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

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Step 2.1

Find the second derivative.

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Step 2.1.1

Find the first derivative.

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Step 2.1.1.1

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 2.1.1.2

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

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Step 2.1.1.2.1

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 2.1.1.2.2

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 2.1.1.2.3

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 2.1.1.2.4

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 2.1.1.2.5

Cancel the common factor of $5$ .

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Step 2.1.1.2.5.1

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 2.1.1.2.5.2

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 2.1.1.3

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

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Step 2.1.1.3.1

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 2.1.1.3.2

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 2.1.1.3.3

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 2.1.1.3.4

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 2.1.1.3.5

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 2.1.1.3.6

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

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Step 2.1.1.3.6.1

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 2.1.1.3.6.2

Cancel the common factors.

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Step 2.1.1.3.6.2.1

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 2.1.1.3.6.2.2

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 2.1.1.3.6.2.3

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 2.1.1.3.6.2.4

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 2.1.1.4

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

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Step 2.1.1.4.1

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 2.1.1.4.2

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 2.1.1.4.3

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 2.1.1.4.4

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 2.1.1.4.5

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 2.1.1.4.6

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

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Step 2.1.1.4.6.1

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 2.1.1.4.6.2

Cancel the common factors.

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Step 2.1.1.4.6.2.1

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 2.1.1.4.6.2.2

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 2.1.1.4.6.2.3

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 2.1.1.4.6.2.4

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 2.1.1.5

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

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Step 2.1.1.5.1

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 2.1.1.5.2

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 2.1.1.5.3

Multiply $2$ by $77$ .

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

Step 2.1.1.6

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

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Step 2.1.1.6.1

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 2.1.1.6.2

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 2.1.1.6.3

Multiply $120$ by $1$ .

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

Step 2.1.2

Find the second derivative.

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Step 2.1.2.1

Differentiate.

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Step 2.1.2.1.1

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 2.1.2.1.2

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 2.1.2.2

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

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Step 2.1.2.2.1

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 2.1.2.2.2

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 2.1.2.2.3

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 2.1.2.3

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

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Step 2.1.2.3.1

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 2.1.2.3.2

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 2.1.2.3.3

Multiply $2$ by $71$ .

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

Step 2.1.2.4

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

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Step 2.1.2.4.1

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 2.1.2.4.2

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 2.1.2.4.3

Multiply $154$ by $1$ .

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

Step 2.1.2.5

Differentiate using the Constant Rule.

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Step 2.1.2.5.1

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 2.1.2.5.2

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

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

Step 2.1.3

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 2.2

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 2.2.1

Set the second derivative equal to $0$ .

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

Step 2.2.2

Factor the left side of the equation.

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Step 2.2.2.1

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

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Step 2.2.2.1.1

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

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

Step 2.2.2.1.2

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

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

Step 2.2.2.1.3

Factor $2$ out of $142 x$ .

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

Step 2.2.2.1.4

Factor $2$ out of $154$ .

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

Step 2.2.2.1.5

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 2.2.2.1.6

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 2.2.2.1.7

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 2.2.2.2

Factor.

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Step 2.2.2.2.1

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

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Step 2.2.2.2.1.1

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 2.2.2.2.1.2

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 2.2.2.2.1.3

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 2.2.2.2.1.3.1

Substitute $- 3.5$ into the polynomial.

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

Step 2.2.2.2.1.3.2

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

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

Step 2.2.2.2.1.3.3

Multiply $2$ by $- 42.875$ .

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

Step 2.2.2.2.1.3.4

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

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

Step 2.2.2.2.1.3.5

Multiply $21$ by $12.25$ .

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

Step 2.2.2.2.1.3.6

Add $- 85.75$ and $257.25$ .

$171.5 + 71 \cdot - 3.5 + 77$

Step 2.2.2.2.1.3.7

Multiply $71$ by $- 3.5$ .

$171.5 - 248.5 + 77$

Step 2.2.2.2.1.3.8

Subtract $248.5$ from $171.5$ .

$- 77 + 77$

Step 2.2.2.2.1.3.9

Add $- 77$ and $77$ .

$0$

Step 2.2.2.2.1.4

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 2.2.2.2.1.5

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

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Step 2.2.2.2.1.5.1

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 2.2.2.2.1.5.2

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 2.2.2.2.1.5.3

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 2.2.2.2.1.5.4

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 2.2.2.2.1.5.5

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 2.2.2.2.1.5.6

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 2.2.2.2.1.5.7

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 2.2.2.2.1.5.8

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 2.2.2.2.1.5.9

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 2.2.2.2.1.5.10

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 2.2.2.2.1.5.11

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 2.2.2.2.1.5.12

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 2.2.2.2.1.5.13

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 2.2.2.2.1.5.14

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 2.2.2.2.1.5.15

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 2.2.2.2.1.5.16

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

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

Step 2.2.2.2.1.6

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 2.2.2.2.2

Remove unnecessary parentheses.

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

Step 2.2.3

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 2.2.4

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

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Step 2.2.4.1

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

$2 x + 7 = 0$

Step 2.2.4.2

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

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Step 2.2.4.2.1

Subtract $7$ from both sides of the equation.

$2 x = - 7$

Step 2.2.4.2.2

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

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Step 2.2.4.2.2.1

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

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

Step 2.2.4.2.2.2

Simplify the left side.

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Step 2.2.4.2.2.2.1

Cancel the common factor of $2$ .

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Step 2.2.4.2.2.2.1.1

Cancel the common factor.

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

Step 2.2.4.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 2.2.4.2.2.3

Simplify the right side.

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Step 2.2.4.2.2.3.1

Move the negative in front of the fraction.

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

Step 2.2.5

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

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Step 2.2.5.1

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

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

Step 2.2.5.2

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

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Step 2.2.5.2.1

Use the quadratic formula to find the solutions.

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

Step 2.2.5.2.2

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 2.2.5.2.3

Simplify.

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Step 2.2.5.2.3.1

Simplify the numerator.

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Step 2.2.5.2.3.1.1

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

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

Step 2.2.5.2.3.1.2

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

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Step 2.2.5.2.3.1.2.1

Multiply $- 4$ by $1$ .

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

Step 2.2.5.2.3.1.2.2

Multiply $- 4$ by $11$ .

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

Step 2.2.5.2.3.1.3

Subtract $44$ from $49$ .

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

Step 2.2.5.2.3.2

Multiply $2$ by $1$ .

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

Step 2.2.5.2.4

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

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Step 2.2.5.2.4.1

Simplify the numerator.

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Step 2.2.5.2.4.1.1

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

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

Step 2.2.5.2.4.1.2

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

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Step 2.2.5.2.4.1.2.1

Multiply $- 4$ by $1$ .

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

Step 2.2.5.2.4.1.2.2

Multiply $- 4$ by $11$ .

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

Step 2.2.5.2.4.1.3

Subtract $44$ from $49$ .

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

Step 2.2.5.2.4.2

Multiply $2$ by $1$ .

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

Step 2.2.5.2.4.3

Change the $\pm$ to $+$ .

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

Step 2.2.5.2.4.4

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

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

Step 2.2.5.2.4.5

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

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

Step 2.2.5.2.4.6

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

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

Step 2.2.5.2.4.7

Move the negative in front of the fraction.

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

Step 2.2.5.2.5

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

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Step 2.2.5.2.5.1

Simplify the numerator.

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Step 2.2.5.2.5.1.1

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

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

Step 2.2.5.2.5.1.2

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

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Step 2.2.5.2.5.1.2.1

Multiply $- 4$ by $1$ .

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

Step 2.2.5.2.5.1.2.2

Multiply $- 4$ by $11$ .

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

Step 2.2.5.2.5.1.3

Subtract $44$ from $49$ .

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

Step 2.2.5.2.5.2

Multiply $2$ by $1$ .

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

Step 2.2.5.2.5.3

Change the $\pm$ to $-$ .

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

Step 2.2.5.2.5.4

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

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

Step 2.2.5.2.5.5

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

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

Step 2.2.5.2.5.6

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

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

Step 2.2.5.2.5.7

Move the negative in front of the fraction.

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

Step 2.2.5.2.6

The final answer is the combination of both solutions.

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

Step 2.2.6

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 3

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.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 4

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

$(- \infty, - \frac{7 + \sqrt{5}}{2}) \cup (- \frac{7 + \sqrt{5}}{2}, - \frac{7}{2}) \cup (- \frac{7}{2}, - \frac{7 - \sqrt{5}}{2}) \cup (- \frac{7 - \sqrt{5}}{2}, \infty)$

Step 5

Substitute any number from the interval $(- \infty, - \frac{7 + \sqrt{5}}{2})$ into the second derivative and evaluate to determine the concavity.

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Step 5.1

Replace the variable $x$ with $- 7$ in the expression.

$f'' (- 7) = 4 {(- 7)}^{3} + 42 {(- 7)}^{2} + 142 (- 7) + 154$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

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

$f'' (- 7) = 4 \cdot - 343 + 42 {(- 7)}^{2} + 142 (- 7) + 154$

Step 5.2.1.2

Multiply $4$ by $- 343$ .

$f'' (- 7) = - 1372 + 42 {(- 7)}^{2} + 142 (- 7) + 154$

Step 5.2.1.3

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

$f'' (- 7) = - 1372 + 42 \cdot 49 + 142 (- 7) + 154$

Step 5.2.1.4

Multiply $42$ by $49$ .

$f'' (- 7) = - 1372 + 2058 + 142 (- 7) + 154$

Step 5.2.1.5

Multiply $142$ by $- 7$ .

$f'' (- 7) = - 1372 + 2058 - 994 + 154$

Step 5.2.2

Simplify by adding and subtracting.

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Step 5.2.2.1

Add $- 1372$ and $2058$ .

$f'' (- 7) = 686 - 994 + 154$

Step 5.2.2.2

Subtract $994$ from $686$ .

$f'' (- 7) = - 308 + 154$

Step 5.2.2.3

Add $- 308$ and $154$ .

$f'' (- 7) = - 154$

Step 5.2.3

The final answer is $- 154$ .

$- 154$

Step 5.3

The graph is concave down on the interval $(- \infty, - \frac{7 + \sqrt{5}}{2})$ because $f'' (- 7)$ is negative.

Concave down on $(- \infty, - \frac{7 + \sqrt{5}}{2})$ since $f'' (x)$ is negative

Step 6

Substitute any number from the interval $(- \frac{7 + \sqrt{5}}{2}, - \frac{7}{2})$ into the second derivative and evaluate to determine the concavity.

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Step 6.1

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

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

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

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

Step 6.2.1.2

Multiply $4$ by $- 64$ .

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

Step 6.2.1.3

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

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

Step 6.2.1.4

Multiply $42$ by $16$ .

$f'' (- 4) = - 256 + 672 + 142 (- 4) + 154$

Step 6.2.1.5

Multiply $142$ by $- 4$ .

$f'' (- 4) = - 256 + 672 - 568 + 154$

Step 6.2.2

Simplify by adding and subtracting.

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Step 6.2.2.1

Add $- 256$ and $672$ .

$f'' (- 4) = 416 - 568 + 154$

Step 6.2.2.2

Subtract $568$ from $416$ .

$f'' (- 4) = - 152 + 154$

Step 6.2.2.3

Add $- 152$ and $154$ .

$f'' (- 4) = 2$

Step 6.2.3

The final answer is $2$ .

$2$

Step 6.3

The graph is concave up on the interval $(- \frac{7 + \sqrt{5}}{2}, - \frac{7}{2})$ because $f'' (- 4)$ is positive.

Concave up on $(- \frac{7 + \sqrt{5}}{2}, - \frac{7}{2})$ since $f'' (x)$ is positive

Step 7

Substitute any number from the interval $(- \frac{7}{2}, - \frac{7 - \sqrt{5}}{2})$ into the second derivative and evaluate to determine the concavity.

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Step 7.1

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

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

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

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

Step 7.2.1.2

Multiply $4$ by $- 27$ .

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

Step 7.2.1.3

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

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

Step 7.2.1.4

Multiply $42$ by $9$ .

$f'' (- 3) = - 108 + 378 + 142 (- 3) + 154$

Step 7.2.1.5

Multiply $142$ by $- 3$ .

$f'' (- 3) = - 108 + 378 - 426 + 154$

Step 7.2.2

Simplify by adding and subtracting.

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Step 7.2.2.1

Add $- 108$ and $378$ .

$f'' (- 3) = 270 - 426 + 154$

Step 7.2.2.2

Subtract $426$ from $270$ .

$f'' (- 3) = - 156 + 154$

Step 7.2.2.3

Add $- 156$ and $154$ .

$f'' (- 3) = - 2$

Step 7.2.3

The final answer is $- 2$ .

$- 2$

Step 7.3

The graph is concave down on the interval $(- \frac{7}{2}, - \frac{7 - \sqrt{5}}{2})$ because $f'' (- 3)$ is negative.

Concave down on $(- \frac{7}{2}, - \frac{7 - \sqrt{5}}{2})$ since $f'' (x)$ is negative

Step 8

Substitute any number from the interval $(- \frac{7 - \sqrt{5}}{2}, \infty)$ into the second derivative and evaluate to determine the concavity.

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Step 8.1

Replace the variable $x$ with $0$ in the expression.

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

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

Raising $0$ to any positive power yields $0$ .

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

Step 8.2.1.2

Multiply $4$ by $0$ .

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

Step 8.2.1.3

Raising $0$ to any positive power yields $0$ .

$f'' (0) = 0 + 42 \cdot 0 + 142 (0) + 154$

Step 8.2.1.4

Multiply $42$ by $0$ .

$f'' (0) = 0 + 0 + 142 (0) + 154$

Step 8.2.1.5

Multiply $142$ by $0$ .

$f'' (0) = 0 + 0 + 0 + 154$

Step 8.2.2

Simplify by adding numbers.

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Step 8.2.2.1

Add $0$ and $0$ .

$f'' (0) = 0 + 0 + 154$

Step 8.2.2.2

Add $0$ and $0$ .

$f'' (0) = 0 + 154$

Step 8.2.2.3

Add $0$ and $154$ .

$f'' (0) = 154$

Step 8.2.3

The final answer is $154$ .

$154$

Step 8.3

The graph is concave up on the interval $(- \frac{7 - \sqrt{5}}{2}, \infty)$ because $f'' (0)$ is positive.

Concave up on $(- \frac{7 - \sqrt{5}}{2}, \infty)$ since $f'' (x)$ is positive

Step 9

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

Concave down on $(- \infty, - \frac{7 + \sqrt{5}}{2})$ since $f'' (x)$ is negative

Concave up on $(- \frac{7 + \sqrt{5}}{2}, - \frac{7}{2})$ since $f'' (x)$ is positive

Concave down on $(- \frac{7}{2}, - \frac{7 - \sqrt{5}}{2})$ since $f'' (x)$ is negative

Concave up on $(- \frac{7 - \sqrt{5}}{2}, \infty)$ since $f'' (x)$ is positive

Step 10