Calculus Examples

$\frac{x}{x^{2} + 27}$

Step 1

Write $\frac{x}{x^{2} + 27}$ as a function.

$f (x) = \frac{x}{x^{2} + 27}$

Step 2

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

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Step 2.1

Find the second derivative.

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Step 2.1.1

Find the first derivative.

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Step 2.1.1.1

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 $f (x) = x$ and $g (x) = x^{2} + 27$ .

$\frac{(x^{2} + 27) \frac{d}{d x} [x] - x \frac{d}{d x} [x^{2} + 27]}{{(x^{2} + 27)}^{2}}$

Step 2.1.1.2

Differentiate.

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Step 2.1.1.2.1

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

$\frac{(x^{2} + 27) \cdot 1 - x \frac{d}{d x} [x^{2} + 27]}{{(x^{2} + 27)}^{2}}$

Step 2.1.1.2.2

Multiply $x^{2} + 27$ by $1$ .

$\frac{x^{2} + 27 - x \frac{d}{d x} [x^{2} + 27]}{{(x^{2} + 27)}^{2}}$

Step 2.1.1.2.3

By the Sum Rule, the derivative of $x^{2} + 27$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [27]$ .

$\frac{x^{2} + 27 - x (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [27])}{{(x^{2} + 27)}^{2}}$

Step 2.1.1.2.4

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

$\frac{x^{2} + 27 - x (2 x + \frac{d}{d x} [27])}{{(x^{2} + 27)}^{2}}$

Step 2.1.1.2.5

Since $27$ is constant with respect to $x$ , the derivative of $27$ with respect to $x$ is $0$ .

$\frac{x^{2} + 27 - x (2 x + 0)}{{(x^{2} + 27)}^{2}}$

Step 2.1.1.2.6

Simplify the expression.

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Step 2.1.1.2.6.1

Add $2 x$ and $0$ .

$\frac{x^{2} + 27 - x (2 x)}{{(x^{2} + 27)}^{2}}$

Step 2.1.1.2.6.2

Multiply $2$ by $- 1$ .

$\frac{x^{2} + 27 - 2 x \cdot x}{{(x^{2} + 27)}^{2}}$

Step 2.1.1.3

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

$\frac{x^{2} + 27 - 2 (x^{1} x)}{{(x^{2} + 27)}^{2}}$

Step 2.1.1.4

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

$\frac{x^{2} + 27 - 2 (x^{1} x^{1})}{{(x^{2} + 27)}^{2}}$

Step 2.1.1.5

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

$\frac{x^{2} + 27 - 2 x^{1 + 1}}{{(x^{2} + 27)}^{2}}$

Step 2.1.1.6

Add $1$ and $1$ .

$\frac{x^{2} + 27 - 2 x^{2}}{{(x^{2} + 27)}^{2}}$

Step 2.1.1.7

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

$\frac{- x^{2} + 27}{{(x^{2} + 27)}^{2}}$

Step 2.1.1.8

Simplify.

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Step 2.1.1.8.1

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

$\frac{- (x^{2}) + 27}{{(x^{2} + 27)}^{2}}$

Step 2.1.1.8.2

Rewrite $27$ as $- 1 (- 27)$ .

$\frac{- (x^{2}) - 1 (- 27)}{{(x^{2} + 27)}^{2}}$

Step 2.1.1.8.3

Factor $- 1$ out of $- (x^{2}) - 1 (- 27)$ .

$\frac{- (x^{2} - 27)}{{(x^{2} + 27)}^{2}}$

Step 2.1.1.8.4

Rewrite $- (x^{2} - 27)$ as $- 1 (x^{2} - 27)$ .

$\frac{- 1 (x^{2} - 27)}{{(x^{2} + 27)}^{2}}$

Step 2.1.1.8.5

Move the negative in front of the fraction.

$f' (x) = - \frac{x^{2} - 27}{{(x^{2} + 27)}^{2}}$

Step 2.1.2

Find the second derivative.

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Step 2.1.2.1

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 $g (x) = \frac{x^{2} - 27}{{(x^{2} + 27)}^{2}}$ .

$- \frac{d}{d x} [\frac{x^{2} - 27}{{(x^{2} + 27)}^{2}}] + \frac{x^{2} - 27}{{(x^{2} + 27)}^{2}} \frac{d}{d x} [- 1]$

Step 2.1.2.2

$- \frac{{(x^{2} + 27)}^{2} \frac{d}{d x} [x^{2} - 27] - (x^{2} - 27) \frac{d}{d x} [{(x^{2} + 27)}^{2}]}{{({(x^{2} + 27)}^{2})}^{2}} + \frac{x^{2} - 27}{{(x^{2} + 27)}^{2}} \frac{d}{d x} [- 1]$

Step 2.1.2.3

Differentiate.

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Step 2.1.2.3.1

Multiply the exponents in ${({(x^{2} + 27)}^{2})}^{2}$ .

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Step 2.1.2.3.1.1

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

$- \frac{{(x^{2} + 27)}^{2} \frac{d}{d x} [x^{2} - 27] - (x^{2} - 27) \frac{d}{d x} [{(x^{2} + 27)}^{2}]}{{(x^{2} + 27)}^{2 \cdot 2}} + \frac{x^{2} - 27}{{(x^{2} + 27)}^{2}} \frac{d}{d x} [- 1]$

Step 2.1.2.3.1.2

Multiply $2$ by $2$ .

$- \frac{{(x^{2} + 27)}^{2} \frac{d}{d x} [x^{2} - 27] - (x^{2} - 27) \frac{d}{d x} [{(x^{2} + 27)}^{2}]}{{(x^{2} + 27)}^{4}} + \frac{x^{2} - 27}{{(x^{2} + 27)}^{2}} \frac{d}{d x} [- 1]$

Step 2.1.2.3.2

By the Sum Rule, the derivative of $x^{2} - 27$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 27]$ .

$- \frac{{(x^{2} + 27)}^{2} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 27]) - (x^{2} - 27) \frac{d}{d x} [{(x^{2} + 27)}^{2}]}{{(x^{2} + 27)}^{4}} + \frac{x^{2} - 27}{{(x^{2} + 27)}^{2}} \frac{d}{d x} [- 1]$

Step 2.1.2.3.3

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

$- \frac{{(x^{2} + 27)}^{2} (2 x + \frac{d}{d x} [- 27]) - (x^{2} - 27) \frac{d}{d x} [{(x^{2} + 27)}^{2}]}{{(x^{2} + 27)}^{4}} + \frac{x^{2} - 27}{{(x^{2} + 27)}^{2}} \frac{d}{d x} [- 1]$

Step 2.1.2.3.4

Since $- 27$ is constant with respect to $x$ , the derivative of $- 27$ with respect to $x$ is $0$ .

$- \frac{{(x^{2} + 27)}^{2} (2 x + 0) - (x^{2} - 27) \frac{d}{d x} [{(x^{2} + 27)}^{2}]}{{(x^{2} + 27)}^{4}} + \frac{x^{2} - 27}{{(x^{2} + 27)}^{2}} \frac{d}{d x} [- 1]$

Step 2.1.2.3.5

Simplify the expression.

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Step 2.1.2.3.5.1

Add $2 x$ and $0$ .

$- \frac{{(x^{2} + 27)}^{2} (2 x) - (x^{2} - 27) \frac{d}{d x} [{(x^{2} + 27)}^{2}]}{{(x^{2} + 27)}^{4}} + \frac{x^{2} - 27}{{(x^{2} + 27)}^{2}} \frac{d}{d x} [- 1]$

Step 2.1.2.3.5.2

Move $2$ to the left of ${(x^{2} + 27)}^{2}$ .

$- \frac{2 \cdot {(x^{2} + 27)}^{2} x - (x^{2} - 27) \frac{d}{d x} [{(x^{2} + 27)}^{2}]}{{(x^{2} + 27)}^{4}} + \frac{x^{2} - 27}{{(x^{2} + 27)}^{2}} \frac{d}{d x} [- 1]$

Step 2.1.2.4

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 $g (x) = x^{2} + 27$ .

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Step 2.1.2.4.1

To apply the Chain Rule, set $u$ as $x^{2} + 27$ .

$- \frac{2 {(x^{2} + 27)}^{2} x - (x^{2} - 27) (\frac{d}{d u} [u^{2}] \frac{d}{d x} [x^{2} + 27])}{{(x^{2} + 27)}^{4}} + \frac{x^{2} - 27}{{(x^{2} + 27)}^{2}} \frac{d}{d x} [- 1]$

Step 2.1.2.4.2

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

$- \frac{2 {(x^{2} + 27)}^{2} x - (x^{2} - 27) (2 u \frac{d}{d x} [x^{2} + 27])}{{(x^{2} + 27)}^{4}} + \frac{x^{2} - 27}{{(x^{2} + 27)}^{2}} \frac{d}{d x} [- 1]$

Step 2.1.2.4.3

Replace all occurrences of $u$ with $x^{2} + 27$ .

$- \frac{2 {(x^{2} + 27)}^{2} x - (x^{2} - 27) (2 (x^{2} + 27) \frac{d}{d x} [x^{2} + 27])}{{(x^{2} + 27)}^{4}} + \frac{x^{2} - 27}{{(x^{2} + 27)}^{2}} \frac{d}{d x} [- 1]$

Step 2.1.2.5

Differentiate.

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Step 2.1.2.5.1

Multiply $2$ by $- 1$ .

$- \frac{2 {(x^{2} + 27)}^{2} x - 2 (x^{2} - 27) ((x^{2} + 27) \frac{d}{d x} [x^{2} + 27])}{{(x^{2} + 27)}^{4}} + \frac{x^{2} - 27}{{(x^{2} + 27)}^{2}} \frac{d}{d x} [- 1]$

Step 2.1.2.5.2

By the Sum Rule, the derivative of $x^{2} + 27$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [27]$ .

$- \frac{2 {(x^{2} + 27)}^{2} x - 2 (x^{2} - 27) ((x^{2} + 27) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [27]))}{{(x^{2} + 27)}^{4}} + \frac{x^{2} - 27}{{(x^{2} + 27)}^{2}} \frac{d}{d x} [- 1]$

Step 2.1.2.5.3

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

$- \frac{2 {(x^{2} + 27)}^{2} x - 2 (x^{2} - 27) ((x^{2} + 27) (2 x + \frac{d}{d x} [27]))}{{(x^{2} + 27)}^{4}} + \frac{x^{2} - 27}{{(x^{2} + 27)}^{2}} \frac{d}{d x} [- 1]$

Step 2.1.2.5.4

Since $27$ is constant with respect to $x$ , the derivative of $27$ with respect to $x$ is $0$ .

$- \frac{2 {(x^{2} + 27)}^{2} x - 2 (x^{2} - 27) ((x^{2} + 27) (2 x + 0))}{{(x^{2} + 27)}^{4}} + \frac{x^{2} - 27}{{(x^{2} + 27)}^{2}} \frac{d}{d x} [- 1]$

Step 2.1.2.5.5

Simplify the expression.

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Step 2.1.2.5.5.1

Add $2 x$ and $0$ .

$- \frac{2 {(x^{2} + 27)}^{2} x - 2 (x^{2} - 27) ((x^{2} + 27) (2 x))}{{(x^{2} + 27)}^{4}} + \frac{x^{2} - 27}{{(x^{2} + 27)}^{2}} \frac{d}{d x} [- 1]$

Step 2.1.2.5.5.2

Move $2$ to the left of $x^{2} + 27$ .

$- \frac{2 {(x^{2} + 27)}^{2} x - 2 (x^{2} - 27) (2 \cdot (x^{2} + 27) x)}{{(x^{2} + 27)}^{4}} + \frac{x^{2} - 27}{{(x^{2} + 27)}^{2}} \frac{d}{d x} [- 1]$

Step 2.1.2.5.5.3

Multiply $2$ by $- 2$ .

$- \frac{2 {(x^{2} + 27)}^{2} x - 4 (x^{2} - 27) ((x^{2} + 27) x)}{{(x^{2} + 27)}^{4}} + \frac{x^{2} - 27}{{(x^{2} + 27)}^{2}} \frac{d}{d x} [- 1]$

Step 2.1.2.5.6

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

$- \frac{2 {(x^{2} + 27)}^{2} x - 4 (x^{2} - 27) ((x^{2} + 27) x)}{{(x^{2} + 27)}^{4}} + \frac{x^{2} - 27}{{(x^{2} + 27)}^{2}} \cdot 0$

Step 2.1.2.5.7

Simplify the expression.

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Step 2.1.2.5.7.1

Multiply $\frac{x^{2} - 27}{{(x^{2} + 27)}^{2}}$ by $0$ .

$- \frac{2 {(x^{2} + 27)}^{2} x - 4 (x^{2} - 27) ((x^{2} + 27) x)}{{(x^{2} + 27)}^{4}} + 0$

Step 2.1.2.5.7.2

Add $- \frac{2 {(x^{2} + 27)}^{2} x - 4 (x^{2} - 27) ((x^{2} + 27) x)}{{(x^{2} + 27)}^{4}}$ and $0$ .

$- \frac{2 {(x^{2} + 27)}^{2} x - 4 (x^{2} - 27) ((x^{2} + 27) x)}{{(x^{2} + 27)}^{4}}$

Step 2.1.2.6

Simplify.

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Step 2.1.2.6.1

Apply the distributive property.

$- \frac{2 {(x^{2} + 27)}^{2} x + (- 4 x^{2} - 4 \cdot - 27) ((x^{2} + 27) x)}{{(x^{2} + 27)}^{4}}$

Step 2.1.2.6.2

Apply the distributive property.

$- \frac{2 {(x^{2} + 27)}^{2} x + (- 4 x^{2} - 4 \cdot - 27) (x^{2} x + 27 x)}{{(x^{2} + 27)}^{4}}$

Step 2.1.2.6.3

Simplify the numerator.

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Step 2.1.2.6.3.1

Simplify each term.

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Step 2.1.2.6.3.1.1

Rewrite ${(x^{2} + 27)}^{2}$ as $(x^{2} + 27) (x^{2} + 27)$ .

$- \frac{2 ((x^{2} + 27) (x^{2} + 27)) x + (- 4 x^{2} - 4 \cdot - 27) (x^{2} x + 27 x)}{{(x^{2} + 27)}^{4}}$

Step 2.1.2.6.3.1.2

Expand $(x^{2} + 27) (x^{2} + 27)$ using the FOIL Method.

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Step 2.1.2.6.3.1.2.1

Apply the distributive property.

$- \frac{2 (x^{2} (x^{2} + 27) + 27 (x^{2} + 27)) x + (- 4 x^{2} - 4 \cdot - 27) (x^{2} x + 27 x)}{{(x^{2} + 27)}^{4}}$

Step 2.1.2.6.3.1.2.2

Apply the distributive property.

$- \frac{2 (x^{2} x^{2} + x^{2} \cdot 27 + 27 (x^{2} + 27)) x + (- 4 x^{2} - 4 \cdot - 27) (x^{2} x + 27 x)}{{(x^{2} + 27)}^{4}}$

Step 2.1.2.6.3.1.2.3

Apply the distributive property.

$- \frac{2 (x^{2} x^{2} + x^{2} \cdot 27 + 27 x^{2} + 27 \cdot 27) x + (- 4 x^{2} - 4 \cdot - 27) (x^{2} x + 27 x)}{{(x^{2} + 27)}^{4}}$

Step 2.1.2.6.3.1.3

Simplify and combine like terms.

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Step 2.1.2.6.3.1.3.1

Simplify each term.

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Step 2.1.2.6.3.1.3.1.1

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

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Step 2.1.2.6.3.1.3.1.1.1

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

$- \frac{2 (x^{2 + 2} + x^{2} \cdot 27 + 27 x^{2} + 27 \cdot 27) x + (- 4 x^{2} - 4 \cdot - 27) (x^{2} x + 27 x)}{{(x^{2} + 27)}^{4}}$

Step 2.1.2.6.3.1.3.1.1.2

Add $2$ and $2$ .

$- \frac{2 (x^{4} + x^{2} \cdot 27 + 27 x^{2} + 27 \cdot 27) x + (- 4 x^{2} - 4 \cdot - 27) (x^{2} x + 27 x)}{{(x^{2} + 27)}^{4}}$

Step 2.1.2.6.3.1.3.1.2

Move $27$ to the left of $x^{2}$ .

$- \frac{2 (x^{4} + 27 \cdot x^{2} + 27 x^{2} + 27 \cdot 27) x + (- 4 x^{2} - 4 \cdot - 27) (x^{2} x + 27 x)}{{(x^{2} + 27)}^{4}}$

Step 2.1.2.6.3.1.3.1.3

Multiply $27$ by $27$ .

$- \frac{2 (x^{4} + 27 x^{2} + 27 x^{2} + 729) x + (- 4 x^{2} - 4 \cdot - 27) (x^{2} x + 27 x)}{{(x^{2} + 27)}^{4}}$

Step 2.1.2.6.3.1.3.2

Add $27 x^{2}$ and $27 x^{2}$ .

$- \frac{2 (x^{4} + 54 x^{2} + 729) x + (- 4 x^{2} - 4 \cdot - 27) (x^{2} x + 27 x)}{{(x^{2} + 27)}^{4}}$

Step 2.1.2.6.3.1.4

Apply the distributive property.

$- \frac{(2 x^{4} + 2 (54 x^{2}) + 2 \cdot 729) x + (- 4 x^{2} - 4 \cdot - 27) (x^{2} x + 27 x)}{{(x^{2} + 27)}^{4}}$

Step 2.1.2.6.3.1.5

Simplify.

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Step 2.1.2.6.3.1.5.1

Multiply $54$ by $2$ .

$- \frac{(2 x^{4} + 108 x^{2} + 2 \cdot 729) x + (- 4 x^{2} - 4 \cdot - 27) (x^{2} x + 27 x)}{{(x^{2} + 27)}^{4}}$

Step 2.1.2.6.3.1.5.2

Multiply $2$ by $729$ .

$- \frac{(2 x^{4} + 108 x^{2} + 1458) x + (- 4 x^{2} - 4 \cdot - 27) (x^{2} x + 27 x)}{{(x^{2} + 27)}^{4}}$

Step 2.1.2.6.3.1.6

Apply the distributive property.

$- \frac{2 x^{4} x + 108 x^{2} x + 1458 x + (- 4 x^{2} - 4 \cdot - 27) (x^{2} x + 27 x)}{{(x^{2} + 27)}^{4}}$

Step 2.1.2.6.3.1.7

Simplify.

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Step 2.1.2.6.3.1.7.1

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

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Step 2.1.2.6.3.1.7.1.1

Move $x$ .

$- \frac{2 (x \cdot x^{4}) + 108 x^{2} x + 1458 x + (- 4 x^{2} - 4 \cdot - 27) (x^{2} x + 27 x)}{{(x^{2} + 27)}^{4}}$

Step 2.1.2.6.3.1.7.1.2

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

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Step 2.1.2.6.3.1.7.1.2.1

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

$- \frac{2 (x^{1} x^{4}) + 108 x^{2} x + 1458 x + (- 4 x^{2} - 4 \cdot - 27) (x^{2} x + 27 x)}{{(x^{2} + 27)}^{4}}$

Step 2.1.2.6.3.1.7.1.2.2

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

$- \frac{2 x^{1 + 4} + 108 x^{2} x + 1458 x + (- 4 x^{2} - 4 \cdot - 27) (x^{2} x + 27 x)}{{(x^{2} + 27)}^{4}}$

Step 2.1.2.6.3.1.7.1.3

Add $1$ and $4$ .

$- \frac{2 x^{5} + 108 x^{2} x + 1458 x + (- 4 x^{2} - 4 \cdot - 27) (x^{2} x + 27 x)}{{(x^{2} + 27)}^{4}}$

Step 2.1.2.6.3.1.7.2

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

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Step 2.1.2.6.3.1.7.2.1

Move $x$ .

$- \frac{2 x^{5} + 108 (x \cdot x^{2}) + 1458 x + (- 4 x^{2} - 4 \cdot - 27) (x^{2} x + 27 x)}{{(x^{2} + 27)}^{4}}$

Step 2.1.2.6.3.1.7.2.2

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

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Step 2.1.2.6.3.1.7.2.2.1

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

$- \frac{2 x^{5} + 108 (x^{1} x^{2}) + 1458 x + (- 4 x^{2} - 4 \cdot - 27) (x^{2} x + 27 x)}{{(x^{2} + 27)}^{4}}$

Step 2.1.2.6.3.1.7.2.2.2

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

$- \frac{2 x^{5} + 108 x^{1 + 2} + 1458 x + (- 4 x^{2} - 4 \cdot - 27) (x^{2} x + 27 x)}{{(x^{2} + 27)}^{4}}$

Step 2.1.2.6.3.1.7.2.3

Add $1$ and $2$ .

$- \frac{2 x^{5} + 108 x^{3} + 1458 x + (- 4 x^{2} - 4 \cdot - 27) (x^{2} x + 27 x)}{{(x^{2} + 27)}^{4}}$

Step 2.1.2.6.3.1.8

Multiply $- 4$ by $- 27$ .

$- \frac{2 x^{5} + 108 x^{3} + 1458 x + (- 4 x^{2} + 108) (x^{2} x + 27 x)}{{(x^{2} + 27)}^{4}}$

Step 2.1.2.6.3.1.9

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

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Step 2.1.2.6.3.1.9.1

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

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Step 2.1.2.6.3.1.9.1.1

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

$- \frac{2 x^{5} + 108 x^{3} + 1458 x + (- 4 x^{2} + 108) (x^{2} x^{1} + 27 x)}{{(x^{2} + 27)}^{4}}$

Step 2.1.2.6.3.1.9.1.2

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

$- \frac{2 x^{5} + 108 x^{3} + 1458 x + (- 4 x^{2} + 108) (x^{2 + 1} + 27 x)}{{(x^{2} + 27)}^{4}}$

Step 2.1.2.6.3.1.9.2

Add $2$ and $1$ .

$- \frac{2 x^{5} + 108 x^{3} + 1458 x + (- 4 x^{2} + 108) (x^{3} + 27 x)}{{(x^{2} + 27)}^{4}}$

Step 2.1.2.6.3.1.10

Expand $(- 4 x^{2} + 108) (x^{3} + 27 x)$ using the FOIL Method.

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Step 2.1.2.6.3.1.10.1

Apply the distributive property.

$- \frac{2 x^{5} + 108 x^{3} + 1458 x - 4 x^{2} (x^{3} + 27 x) + 108 (x^{3} + 27 x)}{{(x^{2} + 27)}^{4}}$

Step 2.1.2.6.3.1.10.2

Apply the distributive property.

$- \frac{2 x^{5} + 108 x^{3} + 1458 x - 4 x^{2} x^{3} - 4 x^{2} (27 x) + 108 (x^{3} + 27 x)}{{(x^{2} + 27)}^{4}}$

Step 2.1.2.6.3.1.10.3

Apply the distributive property.

$- \frac{2 x^{5} + 108 x^{3} + 1458 x - 4 x^{2} x^{3} - 4 x^{2} (27 x) + 108 x^{3} + 108 (27 x)}{{(x^{2} + 27)}^{4}}$

Step 2.1.2.6.3.1.11

Simplify and combine like terms.

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Step 2.1.2.6.3.1.11.1

Simplify each term.

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Step 2.1.2.6.3.1.11.1.1

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

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Step 2.1.2.6.3.1.11.1.1.1

Move $x^{3}$ .

$- \frac{2 x^{5} + 108 x^{3} + 1458 x - 4 (x^{3} x^{2}) - 4 x^{2} (27 x) + 108 x^{3} + 108 (27 x)}{{(x^{2} + 27)}^{4}}$

Step 2.1.2.6.3.1.11.1.1.2

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

$- \frac{2 x^{5} + 108 x^{3} + 1458 x - 4 x^{3 + 2} - 4 x^{2} (27 x) + 108 x^{3} + 108 (27 x)}{{(x^{2} + 27)}^{4}}$

Step 2.1.2.6.3.1.11.1.1.3

Add $3$ and $2$ .

$- \frac{2 x^{5} + 108 x^{3} + 1458 x - 4 x^{5} - 4 x^{2} (27 x) + 108 x^{3} + 108 (27 x)}{{(x^{2} + 27)}^{4}}$

Step 2.1.2.6.3.1.11.1.2

Rewrite using the commutative property of multiplication.

$- \frac{2 x^{5} + 108 x^{3} + 1458 x - 4 x^{5} - 4 \cdot 27 x^{2} x + 108 x^{3} + 108 (27 x)}{{(x^{2} + 27)}^{4}}$

Step 2.1.2.6.3.1.11.1.3

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

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Step 2.1.2.6.3.1.11.1.3.1

Move $x$ .

$- \frac{2 x^{5} + 108 x^{3} + 1458 x - 4 x^{5} - 4 \cdot 27 (x \cdot x^{2}) + 108 x^{3} + 108 (27 x)}{{(x^{2} + 27)}^{4}}$

Step 2.1.2.6.3.1.11.1.3.2

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

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Step 2.1.2.6.3.1.11.1.3.2.1

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

$- \frac{2 x^{5} + 108 x^{3} + 1458 x - 4 x^{5} - 4 \cdot 27 (x^{1} x^{2}) + 108 x^{3} + 108 (27 x)}{{(x^{2} + 27)}^{4}}$

Step 2.1.2.6.3.1.11.1.3.2.2

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

$- \frac{2 x^{5} + 108 x^{3} + 1458 x - 4 x^{5} - 4 \cdot 27 x^{1 + 2} + 108 x^{3} + 108 (27 x)}{{(x^{2} + 27)}^{4}}$

Step 2.1.2.6.3.1.11.1.3.3

Add $1$ and $2$ .

$- \frac{2 x^{5} + 108 x^{3} + 1458 x - 4 x^{5} - 4 \cdot 27 x^{3} + 108 x^{3} + 108 (27 x)}{{(x^{2} + 27)}^{4}}$

Step 2.1.2.6.3.1.11.1.4

Multiply $- 4$ by $27$ .

$- \frac{2 x^{5} + 108 x^{3} + 1458 x - 4 x^{5} - 108 x^{3} + 108 x^{3} + 108 (27 x)}{{(x^{2} + 27)}^{4}}$

Step 2.1.2.6.3.1.11.1.5

Multiply $27$ by $108$ .

$- \frac{2 x^{5} + 108 x^{3} + 1458 x - 4 x^{5} - 108 x^{3} + 108 x^{3} + 2916 x}{{(x^{2} + 27)}^{4}}$

Step 2.1.2.6.3.1.11.2

Add $- 108 x^{3}$ and $108 x^{3}$ .

$- \frac{2 x^{5} + 108 x^{3} + 1458 x - 4 x^{5} + 0 + 2916 x}{{(x^{2} + 27)}^{4}}$

Step 2.1.2.6.3.1.11.3

Add $- 4 x^{5}$ and $0$ .

$- \frac{2 x^{5} + 108 x^{3} + 1458 x - 4 x^{5} + 2916 x}{{(x^{2} + 27)}^{4}}$

Step 2.1.2.6.3.2

Subtract $4 x^{5}$ from $2 x^{5}$ .

$- \frac{- 2 x^{5} + 108 x^{3} + 1458 x + 2916 x}{{(x^{2} + 27)}^{4}}$

Step 2.1.2.6.3.3

Add $1458 x$ and $2916 x$ .

$- \frac{- 2 x^{5} + 108 x^{3} + 4374 x}{{(x^{2} + 27)}^{4}}$

Step 2.1.2.6.4

Simplify the numerator.

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Step 2.1.2.6.4.1

Factor $2 x$ out of $- 2 x^{5} + 108 x^{3} + 4374 x$ .

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Step 2.1.2.6.4.1.1

Factor $2 x$ out of $- 2 x^{5}$ .

$- \frac{2 x (- x^{4}) + 108 x^{3} + 4374 x}{{(x^{2} + 27)}^{4}}$

Step 2.1.2.6.4.1.2

Factor $2 x$ out of $108 x^{3}$ .

$- \frac{2 x (- x^{4}) + 2 x (54 x^{2}) + 4374 x}{{(x^{2} + 27)}^{4}}$

Step 2.1.2.6.4.1.3

Factor $2 x$ out of $4374 x$ .

$- \frac{2 x (- x^{4}) + 2 x (54 x^{2}) + 2 x (2187)}{{(x^{2} + 27)}^{4}}$

Step 2.1.2.6.4.1.4

Factor $2 x$ out of $2 x (- x^{4}) + 2 x (54 x^{2})$ .

$- \frac{2 x (- x^{4} + 54 x^{2}) + 2 x (2187)}{{(x^{2} + 27)}^{4}}$

Step 2.1.2.6.4.1.5

Factor $2 x$ out of $2 x (- x^{4} + 54 x^{2}) + 2 x (2187)$ .

$- \frac{2 x (- x^{4} + 54 x^{2} + 2187)}{{(x^{2} + 27)}^{4}}$

Step 2.1.2.6.4.2

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

$- \frac{2 x (- {(x^{2})}^{2} + 54 x^{2} + 2187)}{{(x^{2} + 27)}^{4}}$

Step 2.1.2.6.4.3

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

$- \frac{2 x (- u^{2} + 54 u + 2187)}{{(x^{2} + 27)}^{4}}$

Step 2.1.2.6.4.4

Factor by grouping.

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Step 2.1.2.6.4.4.1

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 $a \cdot c = - 1 \cdot 2187 = - 2187$ and whose sum is $b = 54$ .

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Step 2.1.2.6.4.4.1.1

Factor $54$ out of $54 u$ .

$- \frac{2 x (- u^{2} + 54 (u) + 2187)}{{(x^{2} + 27)}^{4}}$

Step 2.1.2.6.4.4.1.2

Rewrite $54$ as $- 27$ plus $81$

$- \frac{2 x (- u^{2} + (- 27 + 81) u + 2187)}{{(x^{2} + 27)}^{4}}$

Step 2.1.2.6.4.4.1.3

Apply the distributive property.

$- \frac{2 x (- u^{2} - 27 u + 81 u + 2187)}{{(x^{2} + 27)}^{4}}$

Step 2.1.2.6.4.4.2

Factor out the greatest common factor from each group.

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Step 2.1.2.6.4.4.2.1

Group the first two terms and the last two terms.

$- \frac{2 x ((- u^{2} - 27 u) + 81 u + 2187)}{{(x^{2} + 27)}^{4}}$

Step 2.1.2.6.4.4.2.2

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

$- \frac{2 x (u (- u - 27) - 81 (- u - 27))}{{(x^{2} + 27)}^{4}}$

Step 2.1.2.6.4.4.3

Factor the polynomial by factoring out the greatest common factor, $- u - 27$ .

$- \frac{2 x ((- u - 27) (u - 81))}{{(x^{2} + 27)}^{4}}$

Step 2.1.2.6.4.5

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

$- \frac{2 x ((- x^{2} - 27) (x^{2} - 81))}{{(x^{2} + 27)}^{4}}$

Step 2.1.2.6.4.6

Rewrite $81$ as $9^{2}$ .

$- \frac{2 x ((- x^{2} - 27) (x^{2} - 9^{2}))}{{(x^{2} + 27)}^{4}}$

Step 2.1.2.6.4.7

Factor.

$- \frac{2 x (- x^{2} - 27) (x + 9) (x - 9)}{{(x^{2} + 27)}^{4}}$

Step 2.1.2.6.5

Cancel the common factor of $- x^{2} - 27$ and ${(x^{2} + 27)}^{4}$ .

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Step 2.1.2.6.5.1

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

$- \frac{2 x (- (x^{2}) - 27) (x + 9) (x - 9)}{{(x^{2} + 27)}^{4}}$

Step 2.1.2.6.5.2

Rewrite $- 27$ as $- 1 (27)$ .

$- \frac{2 x (- (x^{2}) - 1 (27)) (x + 9) (x - 9)}{{(x^{2} + 27)}^{4}}$

Step 2.1.2.6.5.3

Factor $- 1$ out of $- (x^{2}) - 1 (27)$ .

$- \frac{2 x (- (x^{2} + 27)) (x + 9) (x - 9)}{{(x^{2} + 27)}^{4}}$

Step 2.1.2.6.5.4

Rewrite $- (x^{2} + 27)$ as $- 1 (x^{2} + 27)$ .

$- \frac{2 x (- 1 (x^{2} + 27)) (x + 9) (x - 9)}{{(x^{2} + 27)}^{4}}$

Step 2.1.2.6.5.5

Factor $x^{2} + 27$ out of $2 x (- 1 (x^{2} + 27)) (x + 9) (x - 9)$ .

$- \frac{(x^{2} + 27) ((2 x \cdot (- 1) (x + 9)) (x - 9))}{{(x^{2} + 27)}^{4}}$

Step 2.1.2.6.5.6

Cancel the common factors.

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Step 2.1.2.6.5.6.1

Factor $x^{2} + 27$ out of ${(x^{2} + 27)}^{4}$ .

$- \frac{(x^{2} + 27) ((2 x \cdot (- 1) (x + 9)) (x - 9))}{(x^{2} + 27) {(x^{2} + 27)}^{3}}$

Step 2.1.2.6.5.6.2

Cancel the common factor.

$- \frac{(x^{2} + 27) ((2 x \cdot (- 1) (x + 9)) (x - 9))}{(x^{2} + 27) {(x^{2} + 27)}^{3}}$

Step 2.1.2.6.5.6.3

Rewrite the expression.

$- \frac{(2 x \cdot (- 1) (x + 9)) (x - 9)}{{(x^{2} + 27)}^{3}}$

Step 2.1.2.6.6

Multiply $- 1$ by $2$ .

$- \frac{- 2 x (x + 9) (x - 9)}{{(x^{2} + 27)}^{3}}$

Step 2.1.2.6.7

Move the negative in front of the fraction.

$- - \frac{(2 x (x + 9)) (x - 9)}{{(x^{2} + 27)}^{3}}$

Step 2.1.2.6.8

Multiply $- - \frac{2 x (x + 9) (x - 9)}{{(x^{2} + 27)}^{3}}$ .

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Step 2.1.2.6.8.1

Multiply $- 1$ by $- 1$ .

$1 \frac{2 x (x + 9) (x - 9)}{{(x^{2} + 27)}^{3}}$

Step 2.1.2.6.8.2

Multiply $\frac{2 x (x + 9) (x - 9)}{{(x^{2} + 27)}^{3}}$ by $1$ .

$f'' (x) = \frac{2 x (x + 9) (x - 9)}{{(x^{2} + 27)}^{3}}$

Step 2.1.3

The second derivative of $f (x)$ with respect to $x$ is $\frac{2 x (x + 9) (x - 9)}{{(x^{2} + 27)}^{3}}$ .

$\frac{2 x (x + 9) (x - 9)}{{(x^{2} + 27)}^{3}}$

Step 2.2

Set the second derivative equal to $0$ then solve the equation $\frac{2 x (x + 9) (x - 9)}{{(x^{2} + 27)}^{3}} = 0$ .

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Step 2.2.1

Set the second derivative equal to $0$ .

$\frac{2 x (x + 9) (x - 9)}{{(x^{2} + 27)}^{3}} = 0$

Step 2.2.2

Set the numerator equal to zero.

$2 x (x + 9) (x - 9) = 0$

Step 2.2.3

Solve the equation for $x$ .

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Step 2.2.3.1

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

$x = 0$

$x + 9 = 0$

$x - 9 = 0$

Step 2.2.3.2

Set $x$ equal to $0$ .

$x = 0$

Step 2.2.3.3

Set $x + 9$ equal to $0$ and solve for $x$ .

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Step 2.2.3.3.1

Set $x + 9$ equal to $0$ .

$x + 9 = 0$

Step 2.2.3.3.2

Subtract $9$ from both sides of the equation.

$x = - 9$

Step 2.2.3.4

Set $x - 9$ equal to $0$ and solve for $x$ .

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Step 2.2.3.4.1

Set $x - 9$ equal to $0$ .

$x - 9 = 0$

Step 2.2.3.4.2

Add $9$ to both sides of the equation.

$x = 9$

Step 2.2.3.5

The final solution is all the values that make $2 x (x + 9) (x - 9) = 0$ true.

$x = 0, - 9, 9$

Step 3

Find the domain of $f (x) = \frac{x}{x^{2} + 27}$ .

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Step 3.1

Set the denominator in $\frac{x}{x^{2} + 27}$ equal to $0$ to find where the expression is undefined.

$x^{2} + 27 = 0$

Step 3.2

Solve for $x$ .

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Step 3.2.1

Subtract $27$ from both sides of the equation.

$x^{2} = - 27$

Step 3.2.2

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

$x = \pm \sqrt{- 27}$

Step 3.2.3

Simplify $\pm \sqrt{- 27}$ .

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Step 3.2.3.1

Rewrite $- 27$ as $- 1 (27)$ .

$x = \pm \sqrt{- 1 (27)}$

Step 3.2.3.2

Rewrite $\sqrt{- 1 (27)}$ as $\sqrt{- 1} \cdot \sqrt{27}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{27}$

Step 3.2.3.3

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

$x = \pm i \cdot \sqrt{27}$

Step 3.2.3.4

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 3.2.3.4.1

Factor $9$ out of $27$ .

$x = \pm i \cdot \sqrt{9 (3)}$

Step 3.2.3.4.2

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

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

Step 3.2.3.5

Pull terms out from under the radical.

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

Step 3.2.3.6

Move $3$ to the left of $i$ .

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

Step 3.2.4

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

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Step 3.2.4.1

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

$x = 3 i \sqrt{3}$

Step 3.2.4.2

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

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

Step 3.2.4.3

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

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

Step 3.3

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 4

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

$(- \infty, - 9) \cup (- 9, 0) \cup (0, 9) \cup (9, \infty)$

Step 5

Substitute any number from the interval $(- \infty, - 9)$ into the second derivative and evaluate to determine the concavity.

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Step 5.1

Replace the variable $x$ with $- 12$ in the expression.

$f'' (- 12) = \frac{2 (- 12) ((- 12) + 9) ((- 12) - 9)}{{({(- 12)}^{2} + 27)}^{3}}$

Step 5.2

Simplify the result.

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Step 5.2.1

Multiply $2$ by $- 12$ .

$f'' (- 12) = \frac{- 24 (- 12 + 9) (- 12 - 9)}{{({(- 12)}^{2} + 27)}^{3}}$

Step 5.2.2

Simplify the denominator.

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Step 5.2.2.1

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

$f'' (- 12) = \frac{- 24 (- 12 + 9) (- 12 - 9)}{{(144 + 27)}^{3}}$

Step 5.2.2.2

Add $144$ and $27$ .

$f'' (- 12) = \frac{- 24 (- 12 + 9) (- 12 - 9)}{171^{3}}$

Step 5.2.2.3

Raise $171$ to the power of $3$ .

$f'' (- 12) = \frac{- 24 (- 12 + 9) (- 12 - 9)}{5000211}$

Step 5.2.3

Simplify the numerator.

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Step 5.2.3.1

Add $- 12$ and $9$ .

$f'' (- 12) = \frac{- 24 \cdot (- 3 (- 12 - 9))}{5000211}$

Step 5.2.3.2

Multiply $- 24$ by $- 3$ .

$f'' (- 12) = \frac{72 (- 12 - 9)}{5000211}$

Step 5.2.3.3

Subtract $9$ from $- 12$ .

$f'' (- 12) = \frac{72 \cdot - 21}{5000211}$

Step 5.2.4

Reduce the expression by cancelling the common factors.

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Step 5.2.4.1

Multiply $72$ by $- 21$ .

$f'' (- 12) = \frac{- 1512}{5000211}$

Step 5.2.4.2

Cancel the common factor of $- 1512$ and $5000211$ .

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Step 5.2.4.2.1

Factor $27$ out of $- 1512$ .

$f'' (- 12) = \frac{27 (- 56)}{5000211}$

Step 5.2.4.2.2

Cancel the common factors.

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Step 5.2.4.2.2.1

Factor $27$ out of $5000211$ .

$f'' (- 12) = \frac{27 \cdot - 56}{27 \cdot 185193}$

Step 5.2.4.2.2.2

Cancel the common factor.

$f'' (- 12) = \frac{27 \cdot - 56}{27 \cdot 185193}$

Step 5.2.4.2.2.3

Rewrite the expression.

$f'' (- 12) = \frac{- 56}{185193}$

Step 5.2.4.3

Move the negative in front of the fraction.

$f'' (- 12) = - \frac{56}{185193}$

Step 5.2.5

The final answer is $- \frac{56}{185193}$ .

$- \frac{56}{185193}$

Step 5.3

The graph is concave down on the interval $(- \infty, - 9)$ because $f'' (- 12)$ is negative.

Concave down on $(- \infty, - 9)$ since $f'' (x)$ is negative

Step 6

Substitute any number from the interval $(- 9, 0)$ into the second derivative and evaluate to determine the concavity.

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Step 6.1

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

$f'' (- 4) = \frac{2 (- 4) ((- 4) + 9) ((- 4) - 9)}{{({(- 4)}^{2} + 27)}^{3}}$

Step 6.2

Simplify the result.

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Step 6.2.1

Multiply $2$ by $- 4$ .

$f'' (- 4) = \frac{- 8 (- 4 + 9) (- 4 - 9)}{{({(- 4)}^{2} + 27)}^{3}}$

Step 6.2.2

Simplify the denominator.

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Step 6.2.2.1

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

$f'' (- 4) = \frac{- 8 (- 4 + 9) (- 4 - 9)}{{(16 + 27)}^{3}}$

Step 6.2.2.2

Add $16$ and $27$ .

$f'' (- 4) = \frac{- 8 (- 4 + 9) (- 4 - 9)}{43^{3}}$

Step 6.2.2.3

Raise $43$ to the power of $3$ .

$f'' (- 4) = \frac{- 8 (- 4 + 9) (- 4 - 9)}{79507}$

Step 6.2.3

Simplify the numerator.

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Step 6.2.3.1

Add $- 4$ and $9$ .

$f'' (- 4) = \frac{- 8 \cdot (5 (- 4 - 9))}{79507}$

Step 6.2.3.2

Multiply $- 8$ by $5$ .

$f'' (- 4) = \frac{- 40 (- 4 - 9)}{79507}$

Step 6.2.3.3

Subtract $9$ from $- 4$ .

$f'' (- 4) = \frac{- 40 \cdot - 13}{79507}$

Step 6.2.4

Multiply $- 40$ by $- 13$ .

$f'' (- 4) = \frac{520}{79507}$

Step 6.2.5

The final answer is $\frac{520}{79507}$ .

$\frac{520}{79507}$

Step 6.3

The graph is concave up on the interval $(- 9, 0)$ because $f'' (- 4)$ is positive.

Concave up on $(- 9, 0)$ since $f'' (x)$ is positive

Step 7

Substitute any number from the interval $(0, 9)$ 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.

$f'' (4) = \frac{2 (4) ((4) + 9) ((4) - 9)}{{({(4)}^{2} + 27)}^{3}}$

Step 7.2

Simplify the result.

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Step 7.2.1

Multiply $2$ by $4$ .

$f'' (4) = \frac{8 (4 + 9) (4 - 9)}{{(4^{2} + 27)}^{3}}$

Step 7.2.2

Simplify the denominator.

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Step 7.2.2.1

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

$f'' (4) = \frac{8 (4 + 9) (4 - 9)}{{(16 + 27)}^{3}}$

Step 7.2.2.2

Add $16$ and $27$ .

$f'' (4) = \frac{8 (4 + 9) (4 - 9)}{43^{3}}$

Step 7.2.2.3

Raise $43$ to the power of $3$ .

$f'' (4) = \frac{8 (4 + 9) (4 - 9)}{79507}$

Step 7.2.3

Simplify the numerator.

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Step 7.2.3.1

Add $4$ and $9$ .

$f'' (4) = \frac{8 \cdot (13 (4 - 9))}{79507}$

Step 7.2.3.2

Multiply $8$ by $13$ .

$f'' (4) = \frac{104 (4 - 9)}{79507}$

Step 7.2.3.3

Subtract $9$ from $4$ .

$f'' (4) = \frac{104 \cdot - 5}{79507}$

Step 7.2.4

Simplify the expression.

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Step 7.2.4.1

Multiply $104$ by $- 5$ .

$f'' (4) = \frac{- 520}{79507}$

Step 7.2.4.2

Move the negative in front of the fraction.

$f'' (4) = - \frac{520}{79507}$

Step 7.2.5

The final answer is $- \frac{520}{79507}$ .

$- \frac{520}{79507}$

Step 7.3

The graph is concave down on the interval $(0, 9)$ because $f'' (4)$ is negative.

Concave down on $(0, 9)$ since $f'' (x)$ is negative

Step 8

Substitute any number from the interval $(9, \infty)$ into the second derivative and evaluate to determine the concavity.

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Step 8.1

Replace the variable $x$ with $12$ in the expression.

$f'' (12) = \frac{2 (12) ((12) + 9) ((12) - 9)}{{({(12)}^{2} + 27)}^{3}}$

Step 8.2

Simplify the result.

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Step 8.2.1

Multiply $2$ by $12$ .

$f'' (12) = \frac{24 (12 + 9) (12 - 9)}{{(12^{2} + 27)}^{3}}$

Step 8.2.2

Simplify the denominator.

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Step 8.2.2.1

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

$f'' (12) = \frac{24 (12 + 9) (12 - 9)}{{(144 + 27)}^{3}}$

Step 8.2.2.2

Add $144$ and $27$ .

$f'' (12) = \frac{24 (12 + 9) (12 - 9)}{171^{3}}$

Step 8.2.2.3

Raise $171$ to the power of $3$ .

$f'' (12) = \frac{24 (12 + 9) (12 - 9)}{5000211}$

Step 8.2.3

Simplify the numerator.

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Step 8.2.3.1

Add $12$ and $9$ .

$f'' (12) = \frac{24 \cdot (21 (12 - 9))}{5000211}$

Step 8.2.3.2

Multiply $24$ by $21$ .

$f'' (12) = \frac{504 (12 - 9)}{5000211}$

Step 8.2.3.3

Subtract $9$ from $12$ .

$f'' (12) = \frac{504 \cdot 3}{5000211}$

Step 8.2.4

Reduce the expression by cancelling the common factors.

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Step 8.2.4.1

Multiply $504$ by $3$ .

$f'' (12) = \frac{1512}{5000211}$

Step 8.2.4.2

Cancel the common factor of $1512$ and $5000211$ .

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Step 8.2.4.2.1

Factor $27$ out of $1512$ .

$f'' (12) = \frac{27 (56)}{5000211}$

Step 8.2.4.2.2

Cancel the common factors.

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Step 8.2.4.2.2.1

Factor $27$ out of $5000211$ .

$f'' (12) = \frac{27 \cdot 56}{27 \cdot 185193}$

Step 8.2.4.2.2.2

Cancel the common factor.

$f'' (12) = \frac{27 \cdot 56}{27 \cdot 185193}$

Step 8.2.4.2.2.3

Rewrite the expression.

$f'' (12) = \frac{56}{185193}$

Step 8.2.5

The final answer is $\frac{56}{185193}$ .

$\frac{56}{185193}$

Step 8.3

The graph is concave up on the interval $(9, \infty)$ because $f'' (12)$ is positive.

Concave up on $(9, \infty)$ since $f'' (x)$ is positive

Step 9

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

Concave down on $(- \infty, - 9)$ since $f'' (x)$ is negative

Concave up on $(- 9, 0)$ since $f'' (x)$ is positive

Concave down on $(0, 9)$ since $f'' (x)$ is negative

Concave up on $(9, \infty)$ since $f'' (x)$ is positive

Step 10