Step 9.2.1.6

Subtract $3.3$ from $4.961$ .

$f'' (1.1) = \frac{2 (1.661 - 1)}{{({1.1}^{2} + 1)}^{3}}$

Step 9.2.1.7

Subtract $1$ from $1.661$ .

$f'' (1.1) = \frac{2 \cdot 0.661}{{({1.1}^{2} + 1)}^{3}}$

Step 9.2.2

Simplify the denominator.

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Step 9.2.2.1

Raise $1.1$ to the power of $2$ .

$f'' (1.1) = \frac{2 \cdot 0.661}{{(1.21 + 1)}^{3}}$

Step 9.2.2.2

Add $1.21$ and $1$ .

$f'' (1.1) = \frac{2 \cdot 0.661}{{2.21}^{3}}$

Step 9.2.2.3

Raise $2.21$ to the power of $3$ .

$f'' (1.1) = \frac{2 \cdot 0.661}{10.793861}$

Step 9.2.3

Simplify the expression.

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Step 9.2.3.1

Multiply $2$ by $0.661$ .

$f'' (1.1) = \frac{1.322}{10.793861}$

Step 9.2.3.2

Divide $1.322$ by $10.793861$ .

$f'' (1.1) = 0.12247702$

Step 9.2.4

The final answer is $0.12247702$ .

$0.12247702$

Step 9.3

At $1.1$ , the second derivative is $0.12247702$ . Since this is positive, the second derivative is increasing on the interval $(1, \infty)$ .

Increasing on $(1, \infty)$ since $f'' (x) > 0$

Step 10

An inflection point is a point on a curve at which the concavity changes sign from plus to minus or from minus to plus. The inflection points in this case are $(- 2 - \sqrt{3}, \frac{1 - \sqrt{3}}{4}), (- 2 + \sqrt{3}, \frac{1 + \sqrt{3}}{4}), (1, 1)$ .

$(- 2 - \sqrt{3}, \frac{1 - \sqrt{3}}{4}), (- 2 + \sqrt{3}, \frac{1 + \sqrt{3}}{4}), (1, 1)$

Step 11