Calculus Examples

$f (x) = \frac{x^{2} + 3 x - 40}{x^{2}}$

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

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x^{2} + 3 x - 40$ and $g (x) = x^{2}$ .

$f' (x) = \frac{x^{2} \frac{d}{d x} (x^{2} + 3 x - 40) - (x^{2} + 3 x - 40) \frac{d}{d x} x^{2}}{{(x^{2})}^{2}}$

Step 1.1.1.2

Differentiate.

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Step 1.1.1.2.1

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

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Step 1.1.1.2.1.1

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

$f' (x) = \frac{x^{2} \frac{d}{d x} (x^{2} + 3 x - 40) - (x^{2} + 3 x - 40) \frac{d}{d x} x^{2}}{x^{2 \cdot 2}}$

Step 1.1.1.2.1.2

Multiply $2$ by $2$ .

$f' (x) = \frac{x^{2} \frac{d}{d x} (x^{2} + 3 x - 40) - (x^{2} + 3 x - 40) \frac{d}{d x} x^{2}}{x^{4}}$

Step 1.1.1.2.2

By the Sum Rule, the derivative of $x^{2} + 3 x - 40$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 40]$ .

$f' (x) = \frac{x^{2} (\frac{d}{d x} (x^{2}) + \frac{d}{d x} (3 x) + \frac{d}{d x} (- 40)) - (x^{2} + 3 x - 40) \frac{d}{d x} x^{2}}{x^{4}}$

Step 1.1.1.2.3

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

$f' (x) = \frac{x^{2} (2 x + \frac{d}{d x} (3 x) + \frac{d}{d x} (- 40)) - (x^{2} + 3 x - 40) \frac{d}{d x} x^{2}}{x^{4}}$

Step 1.1.1.2.4

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

$f' (x) = \frac{x^{2} (2 x + 3 \frac{d}{d x} (x) + \frac{d}{d x} (- 40)) - (x^{2} + 3 x - 40) \frac{d}{d x} x^{2}}{x^{4}}$

Step 1.1.1.2.5

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

$f' (x) = \frac{x^{2} (2 x + 3 \cdot 1 + \frac{d}{d x} (- 40)) - (x^{2} + 3 x - 40) \frac{d}{d x} x^{2}}{x^{4}}$

Step 1.1.1.2.6

Multiply $3$ by $1$ .

$f' (x) = \frac{x^{2} (2 x + 3 + \frac{d}{d x} (- 40)) - (x^{2} + 3 x - 40) \frac{d}{d x} x^{2}}{x^{4}}$

Step 1.1.1.2.7

Since $- 40$ is constant with respect to $x$ , the derivative of $- 40$ with respect to $x$ is $0$ .

$f' (x) = \frac{x^{2} (2 x + 3 + 0) - (x^{2} + 3 x - 40) \frac{d}{d x} x^{2}}{x^{4}}$

Step 1.1.1.2.8

Add $2 x + 3$ and $0$ .

$f' (x) = \frac{x^{2} (2 x + 3) - (x^{2} + 3 x - 40) \frac{d}{d x} x^{2}}{x^{4}}$

Step 1.1.1.2.9

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

$f' (x) = \frac{x^{2} (2 x + 3) - (x^{2} + 3 x - 40) (2 x)}{x^{4}}$

Step 1.1.1.2.10

Simplify with factoring out.

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Step 1.1.1.2.10.1

Multiply $2$ by $- 1$ .

$f' (x) = \frac{x^{2} (2 x + 3) - 2 (x^{2} + 3 x - 40) x}{x^{4}}$

Step 1.1.1.2.10.2

Factor $x$ out of $x^{2} (2 x + 3) - 2 (x^{2} + 3 x - 40) x$ .

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Step 1.1.1.2.10.2.1

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

$f' (x) = \frac{x (x (2 x + 3)) - 2 (x^{2} + 3 x - 40) x}{x^{4}}$

Step 1.1.1.2.10.2.2

Factor $x$ out of $- 2 (x^{2} + 3 x - 40) x$ .

$f' (x) = \frac{x (x (2 x + 3)) + x (- 2 (x^{2} + 3 x - 40))}{x^{4}}$

Step 1.1.1.2.10.2.3

Factor $x$ out of $x (x (2 x + 3)) + x (- 2 (x^{2} + 3 x - 40))$ .

$f' (x) = \frac{x (x (2 x + 3) - 2 (x^{2} + 3 x - 40))}{x^{4}}$

Step 1.1.1.3

Cancel the common factors.

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Step 1.1.1.3.1

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

$f' (x) = \frac{x (x (2 x + 3) - 2 (x^{2} + 3 x - 40))}{x \cdot x^{3}}$

Step 1.1.1.3.2

Cancel the common factor.

$f' (x) = \frac{x (x (2 x + 3) - 2 (x^{2} + 3 x - 40))}{x \cdot x^{3}}$

Step 1.1.1.3.3

Rewrite the expression.

$f' (x) = \frac{x (2 x + 3) - 2 (x^{2} + 3 x - 40)}{x^{3}}$

Step 1.1.1.4

Simplify.

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Step 1.1.1.4.1

Apply the distributive property.

$f' (x) = \frac{x (2 x) + x \cdot 3 - 2 (x^{2} + 3 x - 40)}{x^{3}}$

Step 1.1.1.4.2

Apply the distributive property.

$f' (x) = \frac{x (2 x) + x \cdot 3 - 2 x^{2} - 2 (3 x) - 2 \cdot - 40}{x^{3}}$

Step 1.1.1.4.3

Simplify the numerator.

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Step 1.1.1.4.3.1

Simplify each term.

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Step 1.1.1.4.3.1.1

Rewrite using the commutative property of multiplication.

$f' (x) = \frac{2 x \cdot x + x \cdot 3 - 2 x^{2} - 2 (3 x) - 2 \cdot - 40}{x^{3}}$

Step 1.1.1.4.3.1.2

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

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Step 1.1.1.4.3.1.2.1

Move $x$ .

$f' (x) = \frac{2 (x \cdot x) + x \cdot 3 - 2 x^{2} - 2 (3 x) - 2 \cdot - 40}{x^{3}}$

Step 1.1.1.4.3.1.2.2

Multiply $x$ by $x$ .

$f' (x) = \frac{2 x^{2} + x \cdot 3 - 2 x^{2} - 2 (3 x) - 2 \cdot - 40}{x^{3}}$

Step 1.1.1.4.3.1.3

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

$f' (x) = \frac{2 x^{2} + 3 \cdot x - 2 x^{2} - 2 (3 x) - 2 \cdot - 40}{x^{3}}$

Step 1.1.1.4.3.1.4

Multiply $3$ by $- 2$ .

$f' (x) = \frac{2 x^{2} + 3 x - 2 x^{2} - 6 x - 2 \cdot - 40}{x^{3}}$

Step 1.1.1.4.3.1.5

Multiply $- 2$ by $- 40$ .

$f' (x) = \frac{2 x^{2} + 3 x - 2 x^{2} - 6 x + 80}{x^{3}}$

Step 1.1.1.4.3.2

Combine the opposite terms in $2 x^{2} + 3 x - 2 x^{2} - 6 x + 80$ .

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Step 1.1.1.4.3.2.1

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

$f' (x) = \frac{3 x + 0 - 6 x + 80}{x^{3}}$

Step 1.1.1.4.3.2.2

Add $3 x$ and $0$ .

$f' (x) = \frac{3 x - 6 x + 80}{x^{3}}$

Step 1.1.1.4.3.3

Subtract $6 x$ from $3 x$ .

$f' (x) = \frac{- 3 x + 80}{x^{3}}$

Step 1.1.1.4.4

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

$f' (x) = \frac{- (3 x) + 80}{x^{3}}$

Step 1.1.1.4.5

Rewrite $80$ as $- 1 (- 80)$ .

$f' (x) = \frac{- (3 x) - 1 \cdot - 80}{x^{3}}$

Step 1.1.1.4.6

Factor $- 1$ out of $- (3 x) - 1 (- 80)$ .

$f' (x) = \frac{- (3 x - 80)}{x^{3}}$

Step 1.1.1.4.7

Rewrite $- (3 x - 80)$ as $- 1 (3 x - 80)$ .

$f' (x) = \frac{- 1 (3 x - 80)}{x^{3}}$

Step 1.1.1.4.8

Move the negative in front of the fraction.

$f' (x) = - \frac{3 x - 80}{x^{3}}$

Step 1.1.2

Find the second derivative.

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Step 1.1.2.1

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 $g (x) = \frac{3 x - 80}{x^{3}}$ .

$f'' (x) = - \frac{d}{d x} \cdot \frac{3 x - 80}{x^{3}} + \frac{3 x - 80}{x^{3}} \cdot \frac{d}{d x} (- 1)$

Step 1.1.2.2

$f'' (x) = - \frac{x^{3} \frac{d}{d x} (3 x - 80) - (3 x - 80) \frac{d}{d x} x^{3}}{{(x^{3})}^{2}} + \frac{3 x - 80}{x^{3}} \cdot \frac{d}{d x} (- 1)$

Step 1.1.2.3

Differentiate.

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Step 1.1.2.3.1

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

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Step 1.1.2.3.1.1

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

$f'' (x) = - \frac{x^{3} \frac{d}{d x} (3 x - 80) - (3 x - 80) \frac{d}{d x} x^{3}}{x^{3 \cdot 2}} + \frac{3 x - 80}{x^{3}} \cdot \frac{d}{d x} (- 1)$

Step 1.1.2.3.1.2

Multiply $3$ by $2$ .

$f'' (x) = - \frac{x^{3} \frac{d}{d x} (3 x - 80) - (3 x - 80) \frac{d}{d x} x^{3}}{x^{6}} + \frac{3 x - 80}{x^{3}} \cdot \frac{d}{d x} (- 1)$

Step 1.1.2.3.2

By the Sum Rule, the derivative of $3 x - 80$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [- 80]$ .

$f'' (x) = - \frac{x^{3} (\frac{d}{d x} (3 x) + \frac{d}{d x} (- 80)) - (3 x - 80) \frac{d}{d x} x^{3}}{x^{6}} + \frac{3 x - 80}{x^{3}} \cdot \frac{d}{d x} (- 1)$

Step 1.1.2.3.3

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

$f'' (x) = - \frac{x^{3} (3 \frac{d}{d x} (x) + \frac{d}{d x} (- 80)) - (3 x - 80) \frac{d}{d x} x^{3}}{x^{6}} + \frac{3 x - 80}{x^{3}} \cdot \frac{d}{d x} (- 1)$

Step 1.1.2.3.4

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

$f'' (x) = - \frac{x^{3} (3 \cdot 1 + \frac{d}{d x} (- 80)) - (3 x - 80) \frac{d}{d x} x^{3}}{x^{6}} + \frac{3 x - 80}{x^{3}} \cdot \frac{d}{d x} (- 1)$

Step 1.1.2.3.5

Multiply $3$ by $1$ .

$f'' (x) = - \frac{x^{3} (3 + \frac{d}{d x} (- 80)) - (3 x - 80) \frac{d}{d x} x^{3}}{x^{6}} + \frac{3 x - 80}{x^{3}} \cdot \frac{d}{d x} (- 1)$

Step 1.1.2.3.6

Since $- 80$ is constant with respect to $x$ , the derivative of $- 80$ with respect to $x$ is $0$ .

$f'' (x) = - \frac{x^{3} (3 + 0) - (3 x - 80) \frac{d}{d x} x^{3}}{x^{6}} + \frac{3 x - 80}{x^{3}} \cdot \frac{d}{d x} (- 1)$

Step 1.1.2.3.7

Simplify the expression.

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Step 1.1.2.3.7.1

Add $3$ and $0$ .

$f'' (x) = - \frac{x^{3} \cdot 3 - (3 x - 80) \frac{d}{d x} x^{3}}{x^{6}} + \frac{3 x - 80}{x^{3}} \cdot \frac{d}{d x} (- 1)$

Step 1.1.2.3.7.2

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

$f'' (x) = - \frac{3 \cdot x^{3} - (3 x - 80) \frac{d}{d x} x^{3}}{x^{6}} + \frac{3 x - 80}{x^{3}} \cdot \frac{d}{d x} (- 1)$

Step 1.1.2.3.8

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$f'' (x) = - \frac{3 x^{3} - (3 x - 80) (3 x^{2})}{x^{6}} + \frac{3 x - 80}{x^{3}} \cdot \frac{d}{d x} (- 1)$

Step 1.1.2.3.9

Multiply $3$ by $- 1$ .

$f'' (x) = - \frac{3 x^{3} - 3 (3 x - 80) x^{2}}{x^{6}} + \frac{3 x - 80}{x^{3}} \cdot \frac{d}{d x} (- 1)$

Step 1.1.2.3.10

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

$f'' (x) = - \frac{3 x^{3} - 3 (3 x - 80) x^{2}}{x^{6}} + \frac{3 x - 80}{x^{3}} \cdot 0$

Step 1.1.2.3.11

Simplify the expression.

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Step 1.1.2.3.11.1

Multiply $\frac{3 x - 80}{x^{3}}$ by $0$ .

$f'' (x) = - \frac{3 x^{3} - 3 (3 x - 80) x^{2}}{x^{6}} + 0$

Step 1.1.2.3.11.2

Add $- \frac{3 x^{3} - 3 (3 x - 80) x^{2}}{x^{6}}$ and $0$ .

$f'' (x) = - \frac{3 x^{3} - 3 (3 x - 80) x^{2}}{x^{6}}$

Step 1.1.2.4

Simplify.

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Step 1.1.2.4.1

Apply the distributive property.

$f'' (x) = - \frac{3 x^{3} + (- 3 (3 x) - 3 \cdot - 80) x^{2}}{x^{6}}$

Step 1.1.2.4.2

Apply the distributive property.

$f'' (x) = - \frac{3 x^{3} - 3 (3 x) x^{2} - 3 \cdot (- 80 x^{2})}{x^{6}}$

Step 1.1.2.4.3

Simplify the numerator.

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Step 1.1.2.4.3.1

Simplify each term.

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Step 1.1.2.4.3.1.1

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

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Step 1.1.2.4.3.1.1.1

Move $x^{2}$ .

$f'' (x) = - \frac{3 x^{3} - 3 (3 (x^{2} x)) - 3 \cdot (- 80 x^{2})}{x^{6}}$

Step 1.1.2.4.3.1.1.2

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

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Step 1.1.2.4.3.1.1.2.1

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

$f'' (x) = - \frac{3 x^{3} - 3 (3 (x^{2} x)) - 3 \cdot (- 80 x^{2})}{x^{6}}$

Step 1.1.2.4.3.1.1.2.2

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

$f'' (x) = - \frac{3 x^{3} - 3 (3 x^{2 + 1}) - 3 \cdot (- 80 x^{2})}{x^{6}}$

Step 1.1.2.4.3.1.1.3

Add $2$ and $1$ .

$f'' (x) = - \frac{3 x^{3} - 3 (3 x^{3}) - 3 \cdot (- 80 x^{2})}{x^{6}}$

Step 1.1.2.4.3.1.2

Multiply $3$ by $- 3$ .

$f'' (x) = - \frac{3 x^{3} - 9 x^{3} - 3 \cdot (- 80 x^{2})}{x^{6}}$

Step 1.1.2.4.3.1.3

Multiply $- 3$ by $- 80$ .

$f'' (x) = - \frac{3 x^{3} - 9 x^{3} + 240 x^{2}}{x^{6}}$

Step 1.1.2.4.3.2

Subtract $9 x^{3}$ from $3 x^{3}$ .

$f'' (x) = - \frac{- 6 x^{3} + 240 x^{2}}{x^{6}}$

Step 1.1.2.4.4

Factor $6 x^{2}$ out of $- 6 x^{3} + 240 x^{2}$ .

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Step 1.1.2.4.4.1

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

$f'' (x) = - \frac{6 x^{2} (- x) + 240 x^{2}}{x^{6}}$

Step 1.1.2.4.4.2

Factor $6 x^{2}$ out of $240 x^{2}$ .

$f'' (x) = - \frac{6 x^{2} (- x) + 6 x^{2} (40)}{x^{6}}$

Step 1.1.2.4.4.3

Factor $6 x^{2}$ out of $6 x^{2} (- x) + 6 x^{2} (40)$ .

$f'' (x) = - \frac{6 x^{2} (- x + 40)}{x^{6}}$

Step 1.1.2.4.5

Cancel the common factor of $x^{2}$ and $x^{6}$ .

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Step 1.1.2.4.5.1

Factor $x^{2}$ out of $6 x^{2} (- x + 40)$ .

$f'' (x) = - \frac{x^{2} (6 (- x + 40))}{x^{6}}$

Step 1.1.2.4.5.2

Cancel the common factors.

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Step 1.1.2.4.5.2.1

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

$f'' (x) = - \frac{x^{2} (6 (- x + 40))}{x^{2} x^{4}}$

Step 1.1.2.4.5.2.2

Cancel the common factor.

$f'' (x) = - \frac{x^{2} (6 (- x + 40))}{x^{2} x^{4}}$

Step 1.1.2.4.5.2.3

Rewrite the expression.

$f'' (x) = - \frac{6 (- x + 40)}{x^{4}}$

Step 1.1.2.4.6

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

$f'' (x) = - \frac{6 (- (x) + 40)}{x^{4}}$

Step 1.1.2.4.7

Rewrite $40$ as $- 1 (- 40)$ .

$f'' (x) = - \frac{6 (- (x) - 1 \cdot - 40)}{x^{4}}$

Step 1.1.2.4.8

Factor $- 1$ out of $- (x) - 1 (- 40)$ .

$f'' (x) = - \frac{6 (- (x - 40))}{x^{4}}$

Step 1.1.2.4.9

Rewrite $- (x - 40)$ as $- 1 (x - 40)$ .

$f'' (x) = - \frac{6 (- 1 (x - 40))}{x^{4}}$

Step 1.1.2.4.10

Move the negative in front of the fraction.

$f'' (x) = \frac{6 (x - 40)}{x^{4}}$

Step 1.1.2.4.11

Multiply $- 1$ by $- 1$ .

$f'' (x) = 1 (\frac{6 (x - 40)}{x^{4}})$

Step 1.1.2.4.12

Multiply $\frac{6 (x - 40)}{x^{4}}$ by $1$ .

$f'' (x) = \frac{6 (x - 40)}{x^{4}}$

Step 1.1.3

The second derivative of $f (x)$ with respect to $x$ is $\frac{6 (x - 40)}{x^{4}}$ .

$\frac{6 (x - 40)}{x^{4}}$

Step 1.2

Set the second derivative equal to $0$ then solve the equation $\frac{6 (x - 40)}{x^{4}} = 0$ .

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Step 1.2.1

Set the second derivative equal to $0$ .

$\frac{6 (x - 40)}{x^{4}} = 0$

Step 1.2.2

Set the numerator equal to zero.

$6 (x - 40) = 0$

Step 1.2.3

Solve the equation for $x$ .

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Step 1.2.3.1

Divide each term in $6 (x - 40) = 0$ by $6$ and simplify.

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Step 1.2.3.1.1

Divide each term in $6 (x - 40) = 0$ by $6$ .

$\frac{6 (x - 40)}{6} = \frac{0}{6}$

Step 1.2.3.1.2

Simplify the left side.

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Step 1.2.3.1.2.1

Cancel the common factor of $6$ .

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Step 1.2.3.1.2.1.1

Cancel the common factor.

$\frac{6 (x - 40)}{6} = \frac{0}{6}$

Step 1.2.3.1.2.1.2

Divide $x - 40$ by $1$ .

$x - 40 = \frac{0}{6}$

Step 1.2.3.1.3

Simplify the right side.

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Step 1.2.3.1.3.1

Divide $0$ by $6$ .

$x - 40 = 0$

Step 1.2.3.2

Add $40$ to both sides of the equation.

$x = 40$

Step 2

Find the domain of $f (x) = \frac{x^{2} + 3 x - 40}{x^{2}}$ .

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Step 2.1

Set the denominator in $\frac{x^{2} + 3 x - 40}{x^{2}}$ equal to $0$ to find where the expression is undefined.

$x^{2} = 0$

Step 2.2

Solve for $x$ .

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Step 2.2.1

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

$x = \pm \sqrt{0}$

Step 2.2.2

Simplify $\pm \sqrt{0}$ .

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Step 2.2.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 2.2.2.2

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

$x = \pm 0$

Step 2.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 2.3

The domain is all values of $x$ that make the expression defined.

Interval Notation:

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

Set-Builder Notation:

${x | x \neq 0}$

Interval Notation:

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

Set-Builder Notation:

${x | x \neq 0}$

Step 3

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

$(- \infty, 0) \cup (0, 40) \cup (40, \infty)$

Step 4

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

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Step 4.1

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

$f'' (- 2) = \frac{6 ((- 2) - 40)}{{(- 2)}^{4}}$

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify the expression.

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Step 4.2.1.1

Subtract $40$ from $- 2$ .

$f'' (- 2) = \frac{6 \cdot - 42}{{(- 2)}^{4}}$

Step 4.2.1.2

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

$f'' (- 2) = \frac{6 \cdot - 42}{16}$

Step 4.2.1.3

Multiply $6$ by $- 42$ .

$f'' (- 2) = \frac{- 252}{16}$

Step 4.2.2

Cancel the common factor of $- 252$ and $16$ .

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Step 4.2.2.1

Factor $4$ out of $- 252$ .

$f'' (- 2) = \frac{4 (- 63)}{16}$

Step 4.2.2.2

Cancel the common factors.

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Step 4.2.2.2.1

Factor $4$ out of $16$ .

$f'' (- 2) = \frac{4 \cdot - 63}{4 \cdot 4}$

Step 4.2.2.2.2

Cancel the common factor.

$f'' (- 2) = \frac{4 \cdot - 63}{4 \cdot 4}$

Step 4.2.2.2.3

Rewrite the expression.

$f'' (- 2) = \frac{- 63}{4}$

Step 4.2.3

Move the negative in front of the fraction.

$f'' (- 2) = - \frac{63}{4}$

Step 4.2.4

The final answer is $- \frac{63}{4}$ .

$- \frac{63}{4}$

Step 4.3

The graph is concave down on the interval $(- \infty, 0)$ because $f'' (- 2)$ is negative.

Concave down on $(- \infty, 0)$ since $f'' (x)$ is negative

Step 5

Substitute any number from the interval $(0, 40)$ into the second derivative and evaluate to determine the concavity.

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Step 5.1

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

$f'' (2) = \frac{6 ((2) - 40)}{{(2)}^{4}}$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify the expression.

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Step 5.2.1.1

Subtract $40$ from $2$ .

$f'' (2) = \frac{6 \cdot - 38}{2^{4}}$

Step 5.2.1.2

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

$f'' (2) = \frac{6 \cdot - 38}{16}$

Step 5.2.1.3

Multiply $6$ by $- 38$ .

$f'' (2) = \frac{- 228}{16}$

Step 5.2.2

Cancel the common factor of $- 228$ and $16$ .

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Step 5.2.2.1

Factor $4$ out of $- 228$ .

$f'' (2) = \frac{4 (- 57)}{16}$

Step 5.2.2.2

Cancel the common factors.

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Step 5.2.2.2.1

Factor $4$ out of $16$ .

$f'' (2) = \frac{4 \cdot - 57}{4 \cdot 4}$

Step 5.2.2.2.2

Cancel the common factor.

$f'' (2) = \frac{4 \cdot - 57}{4 \cdot 4}$

Step 5.2.2.2.3

Rewrite the expression.

$f'' (2) = \frac{- 57}{4}$

Step 5.2.3

Move the negative in front of the fraction.

$f'' (2) = - \frac{57}{4}$

Step 5.2.4

The final answer is $- \frac{57}{4}$ .

$- \frac{57}{4}$

Step 5.3

The graph is concave down on the interval $(0, 40)$ because $f'' (2)$ is negative.

Concave down on $(0, 40)$ since $f'' (x)$ is negative

Step 6

Substitute any number from the interval $(40, \infty)$ into the second derivative and evaluate to determine the concavity.

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Step 6.1

Replace the variable $x$ with $42$ in the expression.

$f'' (42) = \frac{6 ((42) - 40)}{{(42)}^{4}}$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify the expression.

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Step 6.2.1.1

Subtract $40$ from $42$ .

$f'' (42) = \frac{6 \cdot 2}{42^{4}}$

Step 6.2.1.2

Raise $42$ to the power of $4$ .

$f'' (42) = \frac{6 \cdot 2}{3111696}$

Step 6.2.1.3

Multiply $6$ by $2$ .

$f'' (42) = \frac{12}{3111696}$

Step 6.2.2

Cancel the common factor of $12$ and $3111696$ .

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Step 6.2.2.1

Factor $12$ out of $12$ .

$f'' (42) = \frac{12 (1)}{3111696}$

Step 6.2.2.2

Cancel the common factors.

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Step 6.2.2.2.1

Factor $12$ out of $3111696$ .

$f'' (42) = \frac{12 \cdot 1}{12 \cdot 259308}$

Step 6.2.2.2.2

Cancel the common factor.

$f'' (42) = \frac{12 \cdot 1}{12 \cdot 259308}$

Step 6.2.2.2.3

Rewrite the expression.

$f'' (42) = \frac{1}{259308}$

Step 6.2.3

The final answer is $\frac{1}{259308}$ .

$\frac{1}{259308}$

Step 6.3

The graph is concave up on the interval $(40, \infty)$ because $f'' (42)$ is positive.

Concave up on $(40, \infty)$ since $f'' (x)$ is positive

Step 7

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

Concave down on $(- \infty, 0)$ since $f'' (x)$ is negative

Concave down on $(0, 40)$ since $f'' (x)$ is negative

Concave up on $(40, \infty)$ since $f'' (x)$ is positive

Step 8