Calculus Examples

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

Step 1

Write $\frac{1}{5} x^{5} + \frac{7}{2} x^{4} + \frac{71}{3} x^{3} + 77 x^{2} + 120 x$ as a function.

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

Step 2

Find the second derivative.

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Step 2.1

Find the first derivative.

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Step 2.1.1

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 2.1.2

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

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Step 2.1.2.1

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 2.1.2.2

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 2.1.2.3

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 2.1.2.4

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 2.1.2.5

Cancel the common factor of $5$ .

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Step 2.1.2.5.1

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 2.1.2.5.2

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 2.1.3

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

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Step 2.1.3.1

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 2.1.3.2

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 2.1.3.3

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 2.1.3.4

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 2.1.3.5

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 2.1.3.6

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

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Step 2.1.3.6.1

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 2.1.3.6.2

Cancel the common factors.

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Step 2.1.3.6.2.1

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 2.1.3.6.2.2

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 2.1.3.6.2.3

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 2.1.3.6.2.4

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 2.1.4

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

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Step 2.1.4.1

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 2.1.4.2

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 2.1.4.3

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 2.1.4.4

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 2.1.4.5

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 2.1.4.6

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

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Step 2.1.4.6.1

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 2.1.4.6.2

Cancel the common factors.

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Step 2.1.4.6.2.1

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 2.1.4.6.2.2

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 2.1.4.6.2.3

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 2.1.4.6.2.4

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 2.1.5

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

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Step 2.1.5.1

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 2.1.5.2

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 2.1.5.3

Multiply $2$ by $77$ .

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

Step 2.1.6

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

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Step 2.1.6.1

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 2.1.6.2

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 2.1.6.3

Multiply $120$ by $1$ .

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

Step 2.2

Find the second derivative.

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Step 2.2.1

Differentiate.

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Step 2.2.1.1

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

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

Step 2.2.1.2

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

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

Step 2.2.2

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

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Step 2.2.2.1

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

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

Step 2.2.2.2

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

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

Step 2.2.2.3

Multiply $3$ by $14$ .

$4 x^{3} + 42 x^{2} + \frac{d}{d x} [71 x^{2}] + \frac{d}{d x} [154 x] + \frac{d}{d x} [120]$

Step 2.2.3

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

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Step 2.2.3.1

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

$4 x^{3} + 42 x^{2} + 71 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [154 x] + \frac{d}{d x} [120]$

Step 2.2.3.2

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

$4 x^{3} + 42 x^{2} + 71 (2 x) + \frac{d}{d x} [154 x] + \frac{d}{d x} [120]$

Step 2.2.3.3

Multiply $2$ by $71$ .

$4 x^{3} + 42 x^{2} + 142 x + \frac{d}{d x} [154 x] + \frac{d}{d x} [120]$

Step 2.2.4

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

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Step 2.2.4.1

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

$4 x^{3} + 42 x^{2} + 142 x + 154 \frac{d}{d x} [x] + \frac{d}{d x} [120]$

Step 2.2.4.2

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

$4 x^{3} + 42 x^{2} + 142 x + 154 \cdot 1 + \frac{d}{d x} [120]$

Step 2.2.4.3

Multiply $154$ by $1$ .

$4 x^{3} + 42 x^{2} + 142 x + 154 + \frac{d}{d x} [120]$

Step 2.2.5

Differentiate using the Constant Rule.

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Step 2.2.5.1

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

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

Step 2.2.5.2

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

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

Step 2.3

The second derivative of $f (x)$ with respect to $x$ is $4 x^{3} + 42 x^{2} + 142 x + 154$ .

$4 x^{3} + 42 x^{2} + 142 x + 154$

Step 3

Set the second derivative equal to $0$ then solve the equation $4 x^{3} + 42 x^{2} + 142 x + 154 = 0$ .

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Step 3.1

Set the second derivative equal to $0$ .

$4 x^{3} + 42 x^{2} + 142 x + 154 = 0$

Step 3.2

Factor the left side of the equation.

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Step 3.2.1

Factor $2$ out of $4 x^{3} + 42 x^{2} + 142 x + 154$ .

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Step 3.2.1.1

Factor $2$ out of $4 x^{3}$ .

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

Step 3.2.1.2

Factor $2$ out of $42 x^{2}$ .

$2 (2 x^{3}) + 2 (21 x^{2}) + 142 x + 154 = 0$

Step 3.2.1.3

Factor $2$ out of $142 x$ .

$2 (2 x^{3}) + 2 (21 x^{2}) + 2 (71 x) + 154 = 0$

Step 3.2.1.4

Factor $2$ out of $154$ .

$2 (2 x^{3}) + 2 (21 x^{2}) + 2 (71 x) + 2 (77) = 0$

Step 3.2.1.5

Factor $2$ out of $2 (2 x^{3}) + 2 (21 x^{2})$ .

$2 (2 x^{3} + 21 x^{2}) + 2 (71 x) + 2 (77) = 0$

Step 3.2.1.6

Factor $2$ out of $2 (2 x^{3} + 21 x^{2}) + 2 (71 x)$ .

$2 (2 x^{3} + 21 x^{2} + 71 x) + 2 (77) = 0$

Step 3.2.1.7

Factor $2$ out of $2 (2 x^{3} + 21 x^{2} + 71 x) + 2 (77)$ .

$2 (2 x^{3} + 21 x^{2} + 71 x + 77) = 0$

Step 3.2.2

Factor.

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Step 3.2.2.1

Factor $2 x^{3} + 21 x^{2} + 71 x + 77$ using the rational roots test.

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Step 3.2.2.1.1

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.

$p = \pm 1, \pm 77, \pm 7, \pm 11$

$q = \pm 1, \pm 2$

Step 3.2.2.1.2

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

$\pm 1, \pm 0.5, \pm 77, \pm 38.5, \pm 7, \pm 3.5, \pm 11, \pm 5.5$

Step 3.2.2.1.3

Substitute $- 3.5$ and simplify the expression. In this case, the expression is equal to $0$ so $- 3.5$ is a root of the polynomial.

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Step 3.2.2.1.3.1

Substitute $- 3.5$ into the polynomial.

$2 {(- 3.5)}^{3} + 21 {(- 3.5)}^{2} + 71 \cdot - 3.5 + 77$

Step 3.2.2.1.3.2

Raise $- 3.5$ to the power of $3$ .

$2 \cdot - 42.875 + 21 {(- 3.5)}^{2} + 71 \cdot - 3.5 + 77$

Step 3.2.2.1.3.3

Multiply $2$ by $- 42.875$ .

$- 85.75 + 21 {(- 3.5)}^{2} + 71 \cdot - 3.5 + 77$

Step 3.2.2.1.3.4

Raise $- 3.5$ to the power of $2$ .

$- 85.75 + 21 \cdot 12.25 + 71 \cdot - 3.5 + 77$

Step 3.2.2.1.3.5

Multiply $21$ by $12.25$ .

$- 85.75 + 257.25 + 71 \cdot - 3.5 + 77$

Step 3.2.2.1.3.6

Add $- 85.75$ and $257.25$ .

$171.5 + 71 \cdot - 3.5 + 77$

Step 3.2.2.1.3.7

Multiply $71$ by $- 3.5$ .

$171.5 - 248.5 + 77$

Step 3.2.2.1.3.8

Subtract $248.5$ from $171.5$ .

$- 77 + 77$

Step 3.2.2.1.3.9

Add $- 77$ and $77$ .

$0$

Step 3.2.2.1.4

Since $- 3.5$ is a known root, divide the polynomial by $2 x + 7$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{2 x^{3} + 21 x^{2} + 71 x + 77}{2 x + 7}$

Step 3.2.2.1.5

Divide $2 x^{3} + 21 x^{2} + 71 x + 77$ by $2 x + 7$ .

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Step 3.2.2.1.5.1

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

$2 x$

$7$

$2 x^{3}$

$21 x^{2}$

$71 x$

$77$

Step 3.2.2.1.5.2

Divide the highest order term in the dividend $2 x^{3}$ by the highest order term in divisor $2 x$ .

					$x^{2}$
	$2 x$	+	$7$		$2 x^{3}$	+	$21 x^{2}$	+	$71 x$	+	$77$

Step 3.2.2.1.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$2 x$	+	$7$		$2 x^{3}$	+	$21 x^{2}$	+	$71 x$	+	$77$
			+	$2 x^{3}$	+	$7 x^{2}$

Step 3.2.2.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $2 x^{3} + 7 x^{2}$

				$x^{2}$
$2 x$	+	$7$		$2 x^{3}$	+	$21 x^{2}$	+	$71 x$	+	$77$
			-	$2 x^{3}$	-	$7 x^{2}$

Step 3.2.2.1.5.5

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

				$x^{2}$
$2 x$	+	$7$		$2 x^{3}$	+	$21 x^{2}$	+	$71 x$	+	$77$
			-	$2 x^{3}$	-	$7 x^{2}$
					+	$14 x^{2}$

Step 3.2.2.1.5.6

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

				$x^{2}$
$2 x$	+	$7$		$2 x^{3}$	+	$21 x^{2}$	+	$71 x$	+	$77$
			-	$2 x^{3}$	-	$7 x^{2}$
					+	$14 x^{2}$	+	$71 x$

Step 3.2.2.1.5.7

Divide the highest order term in the dividend $14 x^{2}$ by the highest order term in divisor $2 x$ .

				$x^{2}$	+	$7 x$
$2 x$	+	$7$		$2 x^{3}$	+	$21 x^{2}$	+	$71 x$	+	$77$
			-	$2 x^{3}$	-	$7 x^{2}$
					+	$14 x^{2}$	+	$71 x$

Step 3.2.2.1.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$7 x$
$2 x$	+	$7$		$2 x^{3}$	+	$21 x^{2}$	+	$71 x$	+	$77$
			-	$2 x^{3}$	-	$7 x^{2}$
					+	$14 x^{2}$	+	$71 x$
					+	$14 x^{2}$	+	$49 x$

Step 3.2.2.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $14 x^{2} + 49 x$

				$x^{2}$	+	$7 x$
$2 x$	+	$7$		$2 x^{3}$	+	$21 x^{2}$	+	$71 x$	+	$77$
			-	$2 x^{3}$	-	$7 x^{2}$
					+	$14 x^{2}$	+	$71 x$
					-	$14 x^{2}$	-	$49 x$

Step 3.2.2.1.5.10

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

				$x^{2}$	+	$7 x$
$2 x$	+	$7$		$2 x^{3}$	+	$21 x^{2}$	+	$71 x$	+	$77$
			-	$2 x^{3}$	-	$7 x^{2}$
					+	$14 x^{2}$	+	$71 x$
					-	$14 x^{2}$	-	$49 x$
							+	$22 x$

Step 3.2.2.1.5.11

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

				$x^{2}$	+	$7 x$
$2 x$	+	$7$		$2 x^{3}$	+	$21 x^{2}$	+	$71 x$	+	$77$
			-	$2 x^{3}$	-	$7 x^{2}$
					+	$14 x^{2}$	+	$71 x$
					-	$14 x^{2}$	-	$49 x$
							+	$22 x$	+	$77$

Step 3.2.2.1.5.12

Divide the highest order term in the dividend $22 x$ by the highest order term in divisor $2 x$ .

				$x^{2}$	+	$7 x$	+	$11$
$2 x$	+	$7$		$2 x^{3}$	+	$21 x^{2}$	+	$71 x$	+	$77$
			-	$2 x^{3}$	-	$7 x^{2}$
					+	$14 x^{2}$	+	$71 x$
					-	$14 x^{2}$	-	$49 x$
							+	$22 x$	+	$77$

Step 3.2.2.1.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$7 x$	+	$11$
$2 x$	+	$7$		$2 x^{3}$	+	$21 x^{2}$	+	$71 x$	+	$77$
			-	$2 x^{3}$	-	$7 x^{2}$
					+	$14 x^{2}$	+	$71 x$
					-	$14 x^{2}$	-	$49 x$
							+	$22 x$	+	$77$
							+	$22 x$	+	$77$

Step 3.2.2.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $22 x + 77$

				$x^{2}$	+	$7 x$	+	$11$
$2 x$	+	$7$		$2 x^{3}$	+	$21 x^{2}$	+	$71 x$	+	$77$
			-	$2 x^{3}$	-	$7 x^{2}$
					+	$14 x^{2}$	+	$71 x$
					-	$14 x^{2}$	-	$49 x$
							+	$22 x$	+	$77$
							-	$22 x$	-	$77$

Step 3.2.2.1.5.15

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

				$x^{2}$	+	$7 x$	+	$11$
$2 x$	+	$7$		$2 x^{3}$	+	$21 x^{2}$	+	$71 x$	+	$77$
			-	$2 x^{3}$	-	$7 x^{2}$
					+	$14 x^{2}$	+	$71 x$
					-	$14 x^{2}$	-	$49 x$
							+	$22 x$	+	$77$
							-	$22 x$	-	$77$
										$0$

Step 3.2.2.1.5.16

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

$x^{2} + 7 x + 11$

Step 3.2.2.1.6

Write $2 x^{3} + 21 x^{2} + 71 x + 77$ as a set of factors.

$2 ((2 x + 7) (x^{2} + 7 x + 11)) = 0$

Step 3.2.2.2

Remove unnecessary parentheses.

$2 (2 x + 7) (x^{2} + 7 x + 11) = 0$

Step 3.3

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

$2 x + 7 = 0$

$x^{2} + 7 x + 11 = 0$

Step 3.4

Set $2 x + 7$ equal to $0$ and solve for $x$ .

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Step 3.4.1

Set $2 x + 7$ equal to $0$ .

$2 x + 7 = 0$

Step 3.4.2

Solve $2 x + 7 = 0$ for $x$ .

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Step 3.4.2.1

Subtract $7$ from both sides of the equation.

$2 x = - 7$

Step 3.4.2.2

Divide each term in $2 x = - 7$ by $2$ and simplify.

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Step 3.4.2.2.1

Divide each term in $2 x = - 7$ by $2$ .

$\frac{2 x}{2} = \frac{- 7}{2}$

Step 3.4.2.2.2

Simplify the left side.

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Step 3.4.2.2.2.1

Cancel the common factor of $2$ .

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Step 3.4.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 7}{2}$

Step 3.4.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 7}{2}$

Step 3.4.2.2.3

Simplify the right side.

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Step 3.4.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{7}{2}$

Step 3.5

Set $x^{2} + 7 x + 11$ equal to $0$ and solve for $x$ .

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Step 3.5.1

Set $x^{2} + 7 x + 11$ equal to $0$ .

$x^{2} + 7 x + 11 = 0$

Step 3.5.2

Solve $x^{2} + 7 x + 11 = 0$ for $x$ .

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Step 3.5.2.1

Use the quadratic formula to find the solutions.

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

Step 3.5.2.2

Substitute the values $a = 1$ , $b = 7$ , and $c = 11$ into the quadratic formula and solve for $x$ .

$\frac{- 7 \pm \sqrt{7^{2} - 4 \cdot (1 \cdot 11)}}{2 \cdot 1}$

Step 3.5.2.3

Simplify.

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Step 3.5.2.3.1

Simplify the numerator.

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Step 3.5.2.3.1.1

Raise $7$ to the power of $2$ .

$x = \frac{- 7 \pm \sqrt{49 - 4 \cdot 1 \cdot 11}}{2 \cdot 1}$

Step 3.5.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 11$ .

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Step 3.5.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 7 \pm \sqrt{49 - 4 \cdot 11}}{2 \cdot 1}$

Step 3.5.2.3.1.2.2

Multiply $- 4$ by $11$ .

$x = \frac{- 7 \pm \sqrt{49 - 44}}{2 \cdot 1}$

Step 3.5.2.3.1.3

Subtract $44$ from $49$ .

$x = \frac{- 7 \pm \sqrt{5}}{2 \cdot 1}$

Step 3.5.2.3.2

Multiply $2$ by $1$ .

$x = \frac{- 7 \pm \sqrt{5}}{2}$

Step 3.5.2.4

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

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Step 3.5.2.4.1

Simplify the numerator.

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Step 3.5.2.4.1.1

Raise $7$ to the power of $2$ .

$x = \frac{- 7 \pm \sqrt{49 - 4 \cdot 1 \cdot 11}}{2 \cdot 1}$

Step 3.5.2.4.1.2

Multiply $- 4 \cdot 1 \cdot 11$ .

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Step 3.5.2.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 7 \pm \sqrt{49 - 4 \cdot 11}}{2 \cdot 1}$

Step 3.5.2.4.1.2.2

Multiply $- 4$ by $11$ .

$x = \frac{- 7 \pm \sqrt{49 - 44}}{2 \cdot 1}$

Step 3.5.2.4.1.3

Subtract $44$ from $49$ .

$x = \frac{- 7 \pm \sqrt{5}}{2 \cdot 1}$

Step 3.5.2.4.2

Multiply $2$ by $1$ .

$x = \frac{- 7 \pm \sqrt{5}}{2}$

Step 3.5.2.4.3

Change the $\pm$ to $+$ .

$x = \frac{- 7 + \sqrt{5}}{2}$

Step 3.5.2.4.4

Rewrite $- 7$ as $- 1 (7)$ .

$x = \frac{- 1 \cdot 7 + \sqrt{5}}{2}$

Step 3.5.2.4.5

Factor $- 1$ out of $\sqrt{5}$ .

$x = \frac{- 1 \cdot 7 - 1 (- \sqrt{5})}{2}$

Step 3.5.2.4.6

Factor $- 1$ out of $- 1 (7) - 1 (- \sqrt{5})$ .

$x = \frac{- 1 (7 - \sqrt{5})}{2}$

Step 3.5.2.4.7

Move the negative in front of the fraction.

$x = - \frac{7 - \sqrt{5}}{2}$

Step 3.5.2.5

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

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Step 3.5.2.5.1

Simplify the numerator.

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Step 3.5.2.5.1.1

Raise $7$ to the power of $2$ .

$x = \frac{- 7 \pm \sqrt{49 - 4 \cdot 1 \cdot 11}}{2 \cdot 1}$

Step 3.5.2.5.1.2

Multiply $- 4 \cdot 1 \cdot 11$ .

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Step 3.5.2.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 7 \pm \sqrt{49 - 4 \cdot 11}}{2 \cdot 1}$

Step 3.5.2.5.1.2.2

Multiply $- 4$ by $11$ .

$x = \frac{- 7 \pm \sqrt{49 - 44}}{2 \cdot 1}$

Step 3.5.2.5.1.3

Subtract $44$ from $49$ .

$x = \frac{- 7 \pm \sqrt{5}}{2 \cdot 1}$

Step 3.5.2.5.2

Multiply $2$ by $1$ .

$x = \frac{- 7 \pm \sqrt{5}}{2}$

Step 3.5.2.5.3

Change the $\pm$ to $-$ .

$x = \frac{- 7 - \sqrt{5}}{2}$

Step 3.5.2.5.4

Rewrite $- 7$ as $- 1 (7)$ .

$x = \frac{- 1 \cdot 7 - \sqrt{5}}{2}$

Step 3.5.2.5.5

Factor $- 1$ out of $- \sqrt{5}$ .

$x = \frac{- 1 \cdot 7 - (\sqrt{5})}{2}$

Step 3.5.2.5.6

Factor $- 1$ out of $- 1 (7) - (\sqrt{5})$ .

$x = \frac{- 1 (7 + \sqrt{5})}{2}$

Step 3.5.2.5.7

Move the negative in front of the fraction.

$x = - \frac{7 + \sqrt{5}}{2}$

Step 3.5.2.6

The final answer is the combination of both solutions.

$x = - \frac{7 - \sqrt{5}}{2}, - \frac{7 + \sqrt{5}}{2}$

Step 3.6

The final solution is all the values that make $2 (2 x + 7) (x^{2} + 7 x + 11) = 0$ true.

$x = - \frac{7}{2}, - \frac{7 - \sqrt{5}}{2}, - \frac{7 + \sqrt{5}}{2}$

Step 4

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

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Step 4.1

Substitute $- \frac{7}{2}$ in $f (x) = \frac{1}{5} x^{5} + \frac{7}{2} x^{4} + \frac{71}{3} x^{3} + 77 x^{2} + 120 x$ to find the value of $y$ .

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Step 4.1.1

Replace the variable $x$ with $- \frac{7}{2}$ in the expression.

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

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

Simplify each term.

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Step 4.1.2.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.1.2.1.1.1

Apply the product rule to $- \frac{7}{2}$ .

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

Step 4.1.2.1.1.2

Apply the product rule to $\frac{7}{2}$ .

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

Step 4.1.2.1.2

Raise $- 1$ to the power of $5$ .

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

Step 4.1.2.1.3

Raise $7$ to the power of $5$ .

$f (- \frac{7}{2}) = \frac{1}{5} \cdot (- \frac{16807}{2^{5}}) + \frac{7}{2} \cdot {(- \frac{7}{2})}^{4} + \frac{71}{3} \cdot {(- \frac{7}{2})}^{3} + 77 {(- \frac{7}{2})}^{2} + 120 (- \frac{7}{2})$

Step 4.1.2.1.4

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

$f (- \frac{7}{2}) = \frac{1}{5} \cdot (- \frac{16807}{32}) + \frac{7}{2} \cdot {(- \frac{7}{2})}^{4} + \frac{71}{3} \cdot {(- \frac{7}{2})}^{3} + 77 {(- \frac{7}{2})}^{2} + 120 (- \frac{7}{2})$

Step 4.1.2.1.5

Multiply $\frac{1}{5} (- \frac{16807}{32})$ .

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Step 4.1.2.1.5.1

Multiply $\frac{1}{5}$ by $\frac{16807}{32}$ .

$f (- \frac{7}{2}) = - \frac{16807}{5 \cdot 32} + \frac{7}{2} \cdot {(- \frac{7}{2})}^{4} + \frac{71}{3} \cdot {(- \frac{7}{2})}^{3} + 77 {(- \frac{7}{2})}^{2} + 120 (- \frac{7}{2})$

Step 4.1.2.1.5.2

Multiply $5$ by $32$ .

$f (- \frac{7}{2}) = - \frac{16807}{160} + \frac{7}{2} \cdot {(- \frac{7}{2})}^{4} + \frac{71}{3} \cdot {(- \frac{7}{2})}^{3} + 77 {(- \frac{7}{2})}^{2} + 120 (- \frac{7}{2})$

Step 4.1.2.1.6

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.1.2.1.6.1

Apply the product rule to $- \frac{7}{2}$ .

$f (- \frac{7}{2}) = - \frac{16807}{160} + \frac{7}{2} \cdot ({(- 1)}^{4} {(\frac{7}{2})}^{4}) + \frac{71}{3} \cdot {(- \frac{7}{2})}^{3} + 77 {(- \frac{7}{2})}^{2} + 120 (- \frac{7}{2})$

Step 4.1.2.1.6.2

Apply the product rule to $\frac{7}{2}$ .

$f (- \frac{7}{2}) = - \frac{16807}{160} + \frac{7}{2} \cdot ({(- 1)}^{4} (\frac{7^{4}}{2^{4}})) + \frac{71}{3} \cdot {(- \frac{7}{2})}^{3} + 77 {(- \frac{7}{2})}^{2} + 120 (- \frac{7}{2})$

Step 4.1.2.1.7

Raise $- 1$ to the power of $4$ .

$f (- \frac{7}{2}) = - \frac{16807}{160} + \frac{7}{2} \cdot (1 (\frac{7^{4}}{2^{4}})) + \frac{71}{3} \cdot {(- \frac{7}{2})}^{3} + 77 {(- \frac{7}{2})}^{2} + 120 (- \frac{7}{2})$

Step 4.1.2.1.8

Multiply $\frac{7^{4}}{2^{4}}$ by $1$ .

$f (- \frac{7}{2}) = - \frac{16807}{160} + \frac{7}{2} \cdot \frac{7^{4}}{2^{4}} + \frac{71}{3} \cdot {(- \frac{7}{2})}^{3} + 77 {(- \frac{7}{2})}^{2} + 120 (- \frac{7}{2})$

Step 4.1.2.1.9

Combine.

$f (- \frac{7}{2}) = - \frac{16807}{160} + \frac{7 \cdot 7^{4}}{2 \cdot 2^{4}} + \frac{71}{3} \cdot {(- \frac{7}{2})}^{3} + 77 {(- \frac{7}{2})}^{2} + 120 (- \frac{7}{2})$

Step 4.1.2.1.10

Multiply $7$ by $7^{4}$ by adding the exponents.

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Step 4.1.2.1.10.1

Multiply $7$ by $7^{4}$ .

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Step 4.1.2.1.10.1.1

Raise $7$ to the power of $1$ .

$f (- \frac{7}{2}) = - \frac{16807}{160} + \frac{7 \cdot 7^{4}}{2 \cdot 2^{4}} + \frac{71}{3} \cdot {(- \frac{7}{2})}^{3} + 77 {(- \frac{7}{2})}^{2} + 120 (- \frac{7}{2})$

Step 4.1.2.1.10.1.2

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

$f (- \frac{7}{2}) = - \frac{16807}{160} + \frac{7^{1 + 4}}{2 \cdot 2^{4}} + \frac{71}{3} \cdot {(- \frac{7}{2})}^{3} + 77 {(- \frac{7}{2})}^{2} + 120 (- \frac{7}{2})$

Step 4.1.2.1.10.2

Add $1$ and $4$ .

$f (- \frac{7}{2}) = - \frac{16807}{160} + \frac{7^{5}}{2 \cdot 2^{4}} + \frac{71}{3} \cdot {(- \frac{7}{2})}^{3} + 77 {(- \frac{7}{2})}^{2} + 120 (- \frac{7}{2})$

Step 4.1.2.1.11

Multiply $2$ by $2^{4}$ by adding the exponents.

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Step 4.1.2.1.11.1

Multiply $2$ by $2^{4}$ .

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Step 4.1.2.1.11.1.1

Raise $2$ to the power of $1$ .

$f (- \frac{7}{2}) = - \frac{16807}{160} + \frac{7^{5}}{2 \cdot 2^{4}} + \frac{71}{3} \cdot {(- \frac{7}{2})}^{3} + 77 {(- \frac{7}{2})}^{2} + 120 (- \frac{7}{2})$

Step 4.1.2.1.11.1.2

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

$f (- \frac{7}{2}) = - \frac{16807}{160} + \frac{7^{5}}{2^{1 + 4}} + \frac{71}{3} \cdot {(- \frac{7}{2})}^{3} + 77 {(- \frac{7}{2})}^{2} + 120 (- \frac{7}{2})$

Step 4.1.2.1.11.2

Add $1$ and $4$ .

$f (- \frac{7}{2}) = - \frac{16807}{160} + \frac{7^{5}}{2^{5}} + \frac{71}{3} \cdot {(- \frac{7}{2})}^{3} + 77 {(- \frac{7}{2})}^{2} + 120 (- \frac{7}{2})$

Step 4.1.2.1.12

Raise $7$ to the power of $5$ .

$f (- \frac{7}{2}) = - \frac{16807}{160} + \frac{16807}{2^{5}} + \frac{71}{3} \cdot {(- \frac{7}{2})}^{3} + 77 {(- \frac{7}{2})}^{2} + 120 (- \frac{7}{2})$

Step 4.1.2.1.13

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

$f (- \frac{7}{2}) = - \frac{16807}{160} + \frac{16807}{32} + \frac{71}{3} \cdot {(- \frac{7}{2})}^{3} + 77 {(- \frac{7}{2})}^{2} + 120 (- \frac{7}{2})$

Step 4.1.2.1.14

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.1.2.1.14.1

Apply the product rule to $- \frac{7}{2}$ .

$f (- \frac{7}{2}) = - \frac{16807}{160} + \frac{16807}{32} + \frac{71}{3} \cdot ({(- 1)}^{3} {(\frac{7}{2})}^{3}) + 77 {(- \frac{7}{2})}^{2} + 120 (- \frac{7}{2})$

Step 4.1.2.1.14.2

Apply the product rule to $\frac{7}{2}$ .

$f (- \frac{7}{2}) = - \frac{16807}{160} + \frac{16807}{32} + \frac{71}{3} \cdot ({(- 1)}^{3} (\frac{7^{3}}{2^{3}})) + 77 {(- \frac{7}{2})}^{2} + 120 (- \frac{7}{2})$

Step 4.1.2.1.15

Raise $- 1$ to the power of $3$ .

$f (- \frac{7}{2}) = - \frac{16807}{160} + \frac{16807}{32} + \frac{71}{3} \cdot (- \frac{7^{3}}{2^{3}}) + 77 {(- \frac{7}{2})}^{2} + 120 (- \frac{7}{2})$

Step 4.1.2.1.16

Raise $7$ to the power of $3$ .

$f (- \frac{7}{2}) = - \frac{16807}{160} + \frac{16807}{32} + \frac{71}{3} \cdot (- \frac{343}{2^{3}}) + 77 {(- \frac{7}{2})}^{2} + 120 (- \frac{7}{2})$

Step 4.1.2.1.17

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

$f (- \frac{7}{2}) = - \frac{16807}{160} + \frac{16807}{32} + \frac{71}{3} \cdot (- \frac{343}{8}) + 77 {(- \frac{7}{2})}^{2} + 120 (- \frac{7}{2})$

Step 4.1.2.1.18

Multiply $\frac{71}{3} (- \frac{343}{8})$ .

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Step 4.1.2.1.18.1

Multiply $\frac{71}{3}$ by $\frac{343}{8}$ .

$f (- \frac{7}{2}) = - \frac{16807}{160} + \frac{16807}{32} - \frac{71 \cdot 343}{3 \cdot 8} + 77 {(- \frac{7}{2})}^{2} + 120 (- \frac{7}{2})$

Step 4.1.2.1.18.2

Multiply $71$ by $343$ .

$f (- \frac{7}{2}) = - \frac{16807}{160} + \frac{16807}{32} - \frac{24353}{3 \cdot 8} + 77 {(- \frac{7}{2})}^{2} + 120 (- \frac{7}{2})$

Step 4.1.2.1.18.3

Multiply $3$ by $8$ .

$f (- \frac{7}{2}) = - \frac{16807}{160} + \frac{16807}{32} - \frac{24353}{24} + 77 {(- \frac{7}{2})}^{2} + 120 (- \frac{7}{2})$

Step 4.1.2.1.19

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.1.2.1.19.1

Apply the product rule to $- \frac{7}{2}$ .

$f (- \frac{7}{2}) = - \frac{16807}{160} + \frac{16807}{32} - \frac{24353}{24} + 77 ({(- 1)}^{2} {(\frac{7}{2})}^{2}) + 120 (- \frac{7}{2})$

Step 4.1.2.1.19.2

Apply the product rule to $\frac{7}{2}$ .

$f (- \frac{7}{2}) = - \frac{16807}{160} + \frac{16807}{32} - \frac{24353}{24} + 77 ({(- 1)}^{2} (\frac{7^{2}}{2^{2}})) + 120 (- \frac{7}{2})$

Step 4.1.2.1.20

Raise $- 1$ to the power of $2$ .

$f (- \frac{7}{2}) = - \frac{16807}{160} + \frac{16807}{32} - \frac{24353}{24} + 77 (1 (\frac{7^{2}}{2^{2}})) + 120 (- \frac{7}{2})$

Step 4.1.2.1.21

Multiply $\frac{7^{2}}{2^{2}}$ by $1$ .

$f (- \frac{7}{2}) = - \frac{16807}{160} + \frac{16807}{32} - \frac{24353}{24} + 77 (\frac{7^{2}}{2^{2}}) + 120 (- \frac{7}{2})$

Step 4.1.2.1.22

Raise $7$ to the power of $2$ .

$f (- \frac{7}{2}) = - \frac{16807}{160} + \frac{16807}{32} - \frac{24353}{24} + 77 (\frac{49}{2^{2}}) + 120 (- \frac{7}{2})$

Step 4.1.2.1.23

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

$f (- \frac{7}{2}) = - \frac{16807}{160} + \frac{16807}{32} - \frac{24353}{24} + 77 (\frac{49}{4}) + 120 (- \frac{7}{2})$

Step 4.1.2.1.24

Multiply $77 (\frac{49}{4})$ .

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Step 4.1.2.1.24.1

Combine $77$ and $\frac{49}{4}$ .

$f (- \frac{7}{2}) = - \frac{16807}{160} + \frac{16807}{32} - \frac{24353}{24} + \frac{77 \cdot 49}{4} + 120 (- \frac{7}{2})$

Step 4.1.2.1.24.2

Multiply $77$ by $49$ .

$f (- \frac{7}{2}) = - \frac{16807}{160} + \frac{16807}{32} - \frac{24353}{24} + \frac{3773}{4} + 120 (- \frac{7}{2})$

Step 4.1.2.1.25

Cancel the common factor of $2$ .

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Step 4.1.2.1.25.1

Move the leading negative in $- \frac{7}{2}$ into the numerator.

$f (- \frac{7}{2}) = - \frac{16807}{160} + \frac{16807}{32} - \frac{24353}{24} + \frac{3773}{4} + 120 (\frac{- 7}{2})$

Step 4.1.2.1.25.2

Factor $2$ out of $120$ .

$f (- \frac{7}{2}) = - \frac{16807}{160} + \frac{16807}{32} - \frac{24353}{24} + \frac{3773}{4} + 2 (60) (\frac{- 7}{2})$

Step 4.1.2.1.25.3

Cancel the common factor.

$f (- \frac{7}{2}) = - \frac{16807}{160} + \frac{16807}{32} - \frac{24353}{24} + \frac{3773}{4} + 2 \cdot (60 (\frac{- 7}{2}))$

Step 4.1.2.1.25.4

Rewrite the expression.

$f (- \frac{7}{2}) = - \frac{16807}{160} + \frac{16807}{32} - \frac{24353}{24} + \frac{3773}{4} + 60 \cdot - 7$

Step 4.1.2.1.26

Multiply $60$ by $- 7$ .

$f (- \frac{7}{2}) = - \frac{16807}{160} + \frac{16807}{32} - \frac{24353}{24} + \frac{3773}{4} - 420$

Step 4.1.2.2

Find the common denominator.

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Step 4.1.2.2.1

Multiply $\frac{16807}{160}$ by $\frac{3}{3}$ .

$f (- \frac{7}{2}) = - (\frac{16807}{160} \cdot \frac{3}{3}) + \frac{16807}{32} - \frac{24353}{24} + \frac{3773}{4} - 420$

Step 4.1.2.2.2

Multiply $\frac{16807}{160}$ by $\frac{3}{3}$ .

$f (- \frac{7}{2}) = - \frac{16807 \cdot 3}{160 \cdot 3} + \frac{16807}{32} - \frac{24353}{24} + \frac{3773}{4} - 420$

Step 4.1.2.2.3

Multiply $\frac{16807}{32}$ by $\frac{15}{15}$ .

$f (- \frac{7}{2}) = - \frac{16807 \cdot 3}{160 \cdot 3} + \frac{16807}{32} \cdot \frac{15}{15} - \frac{24353}{24} + \frac{3773}{4} - 420$

Step 4.1.2.2.4

Multiply $\frac{16807}{32}$ by $\frac{15}{15}$ .

$f (- \frac{7}{2}) = - \frac{16807 \cdot 3}{160 \cdot 3} + \frac{16807 \cdot 15}{32 \cdot 15} - \frac{24353}{24} + \frac{3773}{4} - 420$

Step 4.1.2.2.5

Multiply $\frac{24353}{24}$ by $\frac{20}{20}$ .

$f (- \frac{7}{2}) = - \frac{16807 \cdot 3}{160 \cdot 3} + \frac{16807 \cdot 15}{32 \cdot 15} - (\frac{24353}{24} \cdot \frac{20}{20}) + \frac{3773}{4} - 420$

Step 4.1.2.2.6

Multiply $\frac{24353}{24}$ by $\frac{20}{20}$ .

$f (- \frac{7}{2}) = - \frac{16807 \cdot 3}{160 \cdot 3} + \frac{16807 \cdot 15}{32 \cdot 15} - \frac{24353 \cdot 20}{24 \cdot 20} + \frac{3773}{4} - 420$

Step 4.1.2.2.7

Multiply $\frac{3773}{4}$ by $\frac{120}{120}$ .

$f (- \frac{7}{2}) = - \frac{16807 \cdot 3}{160 \cdot 3} + \frac{16807 \cdot 15}{32 \cdot 15} - \frac{24353 \cdot 20}{24 \cdot 20} + \frac{3773}{4} \cdot \frac{120}{120} - 420$

Step 4.1.2.2.8

Multiply $\frac{3773}{4}$ by $\frac{120}{120}$ .

$f (- \frac{7}{2}) = - \frac{16807 \cdot 3}{160 \cdot 3} + \frac{16807 \cdot 15}{32 \cdot 15} - \frac{24353 \cdot 20}{24 \cdot 20} + \frac{3773 \cdot 120}{4 \cdot 120} - 420$

Step 4.1.2.2.9

Write $- 420$ as a fraction with denominator $1$ .

$f (- \frac{7}{2}) = - \frac{16807 \cdot 3}{160 \cdot 3} + \frac{16807 \cdot 15}{32 \cdot 15} - \frac{24353 \cdot 20}{24 \cdot 20} + \frac{3773 \cdot 120}{4 \cdot 120} + \frac{- 420}{1}$

Step 4.1.2.2.10

Multiply $\frac{- 420}{1}$ by $\frac{480}{480}$ .

$f (- \frac{7}{2}) = - \frac{16807 \cdot 3}{160 \cdot 3} + \frac{16807 \cdot 15}{32 \cdot 15} - \frac{24353 \cdot 20}{24 \cdot 20} + \frac{3773 \cdot 120}{4 \cdot 120} + \frac{- 420}{1} \cdot \frac{480}{480}$

Step 4.1.2.2.11

Multiply $\frac{- 420}{1}$ by $\frac{480}{480}$ .

$f (- \frac{7}{2}) = - \frac{16807 \cdot 3}{160 \cdot 3} + \frac{16807 \cdot 15}{32 \cdot 15} - \frac{24353 \cdot 20}{24 \cdot 20} + \frac{3773 \cdot 120}{4 \cdot 120} + \frac{- 420 \cdot 480}{480}$

Step 4.1.2.2.12

Reorder the factors of $160 \cdot 3$ .

$f (- \frac{7}{2}) = - \frac{16807 \cdot 3}{3 \cdot 160} + \frac{16807 \cdot 15}{32 \cdot 15} - \frac{24353 \cdot 20}{24 \cdot 20} + \frac{3773 \cdot 120}{4 \cdot 120} + \frac{- 420 \cdot 480}{480}$

Step 4.1.2.2.13

Multiply $3$ by $160$ .

$f (- \frac{7}{2}) = - \frac{16807 \cdot 3}{480} + \frac{16807 \cdot 15}{32 \cdot 15} - \frac{24353 \cdot 20}{24 \cdot 20} + \frac{3773 \cdot 120}{4 \cdot 120} + \frac{- 420 \cdot 480}{480}$

Step 4.1.2.2.14

Reorder the factors of $32 \cdot 15$ .

$f (- \frac{7}{2}) = - \frac{16807 \cdot 3}{480} + \frac{16807 \cdot 15}{15 \cdot 32} - \frac{24353 \cdot 20}{24 \cdot 20} + \frac{3773 \cdot 120}{4 \cdot 120} + \frac{- 420 \cdot 480}{480}$

Step 4.1.2.2.15

Multiply $15$ by $32$ .

$f (- \frac{7}{2}) = - \frac{16807 \cdot 3}{480} + \frac{16807 \cdot 15}{480} - \frac{24353 \cdot 20}{24 \cdot 20} + \frac{3773 \cdot 120}{4 \cdot 120} + \frac{- 420 \cdot 480}{480}$

Step 4.1.2.2.16

Reorder the factors of $24 \cdot 20$ .

$f (- \frac{7}{2}) = - \frac{16807 \cdot 3}{480} + \frac{16807 \cdot 15}{480} - \frac{24353 \cdot 20}{20 \cdot 24} + \frac{3773 \cdot 120}{4 \cdot 120} + \frac{- 420 \cdot 480}{480}$

Step 4.1.2.2.17

Multiply $20$ by $24$ .

$f (- \frac{7}{2}) = - \frac{16807 \cdot 3}{480} + \frac{16807 \cdot 15}{480} - \frac{24353 \cdot 20}{480} + \frac{3773 \cdot 120}{4 \cdot 120} + \frac{- 420 \cdot 480}{480}$

Step 4.1.2.2.18

Multiply $4$ by $120$ .

$f (- \frac{7}{2}) = - \frac{16807 \cdot 3}{480} + \frac{16807 \cdot 15}{480} - \frac{24353 \cdot 20}{480} + \frac{3773 \cdot 120}{480} + \frac{- 420 \cdot 480}{480}$

Step 4.1.2.3

Combine the numerators over the common denominator.

$f (- \frac{7}{2}) = \frac{- 16807 \cdot 3 + 16807 \cdot 15 - 24353 \cdot 20 + 3773 \cdot 120 - 420 \cdot 480}{480}$

Step 4.1.2.4

Simplify each term.

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Step 4.1.2.4.1

Multiply $- 16807$ by $3$ .

$f (- \frac{7}{2}) = \frac{- 50421 + 16807 \cdot 15 - 24353 \cdot 20 + 3773 \cdot 120 - 420 \cdot 480}{480}$

Step 4.1.2.4.2

Multiply $16807$ by $15$ .

$f (- \frac{7}{2}) = \frac{- 50421 + 252105 - 24353 \cdot 20 + 3773 \cdot 120 - 420 \cdot 480}{480}$

Step 4.1.2.4.3

Multiply $- 24353$ by $20$ .

$f (- \frac{7}{2}) = \frac{- 50421 + 252105 - 487060 + 3773 \cdot 120 - 420 \cdot 480}{480}$

Step 4.1.2.4.4

Multiply $3773$ by $120$ .

$f (- \frac{7}{2}) = \frac{- 50421 + 252105 - 487060 + 452760 - 420 \cdot 480}{480}$

Step 4.1.2.4.5

Multiply $- 420$ by $480$ .

$f (- \frac{7}{2}) = \frac{- 50421 + 252105 - 487060 + 452760 - 201600}{480}$

Step 4.1.2.5

Reduce the expression by cancelling the common factors.

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Step 4.1.2.5.1

Add $- 50421$ and $252105$ .

$f (- \frac{7}{2}) = \frac{201684 - 487060 + 452760 - 201600}{480}$

Step 4.1.2.5.2

Simplify by adding and subtracting.

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Step 4.1.2.5.2.1

Subtract $487060$ from $201684$ .

$f (- \frac{7}{2}) = \frac{- 285376 + 452760 - 201600}{480}$

Step 4.1.2.5.2.2

Add $- 285376$ and $452760$ .

$f (- \frac{7}{2}) = \frac{167384 - 201600}{480}$

Step 4.1.2.5.2.3

Subtract $201600$ from $167384$ .

$f (- \frac{7}{2}) = \frac{- 34216}{480}$

Step 4.1.2.5.3

Cancel the common factor of $- 34216$ and $480$ .

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Step 4.1.2.5.3.1

Factor $8$ out of $- 34216$ .

$f (- \frac{7}{2}) = \frac{8 (- 4277)}{480}$

Step 4.1.2.5.3.2

Cancel the common factors.

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Step 4.1.2.5.3.2.1

Factor $8$ out of $480$ .

$f (- \frac{7}{2}) = \frac{8 \cdot - 4277}{8 \cdot 60}$

Step 4.1.2.5.3.2.2

Cancel the common factor.

$f (- \frac{7}{2}) = \frac{8 \cdot - 4277}{8 \cdot 60}$

Step 4.1.2.5.3.2.3

Rewrite the expression.

$f (- \frac{7}{2}) = \frac{- 4277}{60}$

Step 4.1.2.5.4

Move the negative in front of the fraction.

$f (- \frac{7}{2}) = - \frac{4277}{60}$

Step 4.1.2.6

The final answer is $- \frac{4277}{60}$ .

$- \frac{4277}{60}$

Step 4.2

The point found by substituting $- \frac{7}{2}$ in $f (x) = \frac{1}{5} x^{5} + \frac{7}{2} x^{4} + \frac{71}{3} x^{3} + 77 x^{2} + 120 x$ is $(- \frac{7}{2}, - \frac{4277}{60})$ . This point can be an inflection point.

$(- \frac{7}{2}, - \frac{4277}{60})$

Step 4.3

Substitute $- 2.38196601$ in $f (x) = \frac{1}{5} x^{5} + \frac{7}{2} x^{4} + \frac{71}{3} x^{3} + 77 x^{2} + 120 x$ to find the value of $y$ .

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Step 4.3.1

Replace the variable $x$ with $- 2.38196601$ in the expression.

$f (- 2.38196601) = \frac{1}{5} \cdot {(- 2.38196601)}^{5} + \frac{7}{2} \cdot {(- 2.38196601)}^{4} + \frac{71}{3} \cdot {(- 2.38196601)}^{3} + 77 {(- 2.38196601)}^{2} + 120 (- 2.38196601)$

Step 4.3.2

Simplify the result.

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Step 4.3.2.1

Simplify each term.

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Step 4.3.2.1.1

Raise $- 2.38196601$ to the power of $5$ .

$f (- 2.38196601) = \frac{1}{5} \cdot - 76.67924018 + \frac{7}{2} \cdot {(- 2.38196601)}^{4} + \frac{71}{3} \cdot {(- 2.38196601)}^{3} + 77 {(- 2.38196601)}^{2} + 120 (- 2.38196601)$

Step 4.3.2.1.2

Combine $\frac{1}{5}$ and $- 76.67924018$ .

$f (- 2.38196601) = \frac{- 76.67924018}{5} + \frac{7}{2} \cdot {(- 2.38196601)}^{4} + \frac{71}{3} \cdot {(- 2.38196601)}^{3} + 77 {(- 2.38196601)}^{2} + 120 (- 2.38196601)$

Step 4.3.2.1.3

Divide $- 76.67924018$ by $5$ .

$f (- 2.38196601) = - 15.33584803 + \frac{7}{2} \cdot {(- 2.38196601)}^{4} + \frac{71}{3} \cdot {(- 2.38196601)}^{3} + 77 {(- 2.38196601)}^{2} + 120 (- 2.38196601)$

Step 4.3.2.1.4

Raise $- 2.38196601$ to the power of $4$ .

$f (- 2.38196601) = - 15.33584803 + \frac{7}{2} \cdot 32.19157612 + \frac{71}{3} \cdot {(- 2.38196601)}^{3} + 77 {(- 2.38196601)}^{2} + 120 (- 2.38196601)$

Step 4.3.2.1.5

Multiply $\frac{7}{2} \cdot 32.19157612$ .

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Step 4.3.2.1.5.1

Combine $\frac{7}{2}$ and $32.19157612$ .

$f (- 2.38196601) = - 15.33584803 + \frac{7 \cdot 32.19157612}{2} + \frac{71}{3} \cdot {(- 2.38196601)}^{3} + 77 {(- 2.38196601)}^{2} + 120 (- 2.38196601)$

Step 4.3.2.1.5.2

Multiply $7$ by $32.19157612$ .

$f (- 2.38196601) = - 15.33584803 + \frac{225.34103288}{2} + \frac{71}{3} \cdot {(- 2.38196601)}^{3} + 77 {(- 2.38196601)}^{2} + 120 (- 2.38196601)$

Step 4.3.2.1.6

Divide $225.34103288$ by $2$ .

$f (- 2.38196601) = - 15.33584803 + 112.67051644 + \frac{71}{3} \cdot {(- 2.38196601)}^{3} + 77 {(- 2.38196601)}^{2} + 120 (- 2.38196601)$

Step 4.3.2.1.7

Raise $- 2.38196601$ to the power of $3$ .

$f (- 2.38196601) = - 15.33584803 + 112.67051644 + \frac{71}{3} \cdot - 13.51470842 + 77 {(- 2.38196601)}^{2} + 120 (- 2.38196601)$

Step 4.3.2.1.8

Multiply $\frac{71}{3} \cdot - 13.51470842$ .

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Step 4.3.2.1.8.1

Combine $\frac{71}{3}$ and $- 13.51470842$ .

$f (- 2.38196601) = - 15.33584803 + 112.67051644 + \frac{71 \cdot - 13.51470842}{3} + 77 {(- 2.38196601)}^{2} + 120 (- 2.38196601)$

Step 4.3.2.1.8.2

Multiply $71$ by $- 13.51470842$ .

$f (- 2.38196601) = - 15.33584803 + 112.67051644 + \frac{- 959.54429835}{3} + 77 {(- 2.38196601)}^{2} + 120 (- 2.38196601)$

Step 4.3.2.1.9

Divide $- 959.54429835$ by $3$ .

$f (- 2.38196601) = - 15.33584803 + 112.67051644 - 319.84809945 + 77 {(- 2.38196601)}^{2} + 120 (- 2.38196601)$

Step 4.3.2.1.10

Raise $- 2.38196601$ to the power of $2$ .

$f (- 2.38196601) = - 15.33584803 + 112.67051644 - 319.84809945 + 77 \cdot 5.67376207 + 120 (- 2.38196601)$

Step 4.3.2.1.11

Multiply $77$ by $5.67376207$ .

$f (- 2.38196601) = - 15.33584803 + 112.67051644 - 319.84809945 + 436.87968006 + 120 (- 2.38196601)$

Step 4.3.2.1.12

Multiply $120$ by $- 2.38196601$ .

$f (- 2.38196601) = - 15.33584803 + 112.67051644 - 319.84809945 + 436.87968006 - 285.83592135$

Step 4.3.2.2

Simplify by adding and subtracting.

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Step 4.3.2.2.1

Add $- 15.33584803$ and $112.67051644$ .

$f (- 2.38196601) = 97.3346684 - 319.84809945 + 436.87968006 - 285.83592135$

Step 4.3.2.2.2

Subtract $319.84809945$ from $97.3346684$ .

$f (- 2.38196601) = - 222.51343104 + 436.87968006 - 285.83592135$

Step 4.3.2.2.3

Add $- 222.51343104$ and $436.87968006$ .

$f (- 2.38196601) = 214.36624901 - 285.83592135$

Step 4.3.2.2.4

Subtract $285.83592135$ from $214.36624901$ .

$f (- 2.38196601) = - 71.46967233$

Step 4.3.2.3

The final answer is $- 71.46967233$ .

$- 71.46967233$

Step 4.4

The point found by substituting $- 2.38196601$ in $f (x) = \frac{1}{5} x^{5} + \frac{7}{2} x^{4} + \frac{71}{3} x^{3} + 77 x^{2} + 120 x$ is $(- 2.38196601, - 71.46967233)$ . This point can be an inflection point.

$(- 2.38196601, - 71.46967233)$

Step 4.5

Substitute $- 4.61803398$ in $f (x) = \frac{1}{5} x^{5} + \frac{7}{2} x^{4} + \frac{71}{3} x^{3} + 77 x^{2} + 120 x$ to find the value of $y$ .

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Step 4.5.1

Replace the variable $x$ with $- 4.61803398$ in the expression.

$f (- 4.61803398) = \frac{1}{5} \cdot {(- 4.61803398)}^{5} + \frac{7}{2} \cdot {(- 4.61803398)}^{4} + \frac{71}{3} \cdot {(- 4.61803398)}^{3} + 77 {(- 4.61803398)}^{2} + 120 (- 4.61803398)$

Step 4.5.2

Simplify the result.

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Step 4.5.2.1

Simplify each term.

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Step 4.5.2.1.1

Raise $- 4.61803398$ to the power of $5$ .

$f (- 4.61803398) = \frac{1}{5} \cdot - 2100.32075981 + \frac{7}{2} \cdot {(- 4.61803398)}^{4} + \frac{71}{3} \cdot {(- 4.61803398)}^{3} + 77 {(- 4.61803398)}^{2} + 120 (- 4.61803398)$

Step 4.5.2.1.2

Combine $\frac{1}{5}$ and $- 2100.32075981$ .

$f (- 4.61803398) = \frac{- 2100.32075981}{5} + \frac{7}{2} \cdot {(- 4.61803398)}^{4} + \frac{71}{3} \cdot {(- 4.61803398)}^{3} + 77 {(- 4.61803398)}^{2} + 120 (- 4.61803398)$

Step 4.5.2.1.3

Divide $- 2100.32075981$ by $5$ .

$f (- 4.61803398) = - 420.06415196 + \frac{7}{2} \cdot {(- 4.61803398)}^{4} + \frac{71}{3} \cdot {(- 4.61803398)}^{3} + 77 {(- 4.61803398)}^{2} + 120 (- 4.61803398)$

Step 4.5.2.1.4

Raise $- 4.61803398$ to the power of $4$ .

$f (- 4.61803398) = - 420.06415196 + \frac{7}{2} \cdot 454.80842387 + \frac{71}{3} \cdot {(- 4.61803398)}^{3} + 77 {(- 4.61803398)}^{2} + 120 (- 4.61803398)$

Step 4.5.2.1.5

Multiply $\frac{7}{2} \cdot 454.80842387$ .

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Step 4.5.2.1.5.1

Combine $\frac{7}{2}$ and $454.80842387$ .

$f (- 4.61803398) = - 420.06415196 + \frac{7 \cdot 454.80842387}{2} + \frac{71}{3} \cdot {(- 4.61803398)}^{3} + 77 {(- 4.61803398)}^{2} + 120 (- 4.61803398)$

Step 4.5.2.1.5.2

Multiply $7$ by $454.80842387$ .

$f (- 4.61803398) = - 420.06415196 + \frac{3183.65896711}{2} + \frac{71}{3} \cdot {(- 4.61803398)}^{3} + 77 {(- 4.61803398)}^{2} + 120 (- 4.61803398)$

Step 4.5.2.1.6

Divide $3183.65896711$ by $2$ .

$f (- 4.61803398) = - 420.06415196 + 1591.82948355 + \frac{71}{3} \cdot {(- 4.61803398)}^{3} + 77 {(- 4.61803398)}^{2} + 120 (- 4.61803398)$

Step 4.5.2.1.7

Raise $- 4.61803398$ to the power of $3$ .

$f (- 4.61803398) = - 420.06415196 + 1591.82948355 + \frac{71}{3} \cdot - 98.48529157 + 77 {(- 4.61803398)}^{2} + 120 (- 4.61803398)$

Step 4.5.2.1.8

Multiply $\frac{71}{3} \cdot - 98.48529157$ .

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Step 4.5.2.1.8.1

Combine $\frac{71}{3}$ and $- 98.48529157$ .

$f (- 4.61803398) = - 420.06415196 + 1591.82948355 + \frac{71 \cdot - 98.48529157}{3} + 77 {(- 4.61803398)}^{2} + 120 (- 4.61803398)$

Step 4.5.2.1.8.2

Multiply $71$ by $- 98.48529157$ .

$f (- 4.61803398) = - 420.06415196 + 1591.82948355 + \frac{- 6992.45570164}{3} + 77 {(- 4.61803398)}^{2} + 120 (- 4.61803398)$

Step 4.5.2.1.9

Divide $- 6992.45570164$ by $3$ .

$f (- 4.61803398) = - 420.06415196 + 1591.82948355 - 2330.81856721 + 77 {(- 4.61803398)}^{2} + 120 (- 4.61803398)$

Step 4.5.2.1.10

Raise $- 4.61803398$ to the power of $2$ .

$f (- 4.61803398) = - 420.06415196 + 1591.82948355 - 2330.81856721 + 77 \cdot 21.32623792 + 120 (- 4.61803398)$

Step 4.5.2.1.11

Multiply $77$ by $21.32623792$ .

$f (- 4.61803398) = - 420.06415196 + 1591.82948355 - 2330.81856721 + 1642.12031993 + 120 (- 4.61803398)$

Step 4.5.2.1.12

Multiply $120$ by $- 4.61803398$ .

$f (- 4.61803398) = - 420.06415196 + 1591.82948355 - 2330.81856721 + 1642.12031993 - 554.16407864$

Step 4.5.2.2

Simplify by adding and subtracting.

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Step 4.5.2.2.1

Add $- 420.06415196$ and $1591.82948355$ .

$f (- 4.61803398) = 1171.76533159 - 2330.81856721 + 1642.12031993 - 554.16407864$

Step 4.5.2.2.2

Subtract $2330.81856721$ from $1171.76533159$ .

$f (- 4.61803398) = - 1159.05323562 + 1642.12031993 - 554.16407864$

Step 4.5.2.2.3

Add $- 1159.05323562$ and $1642.12031993$ .

$f (- 4.61803398) = 483.06708431 - 554.16407864$

Step 4.5.2.2.4

Subtract $554.16407864$ from $483.06708431$ .

$f (- 4.61803398) = - 71.09699433$

Step 4.5.2.3

The final answer is $- 71.09699433$ .

$- 71.09699433$

Step 4.6

The point found by substituting $- 4.61803398$ in $f (x) = \frac{1}{5} x^{5} + \frac{7}{2} x^{4} + \frac{71}{3} x^{3} + 77 x^{2} + 120 x$ is $(- 4.61803398, - 71.09699433)$ . This point can be an inflection point.

$(- 4.61803398, - 71.09699433)$

Step 4.7

Determine the points that could be inflection points.

$(- \frac{7}{2}, - \frac{4277}{60}), (- 2.38196601, - 71.46967233), (- 4.61803398, - 71.09699433)$

Step 5

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

$(- \infty, - 4.61803398) \cup (- 4.61803398, - \frac{7}{2}) \cup (- \frac{7}{2}, - 2.38196601) \cup (- 2.38196601, \infty)$

Step 6

Substitute a value from the interval $(- \infty, - 4.61803398)$ into the second derivative to determine if it is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $- 4.71803398$ in the expression.

$f'' (- 4.71803398) = 4 {(- 4.71803398)}^{3} + 42 {(- 4.71803398)}^{2} + 142 (- 4.71803398) + 154$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Raise $- 4.71803398$ to the power of $3$ .

$f'' (- 4.71803398) = 4 \cdot - 105.02270396 + 42 {(- 4.71803398)}^{2} + 142 (- 4.71803398) + 154$

Step 6.2.1.2

Multiply $4$ by $- 105.02270396$ .

$f'' (- 4.71803398) = - 420.09081587 + 42 {(- 4.71803398)}^{2} + 142 (- 4.71803398) + 154$

Step 6.2.1.3

Raise $- 4.71803398$ to the power of $2$ .

$f'' (- 4.71803398) = - 420.09081587 + 42 \cdot 22.25984471 + 142 (- 4.71803398) + 154$

Step 6.2.1.4

Multiply $42$ by $22.25984471$ .

$f'' (- 4.71803398) = - 420.09081587 + 934.91347819 + 142 (- 4.71803398) + 154$

Step 6.2.1.5

Multiply $142$ by $- 4.71803398$ .

$f'' (- 4.71803398) = - 420.09081587 + 934.91347819 - 669.9608264 + 154$

Step 6.2.2

Simplify by adding and subtracting.

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Step 6.2.2.1

Add $- 420.09081587$ and $934.91347819$ .

$f'' (- 4.71803398) = 514.82266232 - 669.9608264 + 154$

Step 6.2.2.2

Subtract $669.9608264$ from $514.82266232$ .

$f'' (- 4.71803398) = - 155.13816407 + 154$

Step 6.2.2.3

Add $- 155.13816407$ and $154$ .

$f'' (- 4.71803398) = - 1.13816407$

Step 6.2.3

The final answer is $- 1.13816407$ .

$- 1.13816407$

Step 6.3

At $- 4.71803398$ , the second derivative is $- 1.13816407$ . Since this is negative, the second derivative is decreasing on the interval $(- \infty, - 4.61803398)$

Decreasing on $(- \infty, - 4.61803398)$ since $f'' (x) < 0$

Step 7

Substitute a value from the interval $(- 4.61803398, - \frac{7}{2})$ into the second derivative to determine if it is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $- 4.05901699$ in the expression.

$f'' (- 4.05901699) = 4 {(- 4.05901699)}^{3} + 42 {(- 4.05901699)}^{2} + 142 (- 4.05901699) + 154$

Step 7.2

Simplify the result.

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Step 7.2.1

Simplify each term.

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Step 7.2.1.1

Raise $- 4.05901699$ to the power of $3$ .

$f'' (- 4.05901699) = 4 \cdot - 66.87481735 + 42 {(- 4.05901699)}^{2} + 142 (- 4.05901699) + 154$

Step 7.2.1.2

Multiply $4$ by $- 66.87481735$ .

$f'' (- 4.05901699) = - 267.49926941 + 42 {(- 4.05901699)}^{2} + 142 (- 4.05901699) + 154$

Step 7.2.1.3

Raise $- 4.05901699$ to the power of $2$ .

$f'' (- 4.05901699) = - 267.49926941 + 42 \cdot 16.47561896 + 142 (- 4.05901699) + 154$

Step 7.2.1.4

Multiply $42$ by $16.47561896$ .

$f'' (- 4.05901699) = - 267.49926941 + 691.97599634 + 142 (- 4.05901699) + 154$

Step 7.2.1.5

Multiply $142$ by $- 4.05901699$ .

$f'' (- 4.05901699) = - 267.49926941 + 691.97599634 - 576.3804132 + 154$

Step 7.2.2

Simplify by adding and subtracting.

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Step 7.2.2.1

Add $- 267.49926941$ and $691.97599634$ .

$f'' (- 4.05901699) = 424.47672693 - 576.3804132 + 154$

Step 7.2.2.2

Subtract $576.3804132$ from $424.47672693$ .

$f'' (- 4.05901699) = - 151.90368627 + 154$

Step 7.2.2.3

Add $- 151.90368627$ and $154$ .

$f'' (- 4.05901699) = 2.09631372$

Step 7.2.3

The final answer is $2.09631372$ .

$2.09631372$

Step 7.3

At $- 4.05901699$ , the second derivative is $2.09631372$ . Since this is positive, the second derivative is increasing on the interval $(- 4.61803398, - \frac{7}{2})$ .

Increasing on $(- 4.61803398, - \frac{7}{2})$ since $f'' (x) > 0$

Step 8

Substitute a value from the interval $(- \frac{7}{2}, - 2.38196601)$ into the second derivative to determine if it is increasing or decreasing.

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Step 8.1

Replace the variable $x$ with $- 2.940983$ in the expression.

$f'' (- 2.940983) = 4 {(- 2.940983)}^{3} + 42 {(- 2.940983)}^{2} + 142 (- 2.940983) + 154$

Step 8.2

Simplify the result.

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Step 8.2.1

Simplify each term.

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Step 8.2.1.1

Raise $- 2.940983$ to the power of $3$ .

$f'' (- 2.940983) = 4 \cdot - 25.43768264 + 42 {(- 2.940983)}^{2} + 142 (- 2.940983) + 154$

Step 8.2.1.2

Multiply $4$ by $- 25.43768264$ .

$f'' (- 2.940983) = - 101.75073058 + 42 {(- 2.940983)}^{2} + 142 (- 2.940983) + 154$

Step 8.2.1.3

Raise $- 2.940983$ to the power of $2$ .

$f'' (- 2.940983) = - 101.75073058 + 42 \cdot 8.64938103 + 142 (- 2.940983) + 154$

Step 8.2.1.4

Multiply $42$ by $8.64938103$ .

$f'' (- 2.940983) = - 101.75073058 + 363.27400365 + 142 (- 2.940983) + 154$

Step 8.2.1.5

Multiply $142$ by $- 2.940983$ .

$f'' (- 2.940983) = - 101.75073058 + 363.27400365 - 417.61958679 + 154$

Step 8.2.2

Simplify by adding and subtracting.

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Step 8.2.2.1

Add $- 101.75073058$ and $363.27400365$ .

$f'' (- 2.940983) = 261.52327306 - 417.61958679 + 154$

Step 8.2.2.2

Subtract $417.61958679$ from $261.52327306$ .

$f'' (- 2.940983) = - 156.09631372 + 154$

Step 8.2.2.3

Add $- 156.09631372$ and $154$ .

$f'' (- 2.940983) = - 2.09631372$

Step 8.2.3

The final answer is $- 2.09631372$ .

$- 2.09631372$

Step 8.3

At $- 2.940983$ , the second derivative is $- 2.09631372$ . Since this is negative, the second derivative is decreasing on the interval $(- \frac{7}{2}, - 2.38196601)$

Decreasing on $(- \frac{7}{2}, - 2.38196601)$ since $f'' (x) < 0$

Step 9

Substitute a value from the interval $(- 2.38196601, \infty)$ into the second derivative to determine if it is increasing or decreasing.

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Step 9.1

Replace the variable $x$ with $- 2.28196601$ in the expression.

$f'' (- 2.28196601) = 4 {(- 2.28196601)}^{3} + 42 {(- 2.28196601)}^{2} + 142 (- 2.28196601) + 154$

Step 9.2

Simplify the result.

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Step 9.2.1

Simplify each term.

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Step 9.2.1.1

Raise $- 2.28196601$ to the power of $3$ .

$f'' (- 2.28196601) = 4 \cdot - 11.88303878 + 42 {(- 2.28196601)}^{2} + 142 (- 2.28196601) + 154$

Step 9.2.1.2

Multiply $4$ by $- 11.88303878$ .

$f'' (- 2.28196601) = - 47.53215513 + 42 {(- 2.28196601)}^{2} + 142 (- 2.28196601) + 154$

Step 9.2.1.3

Raise $- 2.28196601$ to the power of $2$ .

$f'' (- 2.28196601) = - 47.53215513 + 42 \cdot 5.20736887 + 142 (- 2.28196601) + 154$

Step 9.2.1.4

Multiply $42$ by $5.20736887$ .

$f'' (- 2.28196601) = - 47.53215513 + 218.70949281 + 142 (- 2.28196601) + 154$

Step 9.2.1.5

Multiply $142$ by $- 2.28196601$ .

$f'' (- 2.28196601) = - 47.53215513 + 218.70949281 - 324.03917359 + 154$

Step 9.2.2

Simplify by adding and subtracting.

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Step 9.2.2.1

Add $- 47.53215513$ and $218.70949281$ .

$f'' (- 2.28196601) = 171.17733767 - 324.03917359 + 154$

Step 9.2.2.2

Subtract $324.03917359$ from $171.17733767$ .

$f'' (- 2.28196601) = - 152.86183592 + 154$

Step 9.2.2.3

Add $- 152.86183592$ and $154$ .

$f'' (- 2.28196601) = 1.13816407$

Step 9.2.3

The final answer is $1.13816407$ .

$1.13816407$

Step 9.3

At $- 2.28196601$ , the second derivative is $1.13816407$ . Since this is positive, the second derivative is increasing on the interval $(- 2.38196601, \infty)$ .

Increasing on $(- 2.38196601, \infty)$ since $f'' (x) > 0$

Step 10

An inflection point is a point on a curve at which the concavity changes sign from plus to minus or from minus to plus. The inflection points in this case are $(- 4.61803398, - 71.09699433), (- \frac{7}{2}, - \frac{4277}{60}), (- 2.38196601, - 71.46967233)$ .

$(- 4.61803398, - 71.09699433), (- \frac{7}{2}, - \frac{4277}{60}), (- 2.38196601, - 71.46967233)$

Step 11