Calculus Examples

$\frac{1}{5} x^{5} + \frac{7}{2} x^{4} + \frac{71}{3} x^{3} + 77 x^{2} + 120 x$

Step 1

Find the first derivative of the function.

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Step 1.1

By the Sum Rule, the derivative of $\frac{1}{5} x^{5} + \frac{7}{2} x^{4} + \frac{71}{3} x^{3} + 77 x^{2} + 120 x$ with respect to $x$ is $\frac{d}{d x} [\frac{1}{5} x^{5}] + \frac{d}{d x} [\frac{7}{2} x^{4}] + \frac{d}{d x} [\frac{71}{3} x^{3}] + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$ .

$\frac{d}{d x} [\frac{1}{5} x^{5}] + \frac{d}{d x} [\frac{7}{2} x^{4}] + \frac{d}{d x} [\frac{71}{3} x^{3}] + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 1.2

Evaluate $\frac{d}{d x} [\frac{1}{5} x^{5}]$ .

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Step 1.2.1

Since $\frac{1}{5}$ is constant with respect to $x$ , the derivative of $\frac{1}{5} x^{5}$ with respect to $x$ is $\frac{1}{5} \frac{d}{d x} [x^{5}]$ .

$\frac{1}{5} \frac{d}{d x} [x^{5}] + \frac{d}{d x} [\frac{7}{2} x^{4}] + \frac{d}{d x} [\frac{71}{3} x^{3}] + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$\frac{1}{5} (5 x^{4}) + \frac{d}{d x} [\frac{7}{2} x^{4}] + \frac{d}{d x} [\frac{71}{3} x^{3}] + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 1.2.3

Combine $5$ and $\frac{1}{5}$ .

$\frac{5}{5} x^{4} + \frac{d}{d x} [\frac{7}{2} x^{4}] + \frac{d}{d x} [\frac{71}{3} x^{3}] + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 1.2.4

Combine $\frac{5}{5}$ and $x^{4}$ .

$\frac{5 x^{4}}{5} + \frac{d}{d x} [\frac{7}{2} x^{4}] + \frac{d}{d x} [\frac{71}{3} x^{3}] + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 1.2.5

Cancel the common factor of $5$ .

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Step 1.2.5.1

Cancel the common factor.

$\frac{5 x^{4}}{5} + \frac{d}{d x} [\frac{7}{2} x^{4}] + \frac{d}{d x} [\frac{71}{3} x^{3}] + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 1.2.5.2

Divide $x^{4}$ by $1$ .

$x^{4} + \frac{d}{d x} [\frac{7}{2} x^{4}] + \frac{d}{d x} [\frac{71}{3} x^{3}] + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 1.3

Evaluate $\frac{d}{d x} [\frac{7}{2} x^{4}]$ .

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Step 1.3.1

Since $\frac{7}{2}$ is constant with respect to $x$ , the derivative of $\frac{7}{2} x^{4}$ with respect to $x$ is $\frac{7}{2} \frac{d}{d x} [x^{4}]$ .

$x^{4} + \frac{7}{2} \frac{d}{d x} [x^{4}] + \frac{d}{d x} [\frac{71}{3} x^{3}] + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$x^{4} + \frac{7}{2} (4 x^{3}) + \frac{d}{d x} [\frac{71}{3} x^{3}] + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 1.3.3

Combine $4$ and $\frac{7}{2}$ .

$x^{4} + \frac{4 \cdot 7}{2} x^{3} + \frac{d}{d x} [\frac{71}{3} x^{3}] + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 1.3.4

Multiply $4$ by $7$ .

$x^{4} + \frac{28}{2} x^{3} + \frac{d}{d x} [\frac{71}{3} x^{3}] + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 1.3.5

Combine $\frac{28}{2}$ and $x^{3}$ .

$x^{4} + \frac{28 x^{3}}{2} + \frac{d}{d x} [\frac{71}{3} x^{3}] + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 1.3.6

Cancel the common factor of $28$ and $2$ .

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Step 1.3.6.1

Factor $2$ out of $28 x^{3}$ .

$x^{4} + \frac{2 (14 x^{3})}{2} + \frac{d}{d x} [\frac{71}{3} x^{3}] + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 1.3.6.2

Cancel the common factors.

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Step 1.3.6.2.1

Factor $2$ out of $2$ .

$x^{4} + \frac{2 (14 x^{3})}{2 (1)} + \frac{d}{d x} [\frac{71}{3} x^{3}] + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 1.3.6.2.2

Cancel the common factor.

$x^{4} + \frac{2 (14 x^{3})}{2 \cdot 1} + \frac{d}{d x} [\frac{71}{3} x^{3}] + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 1.3.6.2.3

Rewrite the expression.

$x^{4} + \frac{14 x^{3}}{1} + \frac{d}{d x} [\frac{71}{3} x^{3}] + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 1.3.6.2.4

Divide $14 x^{3}$ by $1$ .

$x^{4} + 14 x^{3} + \frac{d}{d x} [\frac{71}{3} x^{3}] + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 1.4

Evaluate $\frac{d}{d x} [\frac{71}{3} x^{3}]$ .

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Step 1.4.1

Since $\frac{71}{3}$ is constant with respect to $x$ , the derivative of $\frac{71}{3} x^{3}$ with respect to $x$ is $\frac{71}{3} \frac{d}{d x} [x^{3}]$ .

$x^{4} + 14 x^{3} + \frac{71}{3} \frac{d}{d x} [x^{3}] + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 1.4.2

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

$x^{4} + 14 x^{3} + \frac{71}{3} (3 x^{2}) + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 1.4.3

Combine $3$ and $\frac{71}{3}$ .

$x^{4} + 14 x^{3} + \frac{3 \cdot 71}{3} x^{2} + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 1.4.4

Multiply $3$ by $71$ .

$x^{4} + 14 x^{3} + \frac{213}{3} x^{2} + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 1.4.5

Combine $\frac{213}{3}$ and $x^{2}$ .

$x^{4} + 14 x^{3} + \frac{213 x^{2}}{3} + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 1.4.6

Cancel the common factor of $213$ and $3$ .

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Step 1.4.6.1

Factor $3$ out of $213 x^{2}$ .

$x^{4} + 14 x^{3} + \frac{3 (71 x^{2})}{3} + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 1.4.6.2

Cancel the common factors.

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Step 1.4.6.2.1

Factor $3$ out of $3$ .

$x^{4} + 14 x^{3} + \frac{3 (71 x^{2})}{3 (1)} + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 1.4.6.2.2

Cancel the common factor.

$x^{4} + 14 x^{3} + \frac{3 (71 x^{2})}{3 \cdot 1} + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 1.4.6.2.3

Rewrite the expression.

$x^{4} + 14 x^{3} + \frac{71 x^{2}}{1} + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 1.4.6.2.4

Divide $71 x^{2}$ by $1$ .

$x^{4} + 14 x^{3} + 71 x^{2} + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 1.5

Evaluate $\frac{d}{d x} [77 x^{2}]$ .

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Step 1.5.1

Since $77$ is constant with respect to $x$ , the derivative of $77 x^{2}$ with respect to $x$ is $77 \frac{d}{d x} [x^{2}]$ .

$x^{4} + 14 x^{3} + 71 x^{2} + 77 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [120 x]$

Step 1.5.2

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

$x^{4} + 14 x^{3} + 71 x^{2} + 77 (2 x) + \frac{d}{d x} [120 x]$

Step 1.5.3

Multiply $2$ by $77$ .

$x^{4} + 14 x^{3} + 71 x^{2} + 154 x + \frac{d}{d x} [120 x]$

Step 1.6

Evaluate $\frac{d}{d x} [120 x]$ .

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Step 1.6.1

Since $120$ is constant with respect to $x$ , the derivative of $120 x$ with respect to $x$ is $120 \frac{d}{d x} [x]$ .

$x^{4} + 14 x^{3} + 71 x^{2} + 154 x + 120 \frac{d}{d x} [x]$

Step 1.6.2

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

$x^{4} + 14 x^{3} + 71 x^{2} + 154 x + 120 \cdot 1$

Step 1.6.3

Multiply $120$ by $1$ .

$x^{4} + 14 x^{3} + 71 x^{2} + 154 x + 120$

Step 2

Find the second derivative of the function.

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Step 2.1

Differentiate.

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Step 2.1.1

By the Sum Rule, the derivative of $x^{4} + 14 x^{3} + 71 x^{2} + 154 x + 120$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [14 x^{3}] + \frac{d}{d x} [71 x^{2}] + \frac{d}{d x} [154 x] + \frac{d}{d x} [120]$ .

$f'' (x) = \frac{d}{d x} (x^{4}) + \frac{d}{d x} (14 x^{3}) + \frac{d}{d x} (71 x^{2}) + \frac{d}{d x} (154 x) + \frac{d}{d x} (120)$

Step 2.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$f'' (x) = 4 x^{3} + \frac{d}{d x} (14 x^{3}) + \frac{d}{d x} (71 x^{2}) + \frac{d}{d x} (154 x) + \frac{d}{d x} (120)$

Step 2.2

Evaluate $\frac{d}{d x} [14 x^{3}]$ .

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Step 2.2.1

Since $14$ is constant with respect to $x$ , the derivative of $14 x^{3}$ with respect to $x$ is $14 \frac{d}{d x} [x^{3}]$ .

$f'' (x) = 4 x^{3} + 14 \frac{d}{d x} (x^{3}) + \frac{d}{d x} (71 x^{2}) + \frac{d}{d x} (154 x) + \frac{d}{d x} (120)$

Step 2.2.2

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

$f'' (x) = 4 x^{3} + 14 (3 x^{2}) + \frac{d}{d x} (71 x^{2}) + \frac{d}{d x} (154 x) + \frac{d}{d x} (120)$

Step 2.2.3

Multiply $3$ by $14$ .

$f'' (x) = 4 x^{3} + 42 x^{2} + \frac{d}{d x} (71 x^{2}) + \frac{d}{d x} (154 x) + \frac{d}{d x} (120)$

Step 2.3

Evaluate $\frac{d}{d x} [71 x^{2}]$ .

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Step 2.3.1

Since $71$ is constant with respect to $x$ , the derivative of $71 x^{2}$ with respect to $x$ is $71 \frac{d}{d x} [x^{2}]$ .

$f'' (x) = 4 x^{3} + 42 x^{2} + 71 \frac{d}{d x} (x^{2}) + \frac{d}{d x} (154 x) + \frac{d}{d x} (120)$

Step 2.3.2

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

$f'' (x) = 4 x^{3} + 42 x^{2} + 71 (2 x) + \frac{d}{d x} (154 x) + \frac{d}{d x} (120)$

Step 2.3.3

Multiply $2$ by $71$ .

$f'' (x) = 4 x^{3} + 42 x^{2} + 142 x + \frac{d}{d x} (154 x) + \frac{d}{d x} (120)$

Step 2.4

Evaluate $\frac{d}{d x} [154 x]$ .

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Step 2.4.1

Since $154$ is constant with respect to $x$ , the derivative of $154 x$ with respect to $x$ is $154 \frac{d}{d x} [x]$ .

$f'' (x) = 4 x^{3} + 42 x^{2} + 142 x + 154 \frac{d}{d x} (x) + \frac{d}{d x} (120)$

Step 2.4.2

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

$f'' (x) = 4 x^{3} + 42 x^{2} + 142 x + 154 \cdot 1 + \frac{d}{d x} (120)$

Step 2.4.3

Multiply $154$ by $1$ .

$f'' (x) = 4 x^{3} + 42 x^{2} + 142 x + 154 + \frac{d}{d x} (120)$

Step 2.5

Differentiate using the Constant Rule.

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Step 2.5.1

Since $120$ is constant with respect to $x$ , the derivative of $120$ with respect to $x$ is $0$ .

$f'' (x) = 4 x^{3} + 42 x^{2} + 142 x + 154 + 0$

Step 2.5.2

Add $4 x^{3} + 42 x^{2} + 142 x + 154$ and $0$ .

$f'' (x) = 4 x^{3} + 42 x^{2} + 142 x + 154$

Step 3

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

$x^{4} + 14 x^{3} + 71 x^{2} + 154 x + 120 = 0$

Step 4

Find the first derivative.

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Step 4.1

Find the first derivative.

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Step 4.1.1

$\frac{d}{d x} [\frac{1}{5} x^{5}] + \frac{d}{d x} [\frac{7}{2} x^{4}] + \frac{d}{d x} [\frac{71}{3} x^{3}] + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 4.1.2

Evaluate $\frac{d}{d x} [\frac{1}{5} x^{5}]$ .

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Step 4.1.2.1

Since $\frac{1}{5}$ is constant with respect to $x$ , the derivative of $\frac{1}{5} x^{5}$ with respect to $x$ is $\frac{1}{5} \frac{d}{d x} [x^{5}]$ .

$\frac{1}{5} \frac{d}{d x} [x^{5}] + \frac{d}{d x} [\frac{7}{2} x^{4}] + \frac{d}{d x} [\frac{71}{3} x^{3}] + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 4.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$\frac{1}{5} (5 x^{4}) + \frac{d}{d x} [\frac{7}{2} x^{4}] + \frac{d}{d x} [\frac{71}{3} x^{3}] + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 4.1.2.3

Combine $5$ and $\frac{1}{5}$ .

$\frac{5}{5} x^{4} + \frac{d}{d x} [\frac{7}{2} x^{4}] + \frac{d}{d x} [\frac{71}{3} x^{3}] + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 4.1.2.4

Combine $\frac{5}{5}$ and $x^{4}$ .

$\frac{5 x^{4}}{5} + \frac{d}{d x} [\frac{7}{2} x^{4}] + \frac{d}{d x} [\frac{71}{3} x^{3}] + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 4.1.2.5

Cancel the common factor of $5$ .

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Step 4.1.2.5.1

Cancel the common factor.

$\frac{5 x^{4}}{5} + \frac{d}{d x} [\frac{7}{2} x^{4}] + \frac{d}{d x} [\frac{71}{3} x^{3}] + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 4.1.2.5.2

Divide $x^{4}$ by $1$ .

$x^{4} + \frac{d}{d x} [\frac{7}{2} x^{4}] + \frac{d}{d x} [\frac{71}{3} x^{3}] + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 4.1.3

Evaluate $\frac{d}{d x} [\frac{7}{2} x^{4}]$ .

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Step 4.1.3.1

Since $\frac{7}{2}$ is constant with respect to $x$ , the derivative of $\frac{7}{2} x^{4}$ with respect to $x$ is $\frac{7}{2} \frac{d}{d x} [x^{4}]$ .

$x^{4} + \frac{7}{2} \frac{d}{d x} [x^{4}] + \frac{d}{d x} [\frac{71}{3} x^{3}] + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 4.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$x^{4} + \frac{7}{2} (4 x^{3}) + \frac{d}{d x} [\frac{71}{3} x^{3}] + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 4.1.3.3

Combine $4$ and $\frac{7}{2}$ .

$x^{4} + \frac{4 \cdot 7}{2} x^{3} + \frac{d}{d x} [\frac{71}{3} x^{3}] + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 4.1.3.4

Multiply $4$ by $7$ .

$x^{4} + \frac{28}{2} x^{3} + \frac{d}{d x} [\frac{71}{3} x^{3}] + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 4.1.3.5

Combine $\frac{28}{2}$ and $x^{3}$ .

$x^{4} + \frac{28 x^{3}}{2} + \frac{d}{d x} [\frac{71}{3} x^{3}] + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 4.1.3.6

Cancel the common factor of $28$ and $2$ .

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Step 4.1.3.6.1

Factor $2$ out of $28 x^{3}$ .

$x^{4} + \frac{2 (14 x^{3})}{2} + \frac{d}{d x} [\frac{71}{3} x^{3}] + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 4.1.3.6.2

Cancel the common factors.

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Step 4.1.3.6.2.1

Factor $2$ out of $2$ .

$x^{4} + \frac{2 (14 x^{3})}{2 (1)} + \frac{d}{d x} [\frac{71}{3} x^{3}] + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 4.1.3.6.2.2

Cancel the common factor.

$x^{4} + \frac{2 (14 x^{3})}{2 \cdot 1} + \frac{d}{d x} [\frac{71}{3} x^{3}] + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 4.1.3.6.2.3

Rewrite the expression.

$x^{4} + \frac{14 x^{3}}{1} + \frac{d}{d x} [\frac{71}{3} x^{3}] + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 4.1.3.6.2.4

Divide $14 x^{3}$ by $1$ .

$x^{4} + 14 x^{3} + \frac{d}{d x} [\frac{71}{3} x^{3}] + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 4.1.4

Evaluate $\frac{d}{d x} [\frac{71}{3} x^{3}]$ .

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Step 4.1.4.1

Since $\frac{71}{3}$ is constant with respect to $x$ , the derivative of $\frac{71}{3} x^{3}$ with respect to $x$ is $\frac{71}{3} \frac{d}{d x} [x^{3}]$ .

$x^{4} + 14 x^{3} + \frac{71}{3} \frac{d}{d x} [x^{3}] + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 4.1.4.2

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

$x^{4} + 14 x^{3} + \frac{71}{3} (3 x^{2}) + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 4.1.4.3

Combine $3$ and $\frac{71}{3}$ .

$x^{4} + 14 x^{3} + \frac{3 \cdot 71}{3} x^{2} + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 4.1.4.4

Multiply $3$ by $71$ .

$x^{4} + 14 x^{3} + \frac{213}{3} x^{2} + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 4.1.4.5

Combine $\frac{213}{3}$ and $x^{2}$ .

$x^{4} + 14 x^{3} + \frac{213 x^{2}}{3} + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 4.1.4.6

Cancel the common factor of $213$ and $3$ .

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Step 4.1.4.6.1

Factor $3$ out of $213 x^{2}$ .

$x^{4} + 14 x^{3} + \frac{3 (71 x^{2})}{3} + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 4.1.4.6.2

Cancel the common factors.

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Step 4.1.4.6.2.1

Factor $3$ out of $3$ .

$x^{4} + 14 x^{3} + \frac{3 (71 x^{2})}{3 (1)} + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 4.1.4.6.2.2

Cancel the common factor.

$x^{4} + 14 x^{3} + \frac{3 (71 x^{2})}{3 \cdot 1} + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 4.1.4.6.2.3

Rewrite the expression.

$x^{4} + 14 x^{3} + \frac{71 x^{2}}{1} + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 4.1.4.6.2.4

Divide $71 x^{2}$ by $1$ .

$x^{4} + 14 x^{3} + 71 x^{2} + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 4.1.5

Evaluate $\frac{d}{d x} [77 x^{2}]$ .

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Step 4.1.5.1

Since $77$ is constant with respect to $x$ , the derivative of $77 x^{2}$ with respect to $x$ is $77 \frac{d}{d x} [x^{2}]$ .

$x^{4} + 14 x^{3} + 71 x^{2} + 77 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [120 x]$

Step 4.1.5.2

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

$x^{4} + 14 x^{3} + 71 x^{2} + 77 (2 x) + \frac{d}{d x} [120 x]$

Step 4.1.5.3

Multiply $2$ by $77$ .

$x^{4} + 14 x^{3} + 71 x^{2} + 154 x + \frac{d}{d x} [120 x]$

Step 4.1.6

Evaluate $\frac{d}{d x} [120 x]$ .

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Step 4.1.6.1

Since $120$ is constant with respect to $x$ , the derivative of $120 x$ with respect to $x$ is $120 \frac{d}{d x} [x]$ .

$x^{4} + 14 x^{3} + 71 x^{2} + 154 x + 120 \frac{d}{d x} [x]$

Step 4.1.6.2

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

$x^{4} + 14 x^{3} + 71 x^{2} + 154 x + 120 \cdot 1$

Step 4.1.6.3

Multiply $120$ by $1$ .

$f' (x) = x^{4} + 14 x^{3} + 71 x^{2} + 154 x + 120$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $x^{4} + 14 x^{3} + 71 x^{2} + 154 x + 120$ .

$x^{4} + 14 x^{3} + 71 x^{2} + 154 x + 120$

Step 5

Set the first derivative equal to $0$ then solve the equation $x^{4} + 14 x^{3} + 71 x^{2} + 154 x + 120 = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$x^{4} + 14 x^{3} + 71 x^{2} + 154 x + 120 = 0$

Step 5.2

Factor the left side of the equation.

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Step 5.2.1

Factor $x^{4} + 14 x^{3} + 71 x^{2} + 154 x + 120$ using the rational roots test.

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Step 5.2.1.1

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1, \pm 120, \pm 2, \pm 60, \pm 3, \pm 40, \pm 4, \pm 30, \pm 5, \pm 24, \pm 6, \pm 20, \pm 8, \pm 15, \pm 10, \pm 12$

$q = \pm 1$

Step 5.2.1.2

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 120, \pm 2, \pm 60, \pm 3, \pm 40, \pm 4, \pm 30, \pm 5, \pm 24, \pm 6, \pm 20, \pm 8, \pm 15, \pm 10, \pm 12$

Step 5.2.1.3

Substitute $- 2$ and simplify the expression. In this case, the expression is equal to $0$ so $- 2$ is a root of the polynomial.

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Step 5.2.1.3.1

Substitute $- 2$ into the polynomial.

${(- 2)}^{4} + 14 {(- 2)}^{3} + 71 {(- 2)}^{2} + 154 \cdot - 2 + 120$

Step 5.2.1.3.2

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

$16 + 14 {(- 2)}^{3} + 71 {(- 2)}^{2} + 154 \cdot - 2 + 120$

Step 5.2.1.3.3

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

$16 + 14 \cdot - 8 + 71 {(- 2)}^{2} + 154 \cdot - 2 + 120$

Step 5.2.1.3.4

Multiply $14$ by $- 8$ .

$16 - 112 + 71 {(- 2)}^{2} + 154 \cdot - 2 + 120$

Step 5.2.1.3.5

Subtract $112$ from $16$ .

$- 96 + 71 {(- 2)}^{2} + 154 \cdot - 2 + 120$

Step 5.2.1.3.6

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

$- 96 + 71 \cdot 4 + 154 \cdot - 2 + 120$

Step 5.2.1.3.7

Multiply $71$ by $4$ .

$- 96 + 284 + 154 \cdot - 2 + 120$

Step 5.2.1.3.8

Add $- 96$ and $284$ .

$188 + 154 \cdot - 2 + 120$

Step 5.2.1.3.9

Multiply $154$ by $- 2$ .

$188 - 308 + 120$

Step 5.2.1.3.10

Subtract $308$ from $188$ .

$- 120 + 120$

Step 5.2.1.3.11

Add $- 120$ and $120$ .

$0$

Step 5.2.1.4

Since $- 2$ is a known root, divide the polynomial by $x + 2$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{4} + 14 x^{3} + 71 x^{2} + 154 x + 120}{x + 2}$

Step 5.2.1.5

Divide $x^{4} + 14 x^{3} + 71 x^{2} + 154 x + 120$ by $x + 2$ .

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Step 5.2.1.5.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$2$

$x^{4}$

$14 x^{3}$

$71 x^{2}$

$154 x$

$120$

Step 5.2.1.5.2

Divide the highest order term in the dividend $x^{4}$ by the highest order term in divisor $x$ .

$x^{3}$

$x$

$2$

$x^{4}$

$14 x^{3}$

$71 x^{2}$

$154 x$

$120$

Step 5.2.1.5.3

Multiply the new quotient term by the divisor.

$x^{3}$

$x$

$2$

$x^{4}$

$14 x^{3}$

$71 x^{2}$

$154 x$

$120$

$x^{4}$

$2 x^{3}$

Step 5.2.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{4} + 2 x^{3}$

$x^{3}$

$x$

$2$

$x^{4}$

$14 x^{3}$

$71 x^{2}$

$154 x$

$120$

$x^{4}$

$2 x^{3}$

Step 5.2.1.5.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{3}$

$x$

$2$

$x^{4}$

$14 x^{3}$

$71 x^{2}$

$154 x$

$120$

$x^{4}$

$2 x^{3}$

$12 x^{3}$

Step 5.2.1.5.6

Pull the next terms from the original dividend down into the current dividend.

$x^{3}$

$x$

$2$

$x^{4}$

$14 x^{3}$

$71 x^{2}$

$154 x$

$120$

$x^{4}$

$2 x^{3}$

$12 x^{3}$

$71 x^{2}$

Step 5.2.1.5.7

Divide the highest order term in the dividend $12 x^{3}$ by the highest order term in divisor $x$ .

$x^{3}$

$12 x^{2}$

$x$

$2$

$x^{4}$

$14 x^{3}$

$71 x^{2}$

$154 x$

$120$

$x^{4}$

$2 x^{3}$

$12 x^{3}$

$71 x^{2}$

Step 5.2.1.5.8

Multiply the new quotient term by the divisor.

$x^{3}$

$12 x^{2}$

$x$

$2$

$x^{4}$

$14 x^{3}$

$71 x^{2}$

$154 x$

$120$

$x^{4}$

$2 x^{3}$

$12 x^{3}$

$71 x^{2}$

$12 x^{3}$

$24 x^{2}$

Step 5.2.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $12 x^{3} + 24 x^{2}$

$x^{3}$

$12 x^{2}$

$x$

$2$

$x^{4}$

$14 x^{3}$

$71 x^{2}$

$154 x$

$120$

$x^{4}$

$2 x^{3}$

$12 x^{3}$

$71 x^{2}$

$12 x^{3}$

$24 x^{2}$

Step 5.2.1.5.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{3}$

$12 x^{2}$

$x$

$2$

$x^{4}$

$14 x^{3}$

$71 x^{2}$

$154 x$

$120$

$x^{4}$

$2 x^{3}$

$12 x^{3}$

$71 x^{2}$

$12 x^{3}$

$24 x^{2}$

$47 x^{2}$

Step 5.2.1.5.11

Pull the next terms from the original dividend down into the current dividend.

$x^{3}$

$12 x^{2}$

$x$

$2$

$x^{4}$

$14 x^{3}$

$71 x^{2}$

$154 x$

$120$

$x^{4}$

$2 x^{3}$

$12 x^{3}$

$71 x^{2}$

$12 x^{3}$

$24 x^{2}$

$47 x^{2}$

$154 x$

Step 5.2.1.5.12

Divide the highest order term in the dividend $47 x^{2}$ by the highest order term in divisor $x$ .

$x^{3}$

$12 x^{2}$

$47 x$

$x$

$2$

$x^{4}$

$14 x^{3}$

$71 x^{2}$

$154 x$

$120$

$x^{4}$

$2 x^{3}$

$12 x^{3}$

$71 x^{2}$

$12 x^{3}$

$24 x^{2}$

$47 x^{2}$

$154 x$

Step 5.2.1.5.13

Multiply the new quotient term by the divisor.

$x^{3}$

$12 x^{2}$

$47 x$

$x$

$2$

$x^{4}$

$14 x^{3}$

$71 x^{2}$

$154 x$

$120$

$x^{4}$

$2 x^{3}$

$12 x^{3}$

$71 x^{2}$

$12 x^{3}$

$24 x^{2}$

$47 x^{2}$

$154 x$

$47 x^{2}$

$94 x$

Step 5.2.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $47 x^{2} + 94 x$

$x^{3}$

$12 x^{2}$

$47 x$

$x$

$2$

$x^{4}$

$14 x^{3}$

$71 x^{2}$

$154 x$

$120$

$x^{4}$

$2 x^{3}$

$12 x^{3}$

$71 x^{2}$

$12 x^{3}$

$24 x^{2}$

$47 x^{2}$

$154 x$

$47 x^{2}$

$94 x$

Step 5.2.1.5.15

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{3}$

$12 x^{2}$

$47 x$

$x$

$2$

$x^{4}$

$14 x^{3}$

$71 x^{2}$

$154 x$

$120$

$x^{4}$

$2 x^{3}$

$12 x^{3}$

$71 x^{2}$

$12 x^{3}$

$24 x^{2}$

$47 x^{2}$

$154 x$

$47 x^{2}$

$94 x$

$60 x$

Step 5.2.1.5.16

Pull the next terms from the original dividend down into the current dividend.

$x^{3}$

$12 x^{2}$

$47 x$

$x$

$2$

$x^{4}$

$14 x^{3}$

$71 x^{2}$

$154 x$

$120$

$x^{4}$

$2 x^{3}$

$12 x^{3}$

$71 x^{2}$

$12 x^{3}$

$24 x^{2}$

$47 x^{2}$

$154 x$

$47 x^{2}$

$94 x$

$60 x$

$120$

Step 5.2.1.5.17

Divide the highest order term in the dividend $60 x$ by the highest order term in divisor $x$ .

$x^{3}$

$12 x^{2}$

$47 x$

$60$

$x$

$2$

$x^{4}$

$14 x^{3}$

$71 x^{2}$

$154 x$

$120$

$x^{4}$

$2 x^{3}$

$12 x^{3}$

$71 x^{2}$

$12 x^{3}$

$24 x^{2}$

$47 x^{2}$

$154 x$

$47 x^{2}$

$94 x$

$60 x$

$120$

Step 5.2.1.5.18

Multiply the new quotient term by the divisor.

$x^{3}$

$12 x^{2}$

$47 x$

$60$

$x$

$2$

$x^{4}$

$14 x^{3}$

$71 x^{2}$

$154 x$

$120$

$x^{4}$

$2 x^{3}$

$12 x^{3}$

$71 x^{2}$

$12 x^{3}$

$24 x^{2}$

$47 x^{2}$

$154 x$

$47 x^{2}$

$94 x$

$60 x$

$120$

$60 x$

$120$

Step 5.2.1.5.19

The expression needs to be subtracted from the dividend, so change all the signs in $60 x + 120$

$x^{3}$

$12 x^{2}$

$47 x$

$60$

$x$

$2$

$x^{4}$

$14 x^{3}$

$71 x^{2}$

$154 x$

$120$

$x^{4}$

$2 x^{3}$

$12 x^{3}$

$71 x^{2}$

$12 x^{3}$

$24 x^{2}$

$47 x^{2}$

$154 x$

$47 x^{2}$

$94 x$

$60 x$

$120$

$60 x$

$120$

Step 5.2.1.5.20

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{3}$

$12 x^{2}$

$47 x$

$60$

$x$

$2$

$x^{4}$

$14 x^{3}$

$71 x^{2}$

$154 x$

$120$

$x^{4}$

$2 x^{3}$

$12 x^{3}$

$71 x^{2}$

$12 x^{3}$

$24 x^{2}$

$47 x^{2}$

$154 x$

$47 x^{2}$

$94 x$

$60 x$

$120$

$60 x$

$120$

$0$

Step 5.2.1.5.21

Since the remander is $0$ , the final answer is the quotient.

$x^{3} + 12 x^{2} + 47 x + 60$

Step 5.2.1.6

Write $x^{4} + 14 x^{3} + 71 x^{2} + 154 x + 120$ as a set of factors.

$(x + 2) (x^{3} + 12 x^{2} + 47 x + 60) = 0$

Step 5.2.2

Factor $x^{3} + 12 x^{2} + 47 x + 60$ using the rational roots test.

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Step 5.2.2.1

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1, \pm 60, \pm 2, \pm 30, \pm 3, \pm 20, \pm 4, \pm 15, \pm 5, \pm 12, \pm 6, \pm 10$

$q = \pm 1$

Step 5.2.2.2

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 60, \pm 2, \pm 30, \pm 3, \pm 20, \pm 4, \pm 15, \pm 5, \pm 12, \pm 6, \pm 10$

Step 5.2.2.3

Substitute $- 3$ and simplify the expression. In this case, the expression is equal to $0$ so $- 3$ is a root of the polynomial.

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Step 5.2.2.3.1

Substitute $- 3$ into the polynomial.

${(- 3)}^{3} + 12 {(- 3)}^{2} + 47 \cdot - 3 + 60$

Step 5.2.2.3.2

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

$- 27 + 12 {(- 3)}^{2} + 47 \cdot - 3 + 60$

Step 5.2.2.3.3

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

$- 27 + 12 \cdot 9 + 47 \cdot - 3 + 60$

Step 5.2.2.3.4

Multiply $12$ by $9$ .

$- 27 + 108 + 47 \cdot - 3 + 60$

Step 5.2.2.3.5

Add $- 27$ and $108$ .

$81 + 47 \cdot - 3 + 60$

Step 5.2.2.3.6

Multiply $47$ by $- 3$ .

$81 - 141 + 60$

Step 5.2.2.3.7

Subtract $141$ from $81$ .

$- 60 + 60$

Step 5.2.2.3.8

Add $- 60$ and $60$ .

$0$

Step 5.2.2.4

Since $- 3$ is a known root, divide the polynomial by $x + 3$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} + 12 x^{2} + 47 x + 60}{x + 3}$

Step 5.2.2.5

Divide $x^{3} + 12 x^{2} + 47 x + 60$ by $x + 3$ .

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Step 5.2.2.5.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$3$

$x^{3}$

$12 x^{2}$

$47 x$

$60$

Step 5.2.2.5.2

Divide the highest order term in the dividend $x^{3}$ by the highest order term in divisor $x$ .

					$x^{2}$
	$x$	+	$3$		$x^{3}$	+	$12 x^{2}$	+	$47 x$	+	$60$

Step 5.2.2.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$3$		$x^{3}$	+	$12 x^{2}$	+	$47 x$	+	$60$
			+	$x^{3}$	+	$3 x^{2}$

Step 5.2.2.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{3} + 3 x^{2}$

				$x^{2}$
$x$	+	$3$		$x^{3}$	+	$12 x^{2}$	+	$47 x$	+	$60$
			-	$x^{3}$	-	$3 x^{2}$

Step 5.2.2.5.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x^{2}$
$x$	+	$3$		$x^{3}$	+	$12 x^{2}$	+	$47 x$	+	$60$
			-	$x^{3}$	-	$3 x^{2}$
					+	$9 x^{2}$

Step 5.2.2.5.6

Pull the next terms from the original dividend down into the current dividend.

				$x^{2}$
$x$	+	$3$		$x^{3}$	+	$12 x^{2}$	+	$47 x$	+	$60$
			-	$x^{3}$	-	$3 x^{2}$
					+	$9 x^{2}$	+	$47 x$

Step 5.2.2.5.7

Divide the highest order term in the dividend $9 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	+	$9 x$
$x$	+	$3$		$x^{3}$	+	$12 x^{2}$	+	$47 x$	+	$60$
			-	$x^{3}$	-	$3 x^{2}$
					+	$9 x^{2}$	+	$47 x$

Step 5.2.2.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$9 x$
$x$	+	$3$		$x^{3}$	+	$12 x^{2}$	+	$47 x$	+	$60$
			-	$x^{3}$	-	$3 x^{2}$
					+	$9 x^{2}$	+	$47 x$
					+	$9 x^{2}$	+	$27 x$

Step 5.2.2.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $9 x^{2} + 27 x$

				$x^{2}$	+	$9 x$
$x$	+	$3$		$x^{3}$	+	$12 x^{2}$	+	$47 x$	+	$60$
			-	$x^{3}$	-	$3 x^{2}$
					+	$9 x^{2}$	+	$47 x$
					-	$9 x^{2}$	-	$27 x$

Step 5.2.2.5.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x^{2}$	+	$9 x$
$x$	+	$3$		$x^{3}$	+	$12 x^{2}$	+	$47 x$	+	$60$
			-	$x^{3}$	-	$3 x^{2}$
					+	$9 x^{2}$	+	$47 x$
					-	$9 x^{2}$	-	$27 x$
							+	$20 x$

Step 5.2.2.5.11

Pull the next terms from the original dividend down into the current dividend.

				$x^{2}$	+	$9 x$
$x$	+	$3$		$x^{3}$	+	$12 x^{2}$	+	$47 x$	+	$60$
			-	$x^{3}$	-	$3 x^{2}$
					+	$9 x^{2}$	+	$47 x$
					-	$9 x^{2}$	-	$27 x$
							+	$20 x$	+	$60$

Step 5.2.2.5.12

Divide the highest order term in the dividend $20 x$ by the highest order term in divisor $x$ .

				$x^{2}$	+	$9 x$	+	$20$
$x$	+	$3$		$x^{3}$	+	$12 x^{2}$	+	$47 x$	+	$60$
			-	$x^{3}$	-	$3 x^{2}$
					+	$9 x^{2}$	+	$47 x$
					-	$9 x^{2}$	-	$27 x$
							+	$20 x$	+	$60$

Step 5.2.2.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$9 x$	+	$20$
$x$	+	$3$		$x^{3}$	+	$12 x^{2}$	+	$47 x$	+	$60$
			-	$x^{3}$	-	$3 x^{2}$
					+	$9 x^{2}$	+	$47 x$
					-	$9 x^{2}$	-	$27 x$
							+	$20 x$	+	$60$
							+	$20 x$	+	$60$

Step 5.2.2.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $20 x + 60$

				$x^{2}$	+	$9 x$	+	$20$
$x$	+	$3$		$x^{3}$	+	$12 x^{2}$	+	$47 x$	+	$60$
			-	$x^{3}$	-	$3 x^{2}$
					+	$9 x^{2}$	+	$47 x$
					-	$9 x^{2}$	-	$27 x$
							+	$20 x$	+	$60$
							-	$20 x$	-	$60$

Step 5.2.2.5.15

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x^{2}$	+	$9 x$	+	$20$
$x$	+	$3$		$x^{3}$	+	$12 x^{2}$	+	$47 x$	+	$60$
			-	$x^{3}$	-	$3 x^{2}$
					+	$9 x^{2}$	+	$47 x$
					-	$9 x^{2}$	-	$27 x$
							+	$20 x$	+	$60$
							-	$20 x$	-	$60$
										$0$

Step 5.2.2.5.16

Since the remander is $0$ , the final answer is the quotient.

$x^{2} + 9 x + 20$

Step 5.2.2.6

Write $x^{3} + 12 x^{2} + 47 x + 60$ as a set of factors.

$(x + 2) ((x + 3) (x^{2} + 9 x + 20)) = 0$

Step 5.2.3

Factor $x^{2} + 9 x + 20$ using the AC method.

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Step 5.2.3.1

Factor $x^{2} + 9 x + 20$ using the AC method.

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Step 5.2.3.1.1

Factor $x^{2} + 9 x + 20$ using the AC method.

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Step 5.2.3.1.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $20$ and whose sum is $9$ .

$4, 5$

Step 5.2.3.1.1.2

Write the factored form using these integers.

$(x + 2) ((x + 3) ((x + 4) (x + 5))) = 0$

Step 5.2.3.1.2

Remove unnecessary parentheses.

$(x + 2) ((x + 3) (x + 4) (x + 5)) = 0$

Step 5.2.3.2

Remove unnecessary parentheses.

$(x + 2) (x + 3) (x + 4) (x + 5) = 0$

Step 5.3

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

$x + 2 = 0$

$x + 3 = 0$

$x + 4 = 0$

$x + 5 = 0$

Step 5.4

Set $x + 2$ equal to $0$ and solve for $x$ .

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Step 5.4.1

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Step 5.4.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 5.5

Set $x + 3$ equal to $0$ and solve for $x$ .

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Step 5.5.1

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Step 5.5.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 5.6

Set $x + 4$ equal to $0$ and solve for $x$ .

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Step 5.6.1

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Step 5.6.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 5.7

Set $x + 5$ equal to $0$ and solve for $x$ .

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Step 5.7.1

Set $x + 5$ equal to $0$ .

$x + 5 = 0$

Step 5.7.2

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 5.8

The final solution is all the values that make $(x + 2) (x + 3) (x + 4) (x + 5) = 0$ true.

$x = - 2, - 3, - 4, - 5$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

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

Step 7

Critical points to evaluate.

$x = - 2, - 3, - 4, - 5$

Step 8

Evaluate the second derivative at $x = - 2$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$4 {(- 2)}^{3} + 42 {(- 2)}^{2} + 142 (- 2) + 154$

Step 9

Evaluate the second derivative.

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Step 9.1

Simplify each term.

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Step 9.1.1

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

$4 \cdot - 8 + 42 {(- 2)}^{2} + 142 (- 2) + 154$

Step 9.1.2

Multiply $4$ by $- 8$ .

$- 32 + 42 {(- 2)}^{2} + 142 (- 2) + 154$

Step 9.1.3

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

$- 32 + 42 \cdot 4 + 142 (- 2) + 154$

Step 9.1.4

Multiply $42$ by $4$ .

$- 32 + 168 + 142 (- 2) + 154$

Step 9.1.5

Multiply $142$ by $- 2$ .

$- 32 + 168 - 284 + 154$

Step 9.2

Simplify by adding and subtracting.

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Step 9.2.1

Add $- 32$ and $168$ .

$136 - 284 + 154$

Step 9.2.2

Subtract $284$ from $136$ .

$- 148 + 154$

Step 9.2.3

Add $- 148$ and $154$ .

$6$

Step 10

$x = - 2$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = - 2$ is a local minimum

Step 11

Find the y-value when $x = - 2$ .

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Step 11.1

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

$f (- 2) = \frac{1}{5} \cdot {(- 2)}^{5} + \frac{7}{2} \cdot {(- 2)}^{4} + \frac{71}{3} \cdot {(- 2)}^{3} + 77 {(- 2)}^{2} + 120 (- 2)$

Step 11.2

Simplify the result.

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Step 11.2.1

Simplify each term.

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Step 11.2.1.1

Raise $- 2$ to the power of $5$ .

$f (- 2) = \frac{1}{5} \cdot - 32 + \frac{7}{2} \cdot {(- 2)}^{4} + \frac{71}{3} \cdot {(- 2)}^{3} + 77 {(- 2)}^{2} + 120 (- 2)$

Step 11.2.1.2

Combine $\frac{1}{5}$ and $- 32$ .

$f (- 2) = \frac{- 32}{5} + \frac{7}{2} \cdot {(- 2)}^{4} + \frac{71}{3} \cdot {(- 2)}^{3} + 77 {(- 2)}^{2} + 120 (- 2)$

Step 11.2.1.3

Move the negative in front of the fraction.

$f (- 2) = - \frac{32}{5} + \frac{7}{2} \cdot {(- 2)}^{4} + \frac{71}{3} \cdot {(- 2)}^{3} + 77 {(- 2)}^{2} + 120 (- 2)$

Step 11.2.1.4

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

$f (- 2) = - \frac{32}{5} + \frac{7}{2} \cdot 16 + \frac{71}{3} \cdot {(- 2)}^{3} + 77 {(- 2)}^{2} + 120 (- 2)$

Step 11.2.1.5

Cancel the common factor of $2$ .

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Step 11.2.1.5.1

Factor $2$ out of $16$ .

$f (- 2) = - \frac{32}{5} + \frac{7}{2} \cdot (2 (8)) + \frac{71}{3} \cdot {(- 2)}^{3} + 77 {(- 2)}^{2} + 120 (- 2)$

Step 11.2.1.5.2

Cancel the common factor.

$f (- 2) = - \frac{32}{5} + \frac{7}{2} \cdot (2 \cdot 8) + \frac{71}{3} \cdot {(- 2)}^{3} + 77 {(- 2)}^{2} + 120 (- 2)$

Step 11.2.1.5.3

Rewrite the expression.

$f (- 2) = - \frac{32}{5} + 7 \cdot 8 + \frac{71}{3} \cdot {(- 2)}^{3} + 77 {(- 2)}^{2} + 120 (- 2)$

Step 11.2.1.6

Multiply $7$ by $8$ .

$f (- 2) = - \frac{32}{5} + 56 + \frac{71}{3} \cdot {(- 2)}^{3} + 77 {(- 2)}^{2} + 120 (- 2)$

Step 11.2.1.7

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

$f (- 2) = - \frac{32}{5} + 56 + \frac{71}{3} \cdot - 8 + 77 {(- 2)}^{2} + 120 (- 2)$

Step 11.2.1.8

Multiply $\frac{71}{3} \cdot - 8$ .

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Step 11.2.1.8.1

Combine $\frac{71}{3}$ and $- 8$ .

$f (- 2) = - \frac{32}{5} + 56 + \frac{71 \cdot - 8}{3} + 77 {(- 2)}^{2} + 120 (- 2)$

Step 11.2.1.8.2

Multiply $71$ by $- 8$ .

$f (- 2) = - \frac{32}{5} + 56 + \frac{- 568}{3} + 77 {(- 2)}^{2} + 120 (- 2)$

Step 11.2.1.9

Move the negative in front of the fraction.

$f (- 2) = - \frac{32}{5} + 56 - \frac{568}{3} + 77 {(- 2)}^{2} + 120 (- 2)$

Step 11.2.1.10

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

$f (- 2) = - \frac{32}{5} + 56 - \frac{568}{3} + 77 \cdot 4 + 120 (- 2)$

Step 11.2.1.11

Multiply $77$ by $4$ .

$f (- 2) = - \frac{32}{5} + 56 - \frac{568}{3} + 308 + 120 (- 2)$

Step 11.2.1.12

Multiply $120$ by $- 2$ .

$f (- 2) = - \frac{32}{5} + 56 - \frac{568}{3} + 308 - 240$

Step 11.2.2

Find the common denominator.

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Step 11.2.2.1

Multiply $\frac{32}{5}$ by $\frac{3}{3}$ .

$f (- 2) = - (\frac{32}{5} \cdot \frac{3}{3}) + 56 - \frac{568}{3} + 308 - 240$

Step 11.2.2.2

Multiply $\frac{32}{5}$ by $\frac{3}{3}$ .

$f (- 2) = - \frac{32 \cdot 3}{5 \cdot 3} + 56 - \frac{568}{3} + 308 - 240$

Step 11.2.2.3

Write $56$ as a fraction with denominator $1$ .

$f (- 2) = - \frac{32 \cdot 3}{5 \cdot 3} + \frac{56}{1} - \frac{568}{3} + 308 - 240$

Step 11.2.2.4

Multiply $\frac{56}{1}$ by $\frac{15}{15}$ .

$f (- 2) = - \frac{32 \cdot 3}{5 \cdot 3} + \frac{56}{1} \cdot \frac{15}{15} - \frac{568}{3} + 308 - 240$

Step 11.2.2.5

Multiply $\frac{56}{1}$ by $\frac{15}{15}$ .

$f (- 2) = - \frac{32 \cdot 3}{5 \cdot 3} + \frac{56 \cdot 15}{15} - \frac{568}{3} + 308 - 240$

Step 11.2.2.6

Multiply $\frac{568}{3}$ by $\frac{5}{5}$ .

$f (- 2) = - \frac{32 \cdot 3}{5 \cdot 3} + \frac{56 \cdot 15}{15} - (\frac{568}{3} \cdot \frac{5}{5}) + 308 - 240$

Step 11.2.2.7

Multiply $\frac{568}{3}$ by $\frac{5}{5}$ .

$f (- 2) = - \frac{32 \cdot 3}{5 \cdot 3} + \frac{56 \cdot 15}{15} - \frac{568 \cdot 5}{3 \cdot 5} + 308 - 240$

Step 11.2.2.8

Write $308$ as a fraction with denominator $1$ .

$f (- 2) = - \frac{32 \cdot 3}{5 \cdot 3} + \frac{56 \cdot 15}{15} - \frac{568 \cdot 5}{3 \cdot 5} + \frac{308}{1} - 240$

Step 11.2.2.9

Multiply $\frac{308}{1}$ by $\frac{15}{15}$ .

$f (- 2) = - \frac{32 \cdot 3}{5 \cdot 3} + \frac{56 \cdot 15}{15} - \frac{568 \cdot 5}{3 \cdot 5} + \frac{308}{1} \cdot \frac{15}{15} - 240$

Step 11.2.2.10

Multiply $\frac{308}{1}$ by $\frac{15}{15}$ .

$f (- 2) = - \frac{32 \cdot 3}{5 \cdot 3} + \frac{56 \cdot 15}{15} - \frac{568 \cdot 5}{3 \cdot 5} + \frac{308 \cdot 15}{15} - 240$

Step 11.2.2.11

Write $- 240$ as a fraction with denominator $1$ .

$f (- 2) = - \frac{32 \cdot 3}{5 \cdot 3} + \frac{56 \cdot 15}{15} - \frac{568 \cdot 5}{3 \cdot 5} + \frac{308 \cdot 15}{15} + \frac{- 240}{1}$

Step 11.2.2.12

Multiply $\frac{- 240}{1}$ by $\frac{15}{15}$ .

$f (- 2) = - \frac{32 \cdot 3}{5 \cdot 3} + \frac{56 \cdot 15}{15} - \frac{568 \cdot 5}{3 \cdot 5} + \frac{308 \cdot 15}{15} + \frac{- 240}{1} \cdot \frac{15}{15}$

Step 11.2.2.13

Multiply $\frac{- 240}{1}$ by $\frac{15}{15}$ .

$f (- 2) = - \frac{32 \cdot 3}{5 \cdot 3} + \frac{56 \cdot 15}{15} - \frac{568 \cdot 5}{3 \cdot 5} + \frac{308 \cdot 15}{15} + \frac{- 240 \cdot 15}{15}$

Step 11.2.2.14

Reorder the factors of $5 \cdot 3$ .

$f (- 2) = - \frac{32 \cdot 3}{3 \cdot 5} + \frac{56 \cdot 15}{15} - \frac{568 \cdot 5}{3 \cdot 5} + \frac{308 \cdot 15}{15} + \frac{- 240 \cdot 15}{15}$

Step 11.2.2.15

Multiply $3$ by $5$ .

$f (- 2) = - \frac{32 \cdot 3}{15} + \frac{56 \cdot 15}{15} - \frac{568 \cdot 5}{3 \cdot 5} + \frac{308 \cdot 15}{15} + \frac{- 240 \cdot 15}{15}$

Step 11.2.2.16

Multiply $3$ by $5$ .

$f (- 2) = - \frac{32 \cdot 3}{15} + \frac{56 \cdot 15}{15} - \frac{568 \cdot 5}{15} + \frac{308 \cdot 15}{15} + \frac{- 240 \cdot 15}{15}$

Step 11.2.3

Combine the numerators over the common denominator.

$f (- 2) = \frac{- 32 \cdot 3 + 56 \cdot 15 - 568 \cdot 5 + 308 \cdot 15 - 240 \cdot 15}{15}$

Step 11.2.4

Simplify each term.

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Step 11.2.4.1

Multiply $- 32$ by $3$ .

$f (- 2) = \frac{- 96 + 56 \cdot 15 - 568 \cdot 5 + 308 \cdot 15 - 240 \cdot 15}{15}$

Step 11.2.4.2

Multiply $56$ by $15$ .

$f (- 2) = \frac{- 96 + 840 - 568 \cdot 5 + 308 \cdot 15 - 240 \cdot 15}{15}$

Step 11.2.4.3

Multiply $- 568$ by $5$ .

$f (- 2) = \frac{- 96 + 840 - 2840 + 308 \cdot 15 - 240 \cdot 15}{15}$

Step 11.2.4.4

Multiply $308$ by $15$ .

$f (- 2) = \frac{- 96 + 840 - 2840 + 4620 - 240 \cdot 15}{15}$

Step 11.2.4.5

Multiply $- 240$ by $15$ .

$f (- 2) = \frac{- 96 + 840 - 2840 + 4620 - 3600}{15}$

Step 11.2.5

Simplify the expression.

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Step 11.2.5.1

Add $- 96$ and $840$ .

$f (- 2) = \frac{744 - 2840 + 4620 - 3600}{15}$

Step 11.2.5.2

Subtract $2840$ from $744$ .

$f (- 2) = \frac{- 2096 + 4620 - 3600}{15}$

Step 11.2.5.3

Add $- 2096$ and $4620$ .

$f (- 2) = \frac{2524 - 3600}{15}$

Step 11.2.5.4

Subtract $3600$ from $2524$ .

$f (- 2) = \frac{- 1076}{15}$

Step 11.2.5.5

Move the negative in front of the fraction.

$f (- 2) = - \frac{1076}{15}$

Step 11.2.6

The final answer is $- \frac{1076}{15}$ .

$y = - \frac{1076}{15}$

Step 12

Evaluate the second derivative at $x = - 3$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$4 {(- 3)}^{3} + 42 {(- 3)}^{2} + 142 (- 3) + 154$

Step 13

Evaluate the second derivative.

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Step 13.1

Simplify each term.

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Step 13.1.1

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

$4 \cdot - 27 + 42 {(- 3)}^{2} + 142 (- 3) + 154$

Step 13.1.2

Multiply $4$ by $- 27$ .

$- 108 + 42 {(- 3)}^{2} + 142 (- 3) + 154$

Step 13.1.3

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

$- 108 + 42 \cdot 9 + 142 (- 3) + 154$

Step 13.1.4

Multiply $42$ by $9$ .

$- 108 + 378 + 142 (- 3) + 154$

Step 13.1.5

Multiply $142$ by $- 3$ .

$- 108 + 378 - 426 + 154$

Step 13.2

Simplify by adding and subtracting.

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Step 13.2.1

Add $- 108$ and $378$ .

$270 - 426 + 154$

Step 13.2.2

Subtract $426$ from $270$ .

$- 156 + 154$

Step 13.2.3

Add $- 156$ and $154$ .

$- 2$

Step 14

$x = - 3$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = - 3$ is a local maximum

Step 15

Find the y-value when $x = - 3$ .

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Step 15.1

Replace the variable $x$ with $- 3$ in the expression.

$f (- 3) = \frac{1}{5} \cdot {(- 3)}^{5} + \frac{7}{2} \cdot {(- 3)}^{4} + \frac{71}{3} \cdot {(- 3)}^{3} + 77 {(- 3)}^{2} + 120 (- 3)$

Step 15.2

Simplify the result.

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Step 15.2.1

Simplify each term.

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Step 15.2.1.1

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

$f (- 3) = \frac{1}{5} \cdot - 243 + \frac{7}{2} \cdot {(- 3)}^{4} + \frac{71}{3} \cdot {(- 3)}^{3} + 77 {(- 3)}^{2} + 120 (- 3)$

Step 15.2.1.2

Combine $\frac{1}{5}$ and $- 243$ .

$f (- 3) = \frac{- 243}{5} + \frac{7}{2} \cdot {(- 3)}^{4} + \frac{71}{3} \cdot {(- 3)}^{3} + 77 {(- 3)}^{2} + 120 (- 3)$

Step 15.2.1.3

Move the negative in front of the fraction.

$f (- 3) = - \frac{243}{5} + \frac{7}{2} \cdot {(- 3)}^{4} + \frac{71}{3} \cdot {(- 3)}^{3} + 77 {(- 3)}^{2} + 120 (- 3)$

Step 15.2.1.4

Raise $- 3$ to the power of $4$ .

$f (- 3) = - \frac{243}{5} + \frac{7}{2} \cdot 81 + \frac{71}{3} \cdot {(- 3)}^{3} + 77 {(- 3)}^{2} + 120 (- 3)$

Step 15.2.1.5

Multiply $\frac{7}{2} \cdot 81$ .

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Step 15.2.1.5.1

Combine $\frac{7}{2}$ and $81$ .

$f (- 3) = - \frac{243}{5} + \frac{7 \cdot 81}{2} + \frac{71}{3} \cdot {(- 3)}^{3} + 77 {(- 3)}^{2} + 120 (- 3)$

Step 15.2.1.5.2

Multiply $7$ by $81$ .

$f (- 3) = - \frac{243}{5} + \frac{567}{2} + \frac{71}{3} \cdot {(- 3)}^{3} + 77 {(- 3)}^{2} + 120 (- 3)$

Step 15.2.1.6

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

$f (- 3) = - \frac{243}{5} + \frac{567}{2} + \frac{71}{3} \cdot - 27 + 77 {(- 3)}^{2} + 120 (- 3)$

Step 15.2.1.7

Cancel the common factor of $3$ .

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Step 15.2.1.7.1

Factor $3$ out of $- 27$ .

$f (- 3) = - \frac{243}{5} + \frac{567}{2} + \frac{71}{3} \cdot (3 (- 9)) + 77 {(- 3)}^{2} + 120 (- 3)$

Step 15.2.1.7.2

Cancel the common factor.

$f (- 3) = - \frac{243}{5} + \frac{567}{2} + \frac{71}{3} \cdot (3 \cdot - 9) + 77 {(- 3)}^{2} + 120 (- 3)$

Step 15.2.1.7.3

Rewrite the expression.

$f (- 3) = - \frac{243}{5} + \frac{567}{2} + 71 \cdot - 9 + 77 {(- 3)}^{2} + 120 (- 3)$

Step 15.2.1.8

Multiply $71$ by $- 9$ .

$f (- 3) = - \frac{243}{5} + \frac{567}{2} - 639 + 77 {(- 3)}^{2} + 120 (- 3)$

Step 15.2.1.9

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

$f (- 3) = - \frac{243}{5} + \frac{567}{2} - 639 + 77 \cdot 9 + 120 (- 3)$

Step 15.2.1.10

Multiply $77$ by $9$ .

$f (- 3) = - \frac{243}{5} + \frac{567}{2} - 639 + 693 + 120 (- 3)$

Step 15.2.1.11

Multiply $120$ by $- 3$ .

$f (- 3) = - \frac{243}{5} + \frac{567}{2} - 639 + 693 - 360$

Step 15.2.2

Find the common denominator.

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Step 15.2.2.1

Multiply $\frac{243}{5}$ by $\frac{2}{2}$ .

$f (- 3) = - (\frac{243}{5} \cdot \frac{2}{2}) + \frac{567}{2} - 639 + 693 - 360$

Step 15.2.2.2

Multiply $\frac{243}{5}$ by $\frac{2}{2}$ .

$f (- 3) = - \frac{243 \cdot 2}{5 \cdot 2} + \frac{567}{2} - 639 + 693 - 360$

Step 15.2.2.3

Multiply $\frac{567}{2}$ by $\frac{5}{5}$ .

$f (- 3) = - \frac{243 \cdot 2}{5 \cdot 2} + \frac{567}{2} \cdot \frac{5}{5} - 639 + 693 - 360$

Step 15.2.2.4

Multiply $\frac{567}{2}$ by $\frac{5}{5}$ .

$f (- 3) = - \frac{243 \cdot 2}{5 \cdot 2} + \frac{567 \cdot 5}{2 \cdot 5} - 639 + 693 - 360$

Step 15.2.2.5

Write $- 639$ as a fraction with denominator $1$ .

$f (- 3) = - \frac{243 \cdot 2}{5 \cdot 2} + \frac{567 \cdot 5}{2 \cdot 5} + \frac{- 639}{1} + 693 - 360$

Step 15.2.2.6

Multiply $\frac{- 639}{1}$ by $\frac{10}{10}$ .

$f (- 3) = - \frac{243 \cdot 2}{5 \cdot 2} + \frac{567 \cdot 5}{2 \cdot 5} + \frac{- 639}{1} \cdot \frac{10}{10} + 693 - 360$

Step 15.2.2.7

Multiply $\frac{- 639}{1}$ by $\frac{10}{10}$ .

$f (- 3) = - \frac{243 \cdot 2}{5 \cdot 2} + \frac{567 \cdot 5}{2 \cdot 5} + \frac{- 639 \cdot 10}{10} + 693 - 360$

Step 15.2.2.8

Write $693$ as a fraction with denominator $1$ .

$f (- 3) = - \frac{243 \cdot 2}{5 \cdot 2} + \frac{567 \cdot 5}{2 \cdot 5} + \frac{- 639 \cdot 10}{10} + \frac{693}{1} - 360$

Step 15.2.2.9

Multiply $\frac{693}{1}$ by $\frac{10}{10}$ .

$f (- 3) = - \frac{243 \cdot 2}{5 \cdot 2} + \frac{567 \cdot 5}{2 \cdot 5} + \frac{- 639 \cdot 10}{10} + \frac{693}{1} \cdot \frac{10}{10} - 360$

Step 15.2.2.10

Multiply $\frac{693}{1}$ by $\frac{10}{10}$ .

$f (- 3) = - \frac{243 \cdot 2}{5 \cdot 2} + \frac{567 \cdot 5}{2 \cdot 5} + \frac{- 639 \cdot 10}{10} + \frac{693 \cdot 10}{10} - 360$

Step 15.2.2.11

Write $- 360$ as a fraction with denominator $1$ .

$f (- 3) = - \frac{243 \cdot 2}{5 \cdot 2} + \frac{567 \cdot 5}{2 \cdot 5} + \frac{- 639 \cdot 10}{10} + \frac{693 \cdot 10}{10} + \frac{- 360}{1}$

Step 15.2.2.12

Multiply $\frac{- 360}{1}$ by $\frac{10}{10}$ .

$f (- 3) = - \frac{243 \cdot 2}{5 \cdot 2} + \frac{567 \cdot 5}{2 \cdot 5} + \frac{- 639 \cdot 10}{10} + \frac{693 \cdot 10}{10} + \frac{- 360}{1} \cdot \frac{10}{10}$

Step 15.2.2.13

Multiply $\frac{- 360}{1}$ by $\frac{10}{10}$ .

$f (- 3) = - \frac{243 \cdot 2}{5 \cdot 2} + \frac{567 \cdot 5}{2 \cdot 5} + \frac{- 639 \cdot 10}{10} + \frac{693 \cdot 10}{10} + \frac{- 360 \cdot 10}{10}$

Step 15.2.2.14

Reorder the factors of $5 \cdot 2$ .

$f (- 3) = - \frac{243 \cdot 2}{2 \cdot 5} + \frac{567 \cdot 5}{2 \cdot 5} + \frac{- 639 \cdot 10}{10} + \frac{693 \cdot 10}{10} + \frac{- 360 \cdot 10}{10}$

Step 15.2.2.15

Multiply $2$ by $5$ .

$f (- 3) = - \frac{243 \cdot 2}{10} + \frac{567 \cdot 5}{2 \cdot 5} + \frac{- 639 \cdot 10}{10} + \frac{693 \cdot 10}{10} + \frac{- 360 \cdot 10}{10}$

Step 15.2.2.16

Multiply $2$ by $5$ .

$f (- 3) = - \frac{243 \cdot 2}{10} + \frac{567 \cdot 5}{10} + \frac{- 639 \cdot 10}{10} + \frac{693 \cdot 10}{10} + \frac{- 360 \cdot 10}{10}$

Step 15.2.3

Combine the numerators over the common denominator.

$f (- 3) = \frac{- 243 \cdot 2 + 567 \cdot 5 - 639 \cdot 10 + 693 \cdot 10 - 360 \cdot 10}{10}$

Step 15.2.4

Simplify each term.

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Step 15.2.4.1

Multiply $- 243$ by $2$ .

$f (- 3) = \frac{- 486 + 567 \cdot 5 - 639 \cdot 10 + 693 \cdot 10 - 360 \cdot 10}{10}$

Step 15.2.4.2

Multiply $567$ by $5$ .

$f (- 3) = \frac{- 486 + 2835 - 639 \cdot 10 + 693 \cdot 10 - 360 \cdot 10}{10}$

Step 15.2.4.3

Multiply $- 639$ by $10$ .

$f (- 3) = \frac{- 486 + 2835 - 6390 + 693 \cdot 10 - 360 \cdot 10}{10}$

Step 15.2.4.4

Multiply $693$ by $10$ .

$f (- 3) = \frac{- 486 + 2835 - 6390 + 6930 - 360 \cdot 10}{10}$

Step 15.2.4.5

Multiply $- 360$ by $10$ .

$f (- 3) = \frac{- 486 + 2835 - 6390 + 6930 - 3600}{10}$

Step 15.2.5

Simplify the expression.

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Step 15.2.5.1

Add $- 486$ and $2835$ .

$f (- 3) = \frac{2349 - 6390 + 6930 - 3600}{10}$

Step 15.2.5.2

Subtract $6390$ from $2349$ .

$f (- 3) = \frac{- 4041 + 6930 - 3600}{10}$

Step 15.2.5.3

Add $- 4041$ and $6930$ .

$f (- 3) = \frac{2889 - 3600}{10}$

Step 15.2.5.4

Subtract $3600$ from $2889$ .

$f (- 3) = \frac{- 711}{10}$

Step 15.2.5.5

Move the negative in front of the fraction.

$f (- 3) = - \frac{711}{10}$

Step 15.2.6

The final answer is $- \frac{711}{10}$ .

$y = - \frac{711}{10}$

Step 16

Evaluate the second derivative at $x = - 4$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$4 {(- 4)}^{3} + 42 {(- 4)}^{2} + 142 (- 4) + 154$

Step 17

Evaluate the second derivative.

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Step 17.1

Simplify each term.

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Step 17.1.1

Raise $- 4$ to the power of $3$ .

$4 \cdot - 64 + 42 {(- 4)}^{2} + 142 (- 4) + 154$

Step 17.1.2

Multiply $4$ by $- 64$ .

$- 256 + 42 {(- 4)}^{2} + 142 (- 4) + 154$

Step 17.1.3

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

$- 256 + 42 \cdot 16 + 142 (- 4) + 154$

Step 17.1.4

Multiply $42$ by $16$ .

$- 256 + 672 + 142 (- 4) + 154$

Step 17.1.5

Multiply $142$ by $- 4$ .

$- 256 + 672 - 568 + 154$

Step 17.2

Simplify by adding and subtracting.

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Step 17.2.1

Add $- 256$ and $672$ .

$416 - 568 + 154$

Step 17.2.2

Subtract $568$ from $416$ .

$- 152 + 154$

Step 17.2.3

Add $- 152$ and $154$ .

$2$

Step 18

$x = - 4$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = - 4$ is a local minimum

Step 19

Find the y-value when $x = - 4$ .

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Step 19.1

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

$f (- 4) = \frac{1}{5} \cdot {(- 4)}^{5} + \frac{7}{2} \cdot {(- 4)}^{4} + \frac{71}{3} \cdot {(- 4)}^{3} + 77 {(- 4)}^{2} + 120 (- 4)$

Step 19.2

Simplify the result.

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Step 19.2.1

Simplify each term.

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Step 19.2.1.1

Raise $- 4$ to the power of $5$ .

$f (- 4) = \frac{1}{5} \cdot - 1024 + \frac{7}{2} \cdot {(- 4)}^{4} + \frac{71}{3} \cdot {(- 4)}^{3} + 77 {(- 4)}^{2} + 120 (- 4)$

Step 19.2.1.2

Combine $\frac{1}{5}$ and $- 1024$ .

$f (- 4) = \frac{- 1024}{5} + \frac{7}{2} \cdot {(- 4)}^{4} + \frac{71}{3} \cdot {(- 4)}^{3} + 77 {(- 4)}^{2} + 120 (- 4)$

Step 19.2.1.3

Move the negative in front of the fraction.

$f (- 4) = - \frac{1024}{5} + \frac{7}{2} \cdot {(- 4)}^{4} + \frac{71}{3} \cdot {(- 4)}^{3} + 77 {(- 4)}^{2} + 120 (- 4)$

Step 19.2.1.4

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

$f (- 4) = - \frac{1024}{5} + \frac{7}{2} \cdot 256 + \frac{71}{3} \cdot {(- 4)}^{3} + 77 {(- 4)}^{2} + 120 (- 4)$

Step 19.2.1.5

Cancel the common factor of $2$ .

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Step 19.2.1.5.1

Factor $2$ out of $256$ .

$f (- 4) = - \frac{1024}{5} + \frac{7}{2} \cdot (2 (128)) + \frac{71}{3} \cdot {(- 4)}^{3} + 77 {(- 4)}^{2} + 120 (- 4)$

Step 19.2.1.5.2

Cancel the common factor.

$f (- 4) = - \frac{1024}{5} + \frac{7}{2} \cdot (2 \cdot 128) + \frac{71}{3} \cdot {(- 4)}^{3} + 77 {(- 4)}^{2} + 120 (- 4)$

Step 19.2.1.5.3

Rewrite the expression.

$f (- 4) = - \frac{1024}{5} + 7 \cdot 128 + \frac{71}{3} \cdot {(- 4)}^{3} + 77 {(- 4)}^{2} + 120 (- 4)$

Step 19.2.1.6

Multiply $7$ by $128$ .

$f (- 4) = - \frac{1024}{5} + 896 + \frac{71}{3} \cdot {(- 4)}^{3} + 77 {(- 4)}^{2} + 120 (- 4)$

Step 19.2.1.7

Raise $- 4$ to the power of $3$ .

$f (- 4) = - \frac{1024}{5} + 896 + \frac{71}{3} \cdot - 64 + 77 {(- 4)}^{2} + 120 (- 4)$

Step 19.2.1.8

Multiply $\frac{71}{3} \cdot - 64$ .

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Step 19.2.1.8.1

Combine $\frac{71}{3}$ and $- 64$ .

$f (- 4) = - \frac{1024}{5} + 896 + \frac{71 \cdot - 64}{3} + 77 {(- 4)}^{2} + 120 (- 4)$

Step 19.2.1.8.2

Multiply $71$ by $- 64$ .

$f (- 4) = - \frac{1024}{5} + 896 + \frac{- 4544}{3} + 77 {(- 4)}^{2} + 120 (- 4)$

Step 19.2.1.9

Move the negative in front of the fraction.

$f (- 4) = - \frac{1024}{5} + 896 - \frac{4544}{3} + 77 {(- 4)}^{2} + 120 (- 4)$

Step 19.2.1.10

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

$f (- 4) = - \frac{1024}{5} + 896 - \frac{4544}{3} + 77 \cdot 16 + 120 (- 4)$

Step 19.2.1.11

Multiply $77$ by $16$ .

$f (- 4) = - \frac{1024}{5} + 896 - \frac{4544}{3} + 1232 + 120 (- 4)$

Step 19.2.1.12

Multiply $120$ by $- 4$ .

$f (- 4) = - \frac{1024}{5} + 896 - \frac{4544}{3} + 1232 - 480$

Step 19.2.2

Find the common denominator.

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Step 19.2.2.1

Multiply $\frac{1024}{5}$ by $\frac{3}{3}$ .

$f (- 4) = - (\frac{1024}{5} \cdot \frac{3}{3}) + 896 - \frac{4544}{3} + 1232 - 480$

Step 19.2.2.2

Multiply $\frac{1024}{5}$ by $\frac{3}{3}$ .

$f (- 4) = - \frac{1024 \cdot 3}{5 \cdot 3} + 896 - \frac{4544}{3} + 1232 - 480$

Step 19.2.2.3

Write $896$ as a fraction with denominator $1$ .

$f (- 4) = - \frac{1024 \cdot 3}{5 \cdot 3} + \frac{896}{1} - \frac{4544}{3} + 1232 - 480$

Step 19.2.2.4

Multiply $\frac{896}{1}$ by $\frac{15}{15}$ .

$f (- 4) = - \frac{1024 \cdot 3}{5 \cdot 3} + \frac{896}{1} \cdot \frac{15}{15} - \frac{4544}{3} + 1232 - 480$

Step 19.2.2.5

Multiply $\frac{896}{1}$ by $\frac{15}{15}$ .

$f (- 4) = - \frac{1024 \cdot 3}{5 \cdot 3} + \frac{896 \cdot 15}{15} - \frac{4544}{3} + 1232 - 480$

Step 19.2.2.6

Multiply $\frac{4544}{3}$ by $\frac{5}{5}$ .

$f (- 4) = - \frac{1024 \cdot 3}{5 \cdot 3} + \frac{896 \cdot 15}{15} - (\frac{4544}{3} \cdot \frac{5}{5}) + 1232 - 480$

Step 19.2.2.7

Multiply $\frac{4544}{3}$ by $\frac{5}{5}$ .

$f (- 4) = - \frac{1024 \cdot 3}{5 \cdot 3} + \frac{896 \cdot 15}{15} - \frac{4544 \cdot 5}{3 \cdot 5} + 1232 - 480$

Step 19.2.2.8

Write $1232$ as a fraction with denominator $1$ .

$f (- 4) = - \frac{1024 \cdot 3}{5 \cdot 3} + \frac{896 \cdot 15}{15} - \frac{4544 \cdot 5}{3 \cdot 5} + \frac{1232}{1} - 480$

Step 19.2.2.9

Multiply $\frac{1232}{1}$ by $\frac{15}{15}$ .

$f (- 4) = - \frac{1024 \cdot 3}{5 \cdot 3} + \frac{896 \cdot 15}{15} - \frac{4544 \cdot 5}{3 \cdot 5} + \frac{1232}{1} \cdot \frac{15}{15} - 480$

Step 19.2.2.10

Multiply $\frac{1232}{1}$ by $\frac{15}{15}$ .

$f (- 4) = - \frac{1024 \cdot 3}{5 \cdot 3} + \frac{896 \cdot 15}{15} - \frac{4544 \cdot 5}{3 \cdot 5} + \frac{1232 \cdot 15}{15} - 480$

Step 19.2.2.11

Write $- 480$ as a fraction with denominator $1$ .

$f (- 4) = - \frac{1024 \cdot 3}{5 \cdot 3} + \frac{896 \cdot 15}{15} - \frac{4544 \cdot 5}{3 \cdot 5} + \frac{1232 \cdot 15}{15} + \frac{- 480}{1}$

Step 19.2.2.12

Multiply $\frac{- 480}{1}$ by $\frac{15}{15}$ .

$f (- 4) = - \frac{1024 \cdot 3}{5 \cdot 3} + \frac{896 \cdot 15}{15} - \frac{4544 \cdot 5}{3 \cdot 5} + \frac{1232 \cdot 15}{15} + \frac{- 480}{1} \cdot \frac{15}{15}$

Step 19.2.2.13

Multiply $\frac{- 480}{1}$ by $\frac{15}{15}$ .

$f (- 4) = - \frac{1024 \cdot 3}{5 \cdot 3} + \frac{896 \cdot 15}{15} - \frac{4544 \cdot 5}{3 \cdot 5} + \frac{1232 \cdot 15}{15} + \frac{- 480 \cdot 15}{15}$

Step 19.2.2.14

Reorder the factors of $5 \cdot 3$ .

$f (- 4) = - \frac{1024 \cdot 3}{3 \cdot 5} + \frac{896 \cdot 15}{15} - \frac{4544 \cdot 5}{3 \cdot 5} + \frac{1232 \cdot 15}{15} + \frac{- 480 \cdot 15}{15}$

Step 19.2.2.15

Multiply $3$ by $5$ .

$f (- 4) = - \frac{1024 \cdot 3}{15} + \frac{896 \cdot 15}{15} - \frac{4544 \cdot 5}{3 \cdot 5} + \frac{1232 \cdot 15}{15} + \frac{- 480 \cdot 15}{15}$

Step 19.2.2.16

Multiply $3$ by $5$ .

$f (- 4) = - \frac{1024 \cdot 3}{15} + \frac{896 \cdot 15}{15} - \frac{4544 \cdot 5}{15} + \frac{1232 \cdot 15}{15} + \frac{- 480 \cdot 15}{15}$

Step 19.2.3

Combine the numerators over the common denominator.

$f (- 4) = \frac{- 1024 \cdot 3 + 896 \cdot 15 - 4544 \cdot 5 + 1232 \cdot 15 - 480 \cdot 15}{15}$

Step 19.2.4

Simplify each term.

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Step 19.2.4.1

Multiply $- 1024$ by $3$ .

$f (- 4) = \frac{- 3072 + 896 \cdot 15 - 4544 \cdot 5 + 1232 \cdot 15 - 480 \cdot 15}{15}$

Step 19.2.4.2

Multiply $896$ by $15$ .

$f (- 4) = \frac{- 3072 + 13440 - 4544 \cdot 5 + 1232 \cdot 15 - 480 \cdot 15}{15}$

Step 19.2.4.3

Multiply $- 4544$ by $5$ .

$f (- 4) = \frac{- 3072 + 13440 - 22720 + 1232 \cdot 15 - 480 \cdot 15}{15}$

Step 19.2.4.4

Multiply $1232$ by $15$ .

$f (- 4) = \frac{- 3072 + 13440 - 22720 + 18480 - 480 \cdot 15}{15}$

Step 19.2.4.5

Multiply $- 480$ by $15$ .

$f (- 4) = \frac{- 3072 + 13440 - 22720 + 18480 - 7200}{15}$

Step 19.2.5

Simplify the expression.

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Step 19.2.5.1

Add $- 3072$ and $13440$ .

$f (- 4) = \frac{10368 - 22720 + 18480 - 7200}{15}$

Step 19.2.5.2

Subtract $22720$ from $10368$ .

$f (- 4) = \frac{- 12352 + 18480 - 7200}{15}$

Step 19.2.5.3

Add $- 12352$ and $18480$ .

$f (- 4) = \frac{6128 - 7200}{15}$

Step 19.2.5.4

Subtract $7200$ from $6128$ .

$f (- 4) = \frac{- 1072}{15}$

Step 19.2.5.5

Move the negative in front of the fraction.

$f (- 4) = - \frac{1072}{15}$

Step 19.2.6

The final answer is $- \frac{1072}{15}$ .

$y = - \frac{1072}{15}$

Step 20

Evaluate the second derivative at $x = - 5$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$4 {(- 5)}^{3} + 42 {(- 5)}^{2} + 142 (- 5) + 154$

Step 21

Evaluate the second derivative.

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Step 21.1

Simplify each term.

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Step 21.1.1

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

$4 \cdot - 125 + 42 {(- 5)}^{2} + 142 (- 5) + 154$

Step 21.1.2

Multiply $4$ by $- 125$ .

$- 500 + 42 {(- 5)}^{2} + 142 (- 5) + 154$

Step 21.1.3

Raise $- 5$ to the power of $2$ .

$- 500 + 42 \cdot 25 + 142 (- 5) + 154$

Step 21.1.4

Multiply $42$ by $25$ .

$- 500 + 1050 + 142 (- 5) + 154$

Step 21.1.5

Multiply $142$ by $- 5$ .

$- 500 + 1050 - 710 + 154$

Step 21.2

Simplify by adding and subtracting.

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Step 21.2.1

Add $- 500$ and $1050$ .

$550 - 710 + 154$

Step 21.2.2

Subtract $710$ from $550$ .

$- 160 + 154$

Step 21.2.3

Add $- 160$ and $154$ .

$- 6$

Step 22

$x = - 5$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = - 5$ is a local maximum

Step 23

Find the y-value when $x = - 5$ .

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Step 23.1

Replace the variable $x$ with $- 5$ in the expression.

$f (- 5) = \frac{1}{5} \cdot {(- 5)}^{5} + \frac{7}{2} \cdot {(- 5)}^{4} + \frac{71}{3} \cdot {(- 5)}^{3} + 77 {(- 5)}^{2} + 120 (- 5)$

Step 23.2

Simplify the result.

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Step 23.2.1

Simplify each term.

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Step 23.2.1.1

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

$f (- 5) = \frac{1}{5} \cdot - 3125 + \frac{7}{2} \cdot {(- 5)}^{4} + \frac{71}{3} \cdot {(- 5)}^{3} + 77 {(- 5)}^{2} + 120 (- 5)$

Step 23.2.1.2

Cancel the common factor of $5$ .

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Step 23.2.1.2.1

Factor $5$ out of $- 3125$ .

$f (- 5) = \frac{1}{5} \cdot (5 (- 625)) + \frac{7}{2} \cdot {(- 5)}^{4} + \frac{71}{3} \cdot {(- 5)}^{3} + 77 {(- 5)}^{2} + 120 (- 5)$

Step 23.2.1.2.2

Cancel the common factor.

$f (- 5) = \frac{1}{5} \cdot (5 \cdot - 625) + \frac{7}{2} \cdot {(- 5)}^{4} + \frac{71}{3} \cdot {(- 5)}^{3} + 77 {(- 5)}^{2} + 120 (- 5)$

Step 23.2.1.2.3

Rewrite the expression.

$f (- 5) = - 625 + \frac{7}{2} \cdot {(- 5)}^{4} + \frac{71}{3} \cdot {(- 5)}^{3} + 77 {(- 5)}^{2} + 120 (- 5)$

Step 23.2.1.3

Raise $- 5$ to the power of $4$ .

$f (- 5) = - 625 + \frac{7}{2} \cdot 625 + \frac{71}{3} \cdot {(- 5)}^{3} + 77 {(- 5)}^{2} + 120 (- 5)$

Step 23.2.1.4

Multiply $\frac{7}{2} \cdot 625$ .

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Step 23.2.1.4.1

Combine $\frac{7}{2}$ and $625$ .

$f (- 5) = - 625 + \frac{7 \cdot 625}{2} + \frac{71}{3} \cdot {(- 5)}^{3} + 77 {(- 5)}^{2} + 120 (- 5)$

Step 23.2.1.4.2

Multiply $7$ by $625$ .

$f (- 5) = - 625 + \frac{4375}{2} + \frac{71}{3} \cdot {(- 5)}^{3} + 77 {(- 5)}^{2} + 120 (- 5)$

Step 23.2.1.5

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

$f (- 5) = - 625 + \frac{4375}{2} + \frac{71}{3} \cdot - 125 + 77 {(- 5)}^{2} + 120 (- 5)$

Step 23.2.1.6

Multiply $\frac{71}{3} \cdot - 125$ .

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Step 23.2.1.6.1

Combine $\frac{71}{3}$ and $- 125$ .

$f (- 5) = - 625 + \frac{4375}{2} + \frac{71 \cdot - 125}{3} + 77 {(- 5)}^{2} + 120 (- 5)$

Step 23.2.1.6.2

Multiply $71$ by $- 125$ .

$f (- 5) = - 625 + \frac{4375}{2} + \frac{- 8875}{3} + 77 {(- 5)}^{2} + 120 (- 5)$

Step 23.2.1.7

Move the negative in front of the fraction.

$f (- 5) = - 625 + \frac{4375}{2} - \frac{8875}{3} + 77 {(- 5)}^{2} + 120 (- 5)$

Step 23.2.1.8

Raise $- 5$ to the power of $2$ .

$f (- 5) = - 625 + \frac{4375}{2} - \frac{8875}{3} + 77 \cdot 25 + 120 (- 5)$

Step 23.2.1.9

Multiply $77$ by $25$ .

$f (- 5) = - 625 + \frac{4375}{2} - \frac{8875}{3} + 1925 + 120 (- 5)$

Step 23.2.1.10

Multiply $120$ by $- 5$ .

$f (- 5) = - 625 + \frac{4375}{2} - \frac{8875}{3} + 1925 - 600$

Step 23.2.2

Find the common denominator.

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Step 23.2.2.1

Write $- 625$ as a fraction with denominator $1$ .

$f (- 5) = \frac{- 625}{1} + \frac{4375}{2} - \frac{8875}{3} + 1925 - 600$

Step 23.2.2.2

Multiply $\frac{- 625}{1}$ by $\frac{6}{6}$ .

$f (- 5) = \frac{- 625}{1} \cdot \frac{6}{6} + \frac{4375}{2} - \frac{8875}{3} + 1925 - 600$

Step 23.2.2.3

Multiply $\frac{- 625}{1}$ by $\frac{6}{6}$ .

$f (- 5) = \frac{- 625 \cdot 6}{6} + \frac{4375}{2} - \frac{8875}{3} + 1925 - 600$

Step 23.2.2.4

Multiply $\frac{4375}{2}$ by $\frac{3}{3}$ .

$f (- 5) = \frac{- 625 \cdot 6}{6} + \frac{4375}{2} \cdot \frac{3}{3} - \frac{8875}{3} + 1925 - 600$

Step 23.2.2.5

Multiply $\frac{4375}{2}$ by $\frac{3}{3}$ .

$f (- 5) = \frac{- 625 \cdot 6}{6} + \frac{4375 \cdot 3}{2 \cdot 3} - \frac{8875}{3} + 1925 - 600$

Step 23.2.2.6

Multiply $\frac{8875}{3}$ by $\frac{2}{2}$ .

$f (- 5) = \frac{- 625 \cdot 6}{6} + \frac{4375 \cdot 3}{2 \cdot 3} - (\frac{8875}{3} \cdot \frac{2}{2}) + 1925 - 600$

Step 23.2.2.7

Multiply $\frac{8875}{3}$ by $\frac{2}{2}$ .

$f (- 5) = \frac{- 625 \cdot 6}{6} + \frac{4375 \cdot 3}{2 \cdot 3} - \frac{8875 \cdot 2}{3 \cdot 2} + 1925 - 600$

Step 23.2.2.8

Write $1925$ as a fraction with denominator $1$ .

$f (- 5) = \frac{- 625 \cdot 6}{6} + \frac{4375 \cdot 3}{2 \cdot 3} - \frac{8875 \cdot 2}{3 \cdot 2} + \frac{1925}{1} - 600$

Step 23.2.2.9

Multiply $\frac{1925}{1}$ by $\frac{6}{6}$ .

$f (- 5) = \frac{- 625 \cdot 6}{6} + \frac{4375 \cdot 3}{2 \cdot 3} - \frac{8875 \cdot 2}{3 \cdot 2} + \frac{1925}{1} \cdot \frac{6}{6} - 600$

Step 23.2.2.10

Multiply $\frac{1925}{1}$ by $\frac{6}{6}$ .

$f (- 5) = \frac{- 625 \cdot 6}{6} + \frac{4375 \cdot 3}{2 \cdot 3} - \frac{8875 \cdot 2}{3 \cdot 2} + \frac{1925 \cdot 6}{6} - 600$

Step 23.2.2.11

Write $- 600$ as a fraction with denominator $1$ .

$f (- 5) = \frac{- 625 \cdot 6}{6} + \frac{4375 \cdot 3}{2 \cdot 3} - \frac{8875 \cdot 2}{3 \cdot 2} + \frac{1925 \cdot 6}{6} + \frac{- 600}{1}$

Step 23.2.2.12

Multiply $\frac{- 600}{1}$ by $\frac{6}{6}$ .

$f (- 5) = \frac{- 625 \cdot 6}{6} + \frac{4375 \cdot 3}{2 \cdot 3} - \frac{8875 \cdot 2}{3 \cdot 2} + \frac{1925 \cdot 6}{6} + \frac{- 600}{1} \cdot \frac{6}{6}$

Step 23.2.2.13

Multiply $\frac{- 600}{1}$ by $\frac{6}{6}$ .

$f (- 5) = \frac{- 625 \cdot 6}{6} + \frac{4375 \cdot 3}{2 \cdot 3} - \frac{8875 \cdot 2}{3 \cdot 2} + \frac{1925 \cdot 6}{6} + \frac{- 600 \cdot 6}{6}$

Step 23.2.2.14

Multiply $2$ by $3$ .

$f (- 5) = \frac{- 625 \cdot 6}{6} + \frac{4375 \cdot 3}{6} - \frac{8875 \cdot 2}{3 \cdot 2} + \frac{1925 \cdot 6}{6} + \frac{- 600 \cdot 6}{6}$

Step 23.2.2.15

Reorder the factors of $3 \cdot 2$ .

$f (- 5) = \frac{- 625 \cdot 6}{6} + \frac{4375 \cdot 3}{6} - \frac{8875 \cdot 2}{2 \cdot 3} + \frac{1925 \cdot 6}{6} + \frac{- 600 \cdot 6}{6}$

Step 23.2.2.16

Multiply $2$ by $3$ .

$f (- 5) = \frac{- 625 \cdot 6}{6} + \frac{4375 \cdot 3}{6} - \frac{8875 \cdot 2}{6} + \frac{1925 \cdot 6}{6} + \frac{- 600 \cdot 6}{6}$

Step 23.2.3

Combine the numerators over the common denominator.

$f (- 5) = \frac{- 625 \cdot 6 + 4375 \cdot 3 - 8875 \cdot 2 + 1925 \cdot 6 - 600 \cdot 6}{6}$

Step 23.2.4

Simplify each term.

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Step 23.2.4.1

Multiply $- 625$ by $6$ .

$f (- 5) = \frac{- 3750 + 4375 \cdot 3 - 8875 \cdot 2 + 1925 \cdot 6 - 600 \cdot 6}{6}$

Step 23.2.4.2

Multiply $4375$ by $3$ .

$f (- 5) = \frac{- 3750 + 13125 - 8875 \cdot 2 + 1925 \cdot 6 - 600 \cdot 6}{6}$

Step 23.2.4.3

Multiply $- 8875$ by $2$ .

$f (- 5) = \frac{- 3750 + 13125 - 17750 + 1925 \cdot 6 - 600 \cdot 6}{6}$

Step 23.2.4.4

Multiply $1925$ by $6$ .

$f (- 5) = \frac{- 3750 + 13125 - 17750 + 11550 - 600 \cdot 6}{6}$

Step 23.2.4.5

Multiply $- 600$ by $6$ .

$f (- 5) = \frac{- 3750 + 13125 - 17750 + 11550 - 3600}{6}$

Step 23.2.5

Simplify the expression.

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Step 23.2.5.1

Add $- 3750$ and $13125$ .

$f (- 5) = \frac{9375 - 17750 + 11550 - 3600}{6}$

Step 23.2.5.2

Subtract $17750$ from $9375$ .

$f (- 5) = \frac{- 8375 + 11550 - 3600}{6}$

Step 23.2.5.3

Add $- 8375$ and $11550$ .

$f (- 5) = \frac{3175 - 3600}{6}$

Step 23.2.5.4

Subtract $3600$ from $3175$ .

$f (- 5) = \frac{- 425}{6}$

Step 23.2.5.5

Move the negative in front of the fraction.

$f (- 5) = - \frac{425}{6}$

Step 23.2.6

The final answer is $- \frac{425}{6}$ .

$y = - \frac{425}{6}$

Step 24

These are the local extrema for $f (x) = \frac{1}{5} x^{5} + \frac{7}{2} x^{4} + \frac{71}{3} x^{3} + 77 x^{2} + 120 x$ .

$(- 2, - \frac{1076}{15})$ is a local minima

$(- 3, - \frac{711}{10})$ is a local maxima

$(- 4, - \frac{1072}{15})$ is a local minima

$(- 5, - \frac{425}{6})$ is a local maxima

Step 25