Calculus Examples

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

Step 1

Find the $x$ values where the second derivative is equal to $0$ .

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Step 1.1

Find the second derivative.

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Step 1.1.1

Find the first derivative.

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Step 1.1.1.1

Since $6$ is constant with respect to $x$ , the derivative of $\frac{6 x}{1 + 3 x^{2}}$ with respect to $x$ is $6 \frac{d}{d x} [\frac{x}{1 + 3 x^{2}}]$ .

$6 \frac{d}{d x} [\frac{x}{1 + 3 x^{2}}]$

Step 1.1.1.2

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 $f (x) = x$ and $g (x) = 1 + 3 x^{2}$ .

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

Step 1.1.1.3

Differentiate.

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Step 1.1.1.3.1

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

$6 \frac{(1 + 3 x^{2}) \cdot 1 - x \frac{d}{d x} [1 + 3 x^{2}]}{{(1 + 3 x^{2})}^{2}}$

Step 1.1.1.3.2

Multiply $1 + 3 x^{2}$ by $1$ .

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

Step 1.1.1.3.3

By the Sum Rule, the derivative of $1 + 3 x^{2}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [3 x^{2}]$ .

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

Step 1.1.1.3.4

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

$6 \frac{1 + 3 x^{2} - x (0 + \frac{d}{d x} [3 x^{2}])}{{(1 + 3 x^{2})}^{2}}$

Step 1.1.1.3.5

Add $0$ and $\frac{d}{d x} [3 x^{2}]$ .

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

Step 1.1.1.3.6

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{2}$ with respect to $x$ is $3 \frac{d}{d x} [x^{2}]$ .

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

Step 1.1.1.3.7

Multiply $3$ by $- 1$ .

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

Step 1.1.1.3.8

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

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

Step 1.1.1.3.9

Multiply $2$ by $- 3$ .

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

Step 1.1.1.4

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

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

Step 1.1.1.5

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

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

Step 1.1.1.6

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

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

Step 1.1.1.7

Add $1$ and $1$ .

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

Step 1.1.1.8

Subtract $6 x^{2}$ from $3 x^{2}$ .

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

Step 1.1.1.9

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

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

Step 1.1.1.10

Simplify.

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Step 1.1.1.10.1

Apply the distributive property.

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

Step 1.1.1.10.2

Simplify each term.

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Step 1.1.1.10.2.1

Multiply $6$ by $1$ .

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

Step 1.1.1.10.2.2

Multiply $- 3$ by $6$ .

$\frac{6 - 18 x^{2}}{{(1 + 3 x^{2})}^{2}}$

Step 1.1.1.10.3

Reorder terms.

$f' (x) = \frac{- 18 x^{2} + 6}{{(3 x^{2} + 1)}^{2}}$

Step 1.1.2

Find the second derivative.

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Step 1.1.2.1

$\frac{{(3 x^{2} + 1)}^{2} \frac{d}{d x} [- 18 x^{2} + 6] - (- 18 x^{2} + 6) \frac{d}{d x} [{(3 x^{2} + 1)}^{2}]}{{({(3 x^{2} + 1)}^{2})}^{2}}$

Step 1.1.2.2

Differentiate.

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Step 1.1.2.2.1

Multiply the exponents in ${({(3 x^{2} + 1)}^{2})}^{2}$ .

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Step 1.1.2.2.1.1

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

$\frac{{(3 x^{2} + 1)}^{2} \frac{d}{d x} [- 18 x^{2} + 6] - (- 18 x^{2} + 6) \frac{d}{d x} [{(3 x^{2} + 1)}^{2}]}{{(3 x^{2} + 1)}^{2 \cdot 2}}$

Step 1.1.2.2.1.2

Multiply $2$ by $2$ .

$\frac{{(3 x^{2} + 1)}^{2} \frac{d}{d x} [- 18 x^{2} + 6] - (- 18 x^{2} + 6) \frac{d}{d x} [{(3 x^{2} + 1)}^{2}]}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.2.2

By the Sum Rule, the derivative of $- 18 x^{2} + 6$ with respect to $x$ is $\frac{d}{d x} [- 18 x^{2}] + \frac{d}{d x} [6]$ .

$\frac{{(3 x^{2} + 1)}^{2} (\frac{d}{d x} [- 18 x^{2}] + \frac{d}{d x} [6]) - (- 18 x^{2} + 6) \frac{d}{d x} [{(3 x^{2} + 1)}^{2}]}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.2.3

Since $- 18$ is constant with respect to $x$ , the derivative of $- 18 x^{2}$ with respect to $x$ is $- 18 \frac{d}{d x} [x^{2}]$ .

$\frac{{(3 x^{2} + 1)}^{2} (- 18 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [6]) - (- 18 x^{2} + 6) \frac{d}{d x} [{(3 x^{2} + 1)}^{2}]}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.2.4

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

$\frac{{(3 x^{2} + 1)}^{2} (- 18 (2 x) + \frac{d}{d x} [6]) - (- 18 x^{2} + 6) \frac{d}{d x} [{(3 x^{2} + 1)}^{2}]}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.2.5

Multiply $2$ by $- 18$ .

$\frac{{(3 x^{2} + 1)}^{2} (- 36 x + \frac{d}{d x} [6]) - (- 18 x^{2} + 6) \frac{d}{d x} [{(3 x^{2} + 1)}^{2}]}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.2.6

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$\frac{{(3 x^{2} + 1)}^{2} (- 36 x + 0) - (- 18 x^{2} + 6) \frac{d}{d x} [{(3 x^{2} + 1)}^{2}]}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.2.7

Add $- 36 x$ and $0$ .

$\frac{{(3 x^{2} + 1)}^{2} (- 36 x) - (- 18 x^{2} + 6) \frac{d}{d x} [{(3 x^{2} + 1)}^{2}]}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.3

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 $g (x) = 3 x^{2} + 1$ .

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Step 1.1.2.3.1

To apply the Chain Rule, set $u$ as $3 x^{2} + 1$ .

$\frac{{(3 x^{2} + 1)}^{2} (- 36 x) - (- 18 x^{2} + 6) (\frac{d}{d u} [u^{2}] \frac{d}{d x} [3 x^{2} + 1])}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.3.2

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

$\frac{{(3 x^{2} + 1)}^{2} (- 36 x) - (- 18 x^{2} + 6) (2 u \frac{d}{d x} [3 x^{2} + 1])}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.3.3

Replace all occurrences of $u$ with $3 x^{2} + 1$ .

$\frac{{(3 x^{2} + 1)}^{2} (- 36 x) - (- 18 x^{2} + 6) (2 (3 x^{2} + 1) \frac{d}{d x} [3 x^{2} + 1])}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.4

Differentiate.

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Step 1.1.2.4.1

Multiply $2$ by $- 1$ .

$\frac{{(3 x^{2} + 1)}^{2} (- 36 x) - 2 (- 18 x^{2} + 6) ((3 x^{2} + 1) \frac{d}{d x} [3 x^{2} + 1])}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.4.2

By the Sum Rule, the derivative of $3 x^{2} + 1$ with respect to $x$ is $\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [1]$ .

$\frac{{(3 x^{2} + 1)}^{2} (- 36 x) - 2 (- 18 x^{2} + 6) ((3 x^{2} + 1) (\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [1]))}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.4.3

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{2}$ with respect to $x$ is $3 \frac{d}{d x} [x^{2}]$ .

$\frac{{(3 x^{2} + 1)}^{2} (- 36 x) - 2 (- 18 x^{2} + 6) ((3 x^{2} + 1) (3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]))}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.4.4

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

$\frac{{(3 x^{2} + 1)}^{2} (- 36 x) - 2 (- 18 x^{2} + 6) ((3 x^{2} + 1) (3 (2 x) + \frac{d}{d x} [1]))}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.4.5

Multiply $2$ by $3$ .

$\frac{{(3 x^{2} + 1)}^{2} (- 36 x) - 2 (- 18 x^{2} + 6) ((3 x^{2} + 1) (6 x + \frac{d}{d x} [1]))}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.4.6

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

$\frac{{(3 x^{2} + 1)}^{2} (- 36 x) - 2 (- 18 x^{2} + 6) ((3 x^{2} + 1) (6 x + 0))}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.4.7

Simplify the expression.

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Step 1.1.2.4.7.1

Add $6 x$ and $0$ .

$\frac{{(3 x^{2} + 1)}^{2} (- 36 x) - 2 (- 18 x^{2} + 6) ((3 x^{2} + 1) (6 x))}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.4.7.2

Move $6$ to the left of $3 x^{2} + 1$ .

$\frac{{(3 x^{2} + 1)}^{2} (- 36 x) - 2 (- 18 x^{2} + 6) (6 \cdot (3 x^{2} + 1) x)}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.4.7.3

Multiply $6$ by $- 2$ .

$\frac{{(3 x^{2} + 1)}^{2} (- 36 x) - 12 (- 18 x^{2} + 6) ((3 x^{2} + 1) x)}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5

Simplify.

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Step 1.1.2.5.1

Apply the distributive property.

$\frac{{(3 x^{2} + 1)}^{2} (- 36 x) + (- 12 (- 18 x^{2}) - 12 \cdot 6) ((3 x^{2} + 1) x)}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.2

Apply the distributive property.

$\frac{{(3 x^{2} + 1)}^{2} (- 36 x) + (- 12 (- 18 x^{2}) - 12 \cdot 6) (3 x^{2} x + 1 x)}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.3

Simplify the numerator.

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Step 1.1.2.5.3.1

Simplify each term.

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Step 1.1.2.5.3.1.1

Rewrite using the commutative property of multiplication.

$\frac{- 36 {(3 x^{2} + 1)}^{2} x + (- 12 (- 18 x^{2}) - 12 \cdot 6) (3 x^{2} x + 1 x)}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.3.1.2

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

$\frac{- 36 ((3 x^{2} + 1) (3 x^{2} + 1)) x + (- 12 (- 18 x^{2}) - 12 \cdot 6) (3 x^{2} x + 1 x)}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.3.1.3

Expand $(3 x^{2} + 1) (3 x^{2} + 1)$ using the FOIL Method.

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Step 1.1.2.5.3.1.3.1

Apply the distributive property.

$\frac{- 36 (3 x^{2} (3 x^{2} + 1) + 1 (3 x^{2} + 1)) x + (- 12 (- 18 x^{2}) - 12 \cdot 6) (3 x^{2} x + 1 x)}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.3.1.3.2

Apply the distributive property.

$\frac{- 36 (3 x^{2} (3 x^{2}) + 3 x^{2} \cdot 1 + 1 (3 x^{2} + 1)) x + (- 12 (- 18 x^{2}) - 12 \cdot 6) (3 x^{2} x + 1 x)}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.3.1.3.3

Apply the distributive property.

$\frac{- 36 (3 x^{2} (3 x^{2}) + 3 x^{2} \cdot 1 + 1 (3 x^{2}) + 1 \cdot 1) x + (- 12 (- 18 x^{2}) - 12 \cdot 6) (3 x^{2} x + 1 x)}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.3.1.4

Simplify and combine like terms.

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Step 1.1.2.5.3.1.4.1

Simplify each term.

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Step 1.1.2.5.3.1.4.1.1

Rewrite using the commutative property of multiplication.

$\frac{- 36 (3 \cdot 3 x^{2} x^{2} + 3 x^{2} \cdot 1 + 1 (3 x^{2}) + 1 \cdot 1) x + (- 12 (- 18 x^{2}) - 12 \cdot 6) (3 x^{2} x + 1 x)}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.3.1.4.1.2

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

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Step 1.1.2.5.3.1.4.1.2.1

Move $x^{2}$ .

$\frac{- 36 (3 \cdot 3 (x^{2} x^{2}) + 3 x^{2} \cdot 1 + 1 (3 x^{2}) + 1 \cdot 1) x + (- 12 (- 18 x^{2}) - 12 \cdot 6) (3 x^{2} x + 1 x)}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.3.1.4.1.2.2

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

$\frac{- 36 (3 \cdot 3 x^{2 + 2} + 3 x^{2} \cdot 1 + 1 (3 x^{2}) + 1 \cdot 1) x + (- 12 (- 18 x^{2}) - 12 \cdot 6) (3 x^{2} x + 1 x)}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.3.1.4.1.2.3

Add $2$ and $2$ .

$\frac{- 36 (3 \cdot 3 x^{4} + 3 x^{2} \cdot 1 + 1 (3 x^{2}) + 1 \cdot 1) x + (- 12 (- 18 x^{2}) - 12 \cdot 6) (3 x^{2} x + 1 x)}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.3.1.4.1.3

Multiply $3$ by $3$ .

$\frac{- 36 (9 x^{4} + 3 x^{2} \cdot 1 + 1 (3 x^{2}) + 1 \cdot 1) x + (- 12 (- 18 x^{2}) - 12 \cdot 6) (3 x^{2} x + 1 x)}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.3.1.4.1.4

Multiply $3$ by $1$ .

$\frac{- 36 (9 x^{4} + 3 x^{2} + 1 (3 x^{2}) + 1 \cdot 1) x + (- 12 (- 18 x^{2}) - 12 \cdot 6) (3 x^{2} x + 1 x)}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.3.1.4.1.5

Multiply $3 x^{2}$ by $1$ .

$\frac{- 36 (9 x^{4} + 3 x^{2} + 3 x^{2} + 1 \cdot 1) x + (- 12 (- 18 x^{2}) - 12 \cdot 6) (3 x^{2} x + 1 x)}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.3.1.4.1.6

Multiply $1$ by $1$ .

$\frac{- 36 (9 x^{4} + 3 x^{2} + 3 x^{2} + 1) x + (- 12 (- 18 x^{2}) - 12 \cdot 6) (3 x^{2} x + 1 x)}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.3.1.4.2

Add $3 x^{2}$ and $3 x^{2}$ .

$\frac{- 36 (9 x^{4} + 6 x^{2} + 1) x + (- 12 (- 18 x^{2}) - 12 \cdot 6) (3 x^{2} x + 1 x)}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.3.1.5

Apply the distributive property.

$\frac{(- 36 (9 x^{4}) - 36 (6 x^{2}) - 36 \cdot 1) x + (- 12 (- 18 x^{2}) - 12 \cdot 6) (3 x^{2} x + 1 x)}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.3.1.6

Simplify.

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Step 1.1.2.5.3.1.6.1

Multiply $9$ by $- 36$ .

$\frac{(- 324 x^{4} - 36 (6 x^{2}) - 36 \cdot 1) x + (- 12 (- 18 x^{2}) - 12 \cdot 6) (3 x^{2} x + 1 x)}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.3.1.6.2

Multiply $6$ by $- 36$ .

$\frac{(- 324 x^{4} - 216 x^{2} - 36 \cdot 1) x + (- 12 (- 18 x^{2}) - 12 \cdot 6) (3 x^{2} x + 1 x)}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.3.1.6.3

Multiply $- 36$ by $1$ .

$\frac{(- 324 x^{4} - 216 x^{2} - 36) x + (- 12 (- 18 x^{2}) - 12 \cdot 6) (3 x^{2} x + 1 x)}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.3.1.7

Apply the distributive property.

$\frac{- 324 x^{4} x - 216 x^{2} x - 36 x + (- 12 (- 18 x^{2}) - 12 \cdot 6) (3 x^{2} x + 1 x)}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.3.1.8

Simplify.

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Step 1.1.2.5.3.1.8.1

Multiply $x^{4}$ by $x$ by adding the exponents.

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Step 1.1.2.5.3.1.8.1.1

Move $x$ .

$\frac{- 324 (x \cdot x^{4}) - 216 x^{2} x - 36 x + (- 12 (- 18 x^{2}) - 12 \cdot 6) (3 x^{2} x + 1 x)}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.3.1.8.1.2

Multiply $x$ by $x^{4}$ .

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Step 1.1.2.5.3.1.8.1.2.1

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

$\frac{- 324 (x^{1} x^{4}) - 216 x^{2} x - 36 x + (- 12 (- 18 x^{2}) - 12 \cdot 6) (3 x^{2} x + 1 x)}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.3.1.8.1.2.2

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

$\frac{- 324 x^{1 + 4} - 216 x^{2} x - 36 x + (- 12 (- 18 x^{2}) - 12 \cdot 6) (3 x^{2} x + 1 x)}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.3.1.8.1.3

Add $1$ and $4$ .

$\frac{- 324 x^{5} - 216 x^{2} x - 36 x + (- 12 (- 18 x^{2}) - 12 \cdot 6) (3 x^{2} x + 1 x)}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.3.1.8.2

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

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Step 1.1.2.5.3.1.8.2.1

Move $x$ .

$\frac{- 324 x^{5} - 216 (x \cdot x^{2}) - 36 x + (- 12 (- 18 x^{2}) - 12 \cdot 6) (3 x^{2} x + 1 x)}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.3.1.8.2.2

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

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Step 1.1.2.5.3.1.8.2.2.1

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

$\frac{- 324 x^{5} - 216 (x^{1} x^{2}) - 36 x + (- 12 (- 18 x^{2}) - 12 \cdot 6) (3 x^{2} x + 1 x)}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.3.1.8.2.2.2

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

$\frac{- 324 x^{5} - 216 x^{1 + 2} - 36 x + (- 12 (- 18 x^{2}) - 12 \cdot 6) (3 x^{2} x + 1 x)}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.3.1.8.2.3

Add $1$ and $2$ .

$\frac{- 324 x^{5} - 216 x^{3} - 36 x + (- 12 (- 18 x^{2}) - 12 \cdot 6) (3 x^{2} x + 1 x)}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.3.1.9

Simplify each term.

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Step 1.1.2.5.3.1.9.1

Multiply $- 18$ by $- 12$ .

$\frac{- 324 x^{5} - 216 x^{3} - 36 x + (216 x^{2} - 12 \cdot 6) (3 x^{2} x + 1 x)}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.3.1.9.2

Multiply $- 12$ by $6$ .

$\frac{- 324 x^{5} - 216 x^{3} - 36 x + (216 x^{2} - 72) (3 x^{2} x + 1 x)}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.3.1.10

Simplify each term.

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Step 1.1.2.5.3.1.10.1

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

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Step 1.1.2.5.3.1.10.1.1

Move $x$ .

$\frac{- 324 x^{5} - 216 x^{3} - 36 x + (216 x^{2} - 72) (3 (x \cdot x^{2}) + 1 x)}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.3.1.10.1.2

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

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Step 1.1.2.5.3.1.10.1.2.1

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

$\frac{- 324 x^{5} - 216 x^{3} - 36 x + (216 x^{2} - 72) (3 (x^{1} x^{2}) + 1 x)}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.3.1.10.1.2.2

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

$\frac{- 324 x^{5} - 216 x^{3} - 36 x + (216 x^{2} - 72) (3 x^{1 + 2} + 1 x)}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.3.1.10.1.3

Add $1$ and $2$ .

$\frac{- 324 x^{5} - 216 x^{3} - 36 x + (216 x^{2} - 72) (3 x^{3} + 1 x)}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.3.1.10.2

Multiply $x$ by $1$ .

$\frac{- 324 x^{5} - 216 x^{3} - 36 x + (216 x^{2} - 72) (3 x^{3} + x)}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.3.1.11

Expand $(216 x^{2} - 72) (3 x^{3} + x)$ using the FOIL Method.

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Step 1.1.2.5.3.1.11.1

Apply the distributive property.

$\frac{- 324 x^{5} - 216 x^{3} - 36 x + 216 x^{2} (3 x^{3} + x) - 72 (3 x^{3} + x)}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.3.1.11.2

Apply the distributive property.

$\frac{- 324 x^{5} - 216 x^{3} - 36 x + 216 x^{2} (3 x^{3}) + 216 x^{2} x - 72 (3 x^{3} + x)}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.3.1.11.3

Apply the distributive property.

$\frac{- 324 x^{5} - 216 x^{3} - 36 x + 216 x^{2} (3 x^{3}) + 216 x^{2} x - 72 (3 x^{3}) - 72 x}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.3.1.12

Simplify and combine like terms.

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Step 1.1.2.5.3.1.12.1

Simplify each term.

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Step 1.1.2.5.3.1.12.1.1

Rewrite using the commutative property of multiplication.

$\frac{- 324 x^{5} - 216 x^{3} - 36 x + 216 \cdot 3 x^{2} x^{3} + 216 x^{2} x - 72 (3 x^{3}) - 72 x}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.3.1.12.1.2

Multiply $x^{2}$ by $x^{3}$ by adding the exponents.

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Step 1.1.2.5.3.1.12.1.2.1

Move $x^{3}$ .

$\frac{- 324 x^{5} - 216 x^{3} - 36 x + 216 \cdot 3 (x^{3} x^{2}) + 216 x^{2} x - 72 (3 x^{3}) - 72 x}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.3.1.12.1.2.2

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

$\frac{- 324 x^{5} - 216 x^{3} - 36 x + 216 \cdot 3 x^{3 + 2} + 216 x^{2} x - 72 (3 x^{3}) - 72 x}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.3.1.12.1.2.3

Add $3$ and $2$ .

$\frac{- 324 x^{5} - 216 x^{3} - 36 x + 216 \cdot 3 x^{5} + 216 x^{2} x - 72 (3 x^{3}) - 72 x}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.3.1.12.1.3

Multiply $216$ by $3$ .

$\frac{- 324 x^{5} - 216 x^{3} - 36 x + 648 x^{5} + 216 x^{2} x - 72 (3 x^{3}) - 72 x}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.3.1.12.1.4

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

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Step 1.1.2.5.3.1.12.1.4.1

Move $x$ .

$\frac{- 324 x^{5} - 216 x^{3} - 36 x + 648 x^{5} + 216 (x \cdot x^{2}) - 72 (3 x^{3}) - 72 x}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.3.1.12.1.4.2

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

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Step 1.1.2.5.3.1.12.1.4.2.1

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

$\frac{- 324 x^{5} - 216 x^{3} - 36 x + 648 x^{5} + 216 (x^{1} x^{2}) - 72 (3 x^{3}) - 72 x}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.3.1.12.1.4.2.2

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

$\frac{- 324 x^{5} - 216 x^{3} - 36 x + 648 x^{5} + 216 x^{1 + 2} - 72 (3 x^{3}) - 72 x}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.3.1.12.1.4.3

Add $1$ and $2$ .

$\frac{- 324 x^{5} - 216 x^{3} - 36 x + 648 x^{5} + 216 x^{3} - 72 (3 x^{3}) - 72 x}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.3.1.12.1.5

Multiply $3$ by $- 72$ .

$\frac{- 324 x^{5} - 216 x^{3} - 36 x + 648 x^{5} + 216 x^{3} - 216 x^{3} - 72 x}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.3.1.12.2

Subtract $216 x^{3}$ from $216 x^{3}$ .

$\frac{- 324 x^{5} - 216 x^{3} - 36 x + 648 x^{5} + 0 - 72 x}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.3.1.12.3

Add $648 x^{5}$ and $0$ .

$\frac{- 324 x^{5} - 216 x^{3} - 36 x + 648 x^{5} - 72 x}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.3.2

Add $- 324 x^{5}$ and $648 x^{5}$ .

$\frac{324 x^{5} - 216 x^{3} - 36 x - 72 x}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.3.3

Subtract $72 x$ from $- 36 x$ .

$\frac{324 x^{5} - 216 x^{3} - 108 x}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.4

Simplify the numerator.

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Step 1.1.2.5.4.1

Factor $108 x$ out of $324 x^{5} - 216 x^{3} - 108 x$ .

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Step 1.1.2.5.4.1.1

Factor $108 x$ out of $324 x^{5}$ .

$\frac{108 x (3 x^{4}) - 216 x^{3} - 108 x}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.4.1.2

Factor $108 x$ out of $- 216 x^{3}$ .

$\frac{108 x (3 x^{4}) + 108 x (- 2 x^{2}) - 108 x}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.4.1.3

Factor $108 x$ out of $- 108 x$ .

$\frac{108 x (3 x^{4}) + 108 x (- 2 x^{2}) + 108 x (- 1)}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.4.1.4

Factor $108 x$ out of $108 x (3 x^{4}) + 108 x (- 2 x^{2})$ .

$\frac{108 x (3 x^{4} - 2 x^{2}) + 108 x (- 1)}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.4.1.5

Factor $108 x$ out of $108 x (3 x^{4} - 2 x^{2}) + 108 x (- 1)$ .

$\frac{108 x (3 x^{4} - 2 x^{2} - 1)}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.4.2

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

$\frac{108 x (3 {(x^{2})}^{2} - 2 x^{2} - 1)}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.4.3

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$\frac{108 x (3 u^{2} - 2 u - 1)}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.4.4

Factor by grouping.

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Step 1.1.2.5.4.4.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot - 1 = - 3$ and whose sum is $b = - 2$ .

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Step 1.1.2.5.4.4.1.1

Factor $- 2$ out of $- 2 u$ .

$\frac{108 x (3 u^{2} - 2 (u) - 1)}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.4.4.1.2

Rewrite $- 2$ as $1$ plus $- 3$

$\frac{108 x (3 u^{2} + (1 - 3) u - 1)}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.4.4.1.3

Apply the distributive property.

$\frac{108 x (3 u^{2} + 1 u - 3 u - 1)}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.4.4.1.4

Multiply $u$ by $1$ .

$\frac{108 x (3 u^{2} + u - 3 u - 1)}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.4.4.2

Factor out the greatest common factor from each group.

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Step 1.1.2.5.4.4.2.1

Group the first two terms and the last two terms.

$\frac{108 x ((3 u^{2} + u) - 3 u - 1)}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.4.4.2.2

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

$\frac{108 x (u (3 u + 1) - (3 u + 1))}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.4.4.3

Factor the polynomial by factoring out the greatest common factor, $3 u + 1$ .

$\frac{108 x ((3 u + 1) (u - 1))}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.4.5

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

$\frac{108 x ((3 x^{2} + 1) (x^{2} - 1))}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.4.6

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

$\frac{108 x ((3 x^{2} + 1) (x^{2} - 1^{2}))}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.4.7

Factor.

$\frac{108 x (3 x^{2} + 1) (x + 1) (x - 1)}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.5

Cancel the common factor of $3 x^{2} + 1$ and ${(3 x^{2} + 1)}^{4}$ .

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Step 1.1.2.5.5.1

Factor $3 x^{2} + 1$ out of $108 x (3 x^{2} + 1) (x + 1) (x - 1)$ .

$\frac{(3 x^{2} + 1) ((108 x (x + 1)) (x - 1))}{{(3 x^{2} + 1)}^{4}}$

Step 1.1.2.5.5.2

Cancel the common factors.

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Step 1.1.2.5.5.2.1

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

$\frac{(3 x^{2} + 1) ((108 x (x + 1)) (x - 1))}{(3 x^{2} + 1) {(3 x^{2} + 1)}^{3}}$

Step 1.1.2.5.5.2.2

Cancel the common factor.

$\frac{(3 x^{2} + 1) ((108 x (x + 1)) (x - 1))}{(3 x^{2} + 1) {(3 x^{2} + 1)}^{3}}$

Step 1.1.2.5.5.2.3

Rewrite the expression.

$f'' (x) = \frac{(108 x (x + 1)) (x - 1)}{{(3 x^{2} + 1)}^{3}}$

$f'' (x) = \frac{108 x (x + 1) (x - 1)}{{(3 x^{2} + 1)}^{3}}$

Step 1.1.3

The second derivative of $f (x)$ with respect to $x$ is $\frac{108 x (x + 1) (x - 1)}{{(3 x^{2} + 1)}^{3}}$ .

$\frac{108 x (x + 1) (x - 1)}{{(3 x^{2} + 1)}^{3}}$

Step 1.2

Set the second derivative equal to $0$ then solve the equation $\frac{108 x (x + 1) (x - 1)}{{(3 x^{2} + 1)}^{3}} = 0$ .

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Step 1.2.1

Set the second derivative equal to $0$ .

$\frac{108 x (x + 1) (x - 1)}{{(3 x^{2} + 1)}^{3}} = 0$

Step 1.2.2

Set the numerator equal to zero.

$108 x (x + 1) (x - 1) = 0$

Step 1.2.3

Solve the equation for $x$ .

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Step 1.2.3.1

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

$x = 0$

$x + 1 = 0$

$x - 1 = 0$

Step 1.2.3.2

Set $x$ equal to $0$ .

$x = 0$

Step 1.2.3.3

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

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Step 1.2.3.3.1

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

$x + 1 = 0$

Step 1.2.3.3.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 1.2.3.4

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

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Step 1.2.3.4.1

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

$x - 1 = 0$

Step 1.2.3.4.2

Add $1$ to both sides of the equation.

$x = 1$

Step 1.2.3.5

The final solution is all the values that make $108 x (x + 1) (x - 1) = 0$ true.

$x = 0, - 1, 1$

Step 2

Find the domain of $f (x) = \frac{6 x}{1 + 3 x^{2}}$ .

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Step 2.1

Set the denominator in $\frac{6 x}{1 + 3 x^{2}}$ equal to $0$ to find where the expression is undefined.

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

Step 2.2

Solve for $x$ .

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Step 2.2.1

Subtract $1$ from both sides of the equation.

$3 x^{2} = - 1$

Step 2.2.2

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

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Step 2.2.2.1

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

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

Step 2.2.2.2

Simplify the left side.

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Step 2.2.2.2.1

Cancel the common factor of $3$ .

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Step 2.2.2.2.1.1

Cancel the common factor.

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

Step 2.2.2.2.1.2

Divide $x^{2}$ by $1$ .

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

Step 2.2.2.3

Simplify the right side.

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Step 2.2.2.3.1

Move the negative in front of the fraction.

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

Step 2.2.3

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

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

Step 2.2.4

Simplify $\pm \sqrt{- \frac{1}{3}}$ .

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Step 2.2.4.1

Rewrite $- \frac{1}{3}$ as $i^{2} \frac{1^{2}}{3}$ .

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Step 2.2.4.1.1

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

$x = \pm \sqrt{i^{2} \frac{1}{3}}$

Step 2.2.4.1.2

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

$x = \pm \sqrt{i^{2} \frac{1^{2}}{3}}$

Step 2.2.4.2

Pull terms out from under the radical.

$x = \pm i \sqrt{\frac{1^{2}}{3}}$

Step 2.2.4.3

One to any power is one.

$x = \pm i \sqrt{\frac{1}{3}}$

Step 2.2.4.4

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

$x = \pm i \frac{\sqrt{1}}{\sqrt{3}}$

Step 2.2.4.5

Any root of $1$ is $1$ .

$x = \pm i \frac{1}{\sqrt{3}}$

Step 2.2.4.6

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

$x = \pm i (\frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})$

Step 2.2.4.7

Combine and simplify the denominator.

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Step 2.2.4.7.1

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

$x = \pm i \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 2.2.4.7.2

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

$x = \pm i \frac{\sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 2.2.4.7.3

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

$x = \pm i \frac{\sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 2.2.4.7.4

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

$x = \pm i \frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 2.2.4.7.5

Add $1$ and $1$ .

$x = \pm i \frac{\sqrt{3}}{{\sqrt{3}}^{2}}$

Step 2.2.4.7.6

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

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Step 2.2.4.7.6.1

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

$x = \pm i \frac{\sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 2.2.4.7.6.2

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

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

Step 2.2.4.7.6.3

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

$x = \pm i \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Step 2.2.4.7.6.4

Cancel the common factor of $2$ .

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Step 2.2.4.7.6.4.1

Cancel the common factor.

$x = \pm i \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Step 2.2.4.7.6.4.2

Rewrite the expression.

$x = \pm i \frac{\sqrt{3}}{3^{1}}$

Step 2.2.4.7.6.5

Evaluate the exponent.

$x = \pm i \frac{\sqrt{3}}{3}$

Step 2.2.4.8

Combine $i$ and $\frac{\sqrt{3}}{3}$ .

$x = \pm \frac{i \sqrt{3}}{3}$

Step 2.2.5

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

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Step 2.2.5.1

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

$x = \frac{i \sqrt{3}}{3}$

Step 2.2.5.2

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

$x = - \frac{i \sqrt{3}}{3}$

Step 2.2.5.3

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

$x = \frac{i \sqrt{3}}{3}, - \frac{i \sqrt{3}}{3}$

Step 2.3

The domain is all real numbers.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 3

Create intervals around the $x$ -values where the second derivative is zero or undefined.

$(- \infty, - 1) \cup (- 1, 0) \cup (0, 1) \cup (1, \infty)$

Step 4

Substitute any number from the interval $(- \infty, - 1)$ into the second derivative and evaluate to determine the concavity.

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Step 4.1

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

$f'' (- 4) = \frac{108 (- 4) ((- 4) + 1) ((- 4) - 1)}{{(3 {(- 4)}^{2} + 1)}^{3}}$

Step 4.2

Simplify the result.

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Step 4.2.1

Multiply $108$ by $- 4$ .

$f'' (- 4) = \frac{- 432 (- 4 + 1) (- 4 - 1)}{{(3 {(- 4)}^{2} + 1)}^{3}}$

Step 4.2.2

Simplify the denominator.

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Step 4.2.2.1

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

$f'' (- 4) = \frac{- 432 (- 4 + 1) (- 4 - 1)}{{(3 \cdot 16 + 1)}^{3}}$

Step 4.2.2.2

Multiply $3$ by $16$ .

$f'' (- 4) = \frac{- 432 (- 4 + 1) (- 4 - 1)}{{(48 + 1)}^{3}}$

Step 4.2.2.3

Add $48$ and $1$ .

$f'' (- 4) = \frac{- 432 (- 4 + 1) (- 4 - 1)}{49^{3}}$

Step 4.2.2.4

Raise $49$ to the power of $3$ .

$f'' (- 4) = \frac{- 432 (- 4 + 1) (- 4 - 1)}{117649}$

Step 4.2.3

Simplify the numerator.

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Step 4.2.3.1

Add $- 4$ and $1$ .

$f'' (- 4) = \frac{- 432 \cdot (- 3 (- 4 - 1))}{117649}$

Step 4.2.3.2

Multiply $- 432$ by $- 3$ .

$f'' (- 4) = \frac{1296 (- 4 - 1)}{117649}$

Step 4.2.3.3

Subtract $1$ from $- 4$ .

$f'' (- 4) = \frac{1296 \cdot - 5}{117649}$

Step 4.2.4

Simplify the expression.

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Step 4.2.4.1

Multiply $1296$ by $- 5$ .

$f'' (- 4) = \frac{- 6480}{117649}$

Step 4.2.4.2

Move the negative in front of the fraction.

$f'' (- 4) = - \frac{6480}{117649}$

Step 4.2.5

The final answer is $- \frac{6480}{117649}$ .

$- \frac{6480}{117649}$

Step 4.3

The graph is concave down on the interval $(- \infty, - 1)$ because $f'' (- 4)$ is negative.

Concave down on $(- \infty, - 1)$ since $f'' (x)$ is negative

Step 5

Substitute any number from the interval $(- 1, 0)$ into the second derivative and evaluate to determine the concavity.

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Step 5.1

Replace the variable $x$ with $- 0.5$ in the expression.

$f'' (- 0.5) = \frac{108 (- 0.5) ((- 0.5) + 1) ((- 0.5) - 1)}{{(3 {(- 0.5)}^{2} + 1)}^{3}}$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify the numerator.

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Step 5.2.1.1

Multiply $108$ by $- 0.5$ .

$f'' (- 0.5) = \frac{- 54 (- 0.5 + 1) (- 0.5 - 1)}{{(3 {(- 0.5)}^{2} + 1)}^{3}}$

Step 5.2.1.2

Multiply $- 54$ by $- 0.5 + 1$ .

$f'' (- 0.5) = \frac{- 27 (- 0.5 - 1)}{{(3 {(- 0.5)}^{2} + 1)}^{3}}$

Step 5.2.1.3

Multiply $- 27$ by $- 0.5 - 1$ .

$f'' (- 0.5) = \frac{40.5}{{(3 {(- 0.5)}^{2} + 1)}^{3}}$

Step 5.2.2

Simplify the denominator.

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Step 5.2.2.1

Raise $- 0.5$ to the power of $2$ .

$f'' (- 0.5) = \frac{40.5}{{(3 \cdot 0.25 + 1)}^{3}}$

Step 5.2.2.2

Multiply $3$ by $0.25$ .

$f'' (- 0.5) = \frac{40.5}{{(0.75 + 1)}^{3}}$

Step 5.2.2.3

Add $0.75$ and $1$ .

$f'' (- 0.5) = \frac{40.5}{{1.75}^{3}}$

Step 5.2.2.4

Raise $1.75$ to the power of $3$ .

$f'' (- 0.5) = \frac{40.5}{5.359375}$

Step 5.2.3

Divide $40.5$ by $5.359375$ .

$f'' (- 0.5) = 7.55685131$

Step 5.2.4

The final answer is $7.55685131$ .

$7.55685131$

Step 5.3

The graph is concave up on the interval $(- 1, 0)$ because $f'' (- 0.5)$ is positive.

Concave up on $(- 1, 0)$ since $f'' (x)$ is positive

Step 6

Substitute any number from the interval $(0, 1)$ into the second derivative and evaluate to determine the concavity.

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Step 6.1

Replace the variable $x$ with $0.5$ in the expression.

$f'' (0.5) = \frac{108 (0.5) ((0.5) + 1) ((0.5) - 1)}{{(3 {(0.5)}^{2} + 1)}^{3}}$

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

Multiply $108$ by $0.5$ .

$f'' (0.5) = \frac{54 (0.5 + 1) (0.5 - 1)}{{(3 \cdot {0.5}^{2} + 1)}^{3}}$

Step 6.2.1.2

Multiply $54$ by $0.5 + 1$ .

$f'' (0.5) = \frac{81 (0.5 - 1)}{{(3 \cdot {0.5}^{2} + 1)}^{3}}$

Step 6.2.1.3

Multiply $81$ by $0.5 - 1$ .

$f'' (0.5) = \frac{- 40.5}{{(3 \cdot {0.5}^{2} + 1)}^{3}}$

Step 6.2.2

Simplify the denominator.

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Step 6.2.2.1

Raise $0.5$ to the power of $2$ .

$f'' (0.5) = \frac{- 40.5}{{(3 \cdot 0.25 + 1)}^{3}}$

Step 6.2.2.2

Multiply $3$ by $0.25$ .

$f'' (0.5) = \frac{- 40.5}{{(0.75 + 1)}^{3}}$

Step 6.2.2.3

Add $0.75$ and $1$ .

$f'' (0.5) = \frac{- 40.5}{{1.75}^{3}}$

Step 6.2.2.4

Raise $1.75$ to the power of $3$ .

$f'' (0.5) = \frac{- 40.5}{5.359375}$

Step 6.2.3

Divide $- 40.5$ by $5.359375$ .

$f'' (0.5) = - 7.55685131$

Step 6.2.4

The final answer is $- 7.55685131$ .

$- 7.55685131$

Step 6.3

The graph is concave down on the interval $(0, 1)$ because $f'' (0.5)$ is negative.

Concave down on $(0, 1)$ since $f'' (x)$ is negative

Step 7

Substitute any number from the interval $(1, \infty)$ into the second derivative and evaluate to determine the concavity.

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Step 7.1

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

$f'' (4) = \frac{108 (4) ((4) + 1) ((4) - 1)}{{(3 {(4)}^{2} + 1)}^{3}}$

Step 7.2

Simplify the result.

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Step 7.2.1

Multiply $108$ by $4$ .

$f'' (4) = \frac{432 (4 + 1) (4 - 1)}{{(3 \cdot 4^{2} + 1)}^{3}}$

Step 7.2.2

Simplify the denominator.

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Step 7.2.2.1

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

$f'' (4) = \frac{432 (4 + 1) (4 - 1)}{{(3 \cdot 16 + 1)}^{3}}$

Step 7.2.2.2

Multiply $3$ by $16$ .

$f'' (4) = \frac{432 (4 + 1) (4 - 1)}{{(48 + 1)}^{3}}$

Step 7.2.2.3

Add $48$ and $1$ .

$f'' (4) = \frac{432 (4 + 1) (4 - 1)}{49^{3}}$

Step 7.2.2.4

Raise $49$ to the power of $3$ .

$f'' (4) = \frac{432 (4 + 1) (4 - 1)}{117649}$

Step 7.2.3

Simplify the numerator.

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Step 7.2.3.1

Add $4$ and $1$ .

$f'' (4) = \frac{432 \cdot (5 (4 - 1))}{117649}$

Step 7.2.3.2

Multiply $432$ by $5$ .

$f'' (4) = \frac{2160 (4 - 1)}{117649}$

Step 7.2.3.3

Subtract $1$ from $4$ .

$f'' (4) = \frac{2160 \cdot 3}{117649}$

Step 7.2.4

Multiply $2160$ by $3$ .

$f'' (4) = \frac{6480}{117649}$

Step 7.2.5

The final answer is $\frac{6480}{117649}$ .

$\frac{6480}{117649}$

Step 7.3

The graph is concave up on the interval $(1, \infty)$ because $f'' (4)$ is positive.

Concave up on $(1, \infty)$ since $f'' (x)$ is positive

Step 8

The graph is concave down when the second derivative is negative and concave up when the second derivative is positive.

Concave down on $(- \infty, - 1)$ since $f'' (x)$ is negative

Concave up on $(- 1, 0)$ since $f'' (x)$ is positive

Concave down on $(0, 1)$ since $f'' (x)$ is negative

Concave up on $(1, \infty)$ since $f'' (x)$ is positive

Step 9