Calculus Examples

$f (x) = \frac{3}{10} x^{5} - 13 x^{3}$

Step 1

Find the second derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

By the Sum Rule, the derivative of $\frac{3}{10} x^{5} - 13 x^{3}$ with respect to $x$ is $\frac{d}{d x} [\frac{3}{10} x^{5}] + \frac{d}{d x} [- 13 x^{3}]$ .

$\frac{d}{d x} [\frac{3}{10} x^{5}] + \frac{d}{d x} [- 13 x^{3}]$

Step 1.1.2

Evaluate $\frac{d}{d x} [\frac{3}{10} x^{5}]$ .

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Step 1.1.2.1

Since $\frac{3}{10}$ is constant with respect to $x$ , the derivative of $\frac{3}{10} x^{5}$ with respect to $x$ is $\frac{3}{10} \frac{d}{d x} [x^{5}]$ .

$\frac{3}{10} \frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 13 x^{3}]$

Step 1.1.2.2

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

$\frac{3}{10} (5 x^{4}) + \frac{d}{d x} [- 13 x^{3}]$

Step 1.1.2.3

Combine $5$ and $\frac{3}{10}$ .

$\frac{5 \cdot 3}{10} x^{4} + \frac{d}{d x} [- 13 x^{3}]$

Step 1.1.2.4

Multiply $5$ by $3$ .

$\frac{15}{10} x^{4} + \frac{d}{d x} [- 13 x^{3}]$

Step 1.1.2.5

Combine $\frac{15}{10}$ and $x^{4}$ .

$\frac{15 x^{4}}{10} + \frac{d}{d x} [- 13 x^{3}]$

Step 1.1.2.6

Cancel the common factor of $15$ and $10$ .

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Step 1.1.2.6.1

Factor $5$ out of $15 x^{4}$ .

$\frac{5 (3 x^{4})}{10} + \frac{d}{d x} [- 13 x^{3}]$

Step 1.1.2.6.2

Cancel the common factors.

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Step 1.1.2.6.2.1

Factor $5$ out of $10$ .

$\frac{5 (3 x^{4})}{5 (2)} + \frac{d}{d x} [- 13 x^{3}]$

Step 1.1.2.6.2.2

Cancel the common factor.

$\frac{5 (3 x^{4})}{5 \cdot 2} + \frac{d}{d x} [- 13 x^{3}]$

Step 1.1.2.6.2.3

Rewrite the expression.

$\frac{3 x^{4}}{2} + \frac{d}{d x} [- 13 x^{3}]$

Step 1.1.3

Evaluate $\frac{d}{d x} [- 13 x^{3}]$ .

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Step 1.1.3.1

Since $- 13$ is constant with respect to $x$ , the derivative of $- 13 x^{3}$ with respect to $x$ is $- 13 \frac{d}{d x} [x^{3}]$ .

$\frac{3 x^{4}}{2} - 13 \frac{d}{d x} [x^{3}]$

Step 1.1.3.2

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

$\frac{3 x^{4}}{2} - 13 (3 x^{2})$

Step 1.1.3.3

Multiply $3$ by $- 13$ .

$f' (x) = \frac{3 x^{4}}{2} - 39 x^{2}$

Step 1.2

Find the second derivative.

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Step 1.2.1

By the Sum Rule, the derivative of $\frac{3 x^{4}}{2} - 39 x^{2}$ with respect to $x$ is $\frac{d}{d x} [\frac{3 x^{4}}{2}] + \frac{d}{d x} [- 39 x^{2}]$ .

$\frac{d}{d x} [\frac{3 x^{4}}{2}] + \frac{d}{d x} [- 39 x^{2}]$

Step 1.2.2

Evaluate $\frac{d}{d x} [\frac{3 x^{4}}{2}]$ .

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Step 1.2.2.1

Since $\frac{3}{2}$ is constant with respect to $x$ , the derivative of $\frac{3 x^{4}}{2}$ with respect to $x$ is $\frac{3}{2} \frac{d}{d x} [x^{4}]$ .

$\frac{3}{2} \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 39 x^{2}]$

Step 1.2.2.2

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

$\frac{3}{2} (4 x^{3}) + \frac{d}{d x} [- 39 x^{2}]$

Step 1.2.2.3

Combine $4$ and $\frac{3}{2}$ .

$\frac{4 \cdot 3}{2} x^{3} + \frac{d}{d x} [- 39 x^{2}]$

Step 1.2.2.4

Multiply $4$ by $3$ .

$\frac{12}{2} x^{3} + \frac{d}{d x} [- 39 x^{2}]$

Step 1.2.2.5

Combine $\frac{12}{2}$ and $x^{3}$ .

$\frac{12 x^{3}}{2} + \frac{d}{d x} [- 39 x^{2}]$

Step 1.2.2.6

Cancel the common factor of $12$ and $2$ .

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Step 1.2.2.6.1

Factor $2$ out of $12 x^{3}$ .

$\frac{2 (6 x^{3})}{2} + \frac{d}{d x} [- 39 x^{2}]$

Step 1.2.2.6.2

Cancel the common factors.

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Step 1.2.2.6.2.1

Factor $2$ out of $2$ .

$\frac{2 (6 x^{3})}{2 (1)} + \frac{d}{d x} [- 39 x^{2}]$

Step 1.2.2.6.2.2

Cancel the common factor.

$\frac{2 (6 x^{3})}{2 \cdot 1} + \frac{d}{d x} [- 39 x^{2}]$

Step 1.2.2.6.2.3

Rewrite the expression.

$\frac{6 x^{3}}{1} + \frac{d}{d x} [- 39 x^{2}]$

Step 1.2.2.6.2.4

Divide $6 x^{3}$ by $1$ .

$6 x^{3} + \frac{d}{d x} [- 39 x^{2}]$

Step 1.2.3

Evaluate $\frac{d}{d x} [- 39 x^{2}]$ .

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Step 1.2.3.1

Since $- 39$ is constant with respect to $x$ , the derivative of $- 39 x^{2}$ with respect to $x$ is $- 39 \frac{d}{d x} [x^{2}]$ .

$6 x^{3} - 39 \frac{d}{d x} [x^{2}]$

Step 1.2.3.2

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

$6 x^{3} - 39 (2 x)$

Step 1.2.3.3

Multiply $2$ by $- 39$ .

$f'' (x) = 6 x^{3} - 78 x$

Step 1.3

The second derivative of $f (x)$ with respect to $x$ is $6 x^{3} - 78 x$ .

$6 x^{3} - 78 x$

Step 2

Set the second derivative equal to $0$ then solve the equation $6 x^{3} - 78 x = 0$ .

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Step 2.1

Set the second derivative equal to $0$ .

$6 x^{3} - 78 x = 0$

Step 2.2

Factor $6 x$ out of $6 x^{3} - 78 x$ .

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Step 2.2.1

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

$6 x (x^{2}) - 78 x = 0$

Step 2.2.2

Factor $6 x$ out of $- 78 x$ .

$6 x (x^{2}) + 6 x (- 13) = 0$

Step 2.2.3

Factor $6 x$ out of $6 x (x^{2}) + 6 x (- 13)$ .

$6 x (x^{2} - 13) = 0$

Step 2.3

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

$x = 0$

$x^{2} - 13 = 0$

Step 2.4

Set $x$ equal to $0$ .

$x = 0$

Step 2.5

Set $x^{2} - 13$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $x^{2} - 13$ equal to $0$ .

$x^{2} - 13 = 0$

Step 2.5.2

Solve $x^{2} - 13 = 0$ for $x$ .

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Step 2.5.2.1

Add $13$ to both sides of the equation.

$x^{2} = 13$

Step 2.5.2.2

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

$x = \pm \sqrt{13}$

Step 2.5.2.3

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

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Step 2.5.2.3.1

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

$x = \sqrt{13}$

Step 2.5.2.3.2

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

$x = - \sqrt{13}$

Step 2.5.2.3.3

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

$x = \sqrt{13}, - \sqrt{13}$

Step 2.6

The final solution is all the values that make $6 x (x^{2} - 13) = 0$ true.

$x = 0, \sqrt{13}, - \sqrt{13}$

Step 3

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

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Step 3.1

Substitute $0$ in $f (x) = \frac{3}{10} x^{5} - 13 x^{3}$ to find the value of $y$ .

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Step 3.1.1

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

$f (0) = \frac{3}{10} \cdot {(0)}^{5} - 13 {(0)}^{3}$

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

Simplify each term.

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Step 3.1.2.1.1

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

$f (0) = \frac{3}{10} \cdot 0 - 13 {(0)}^{3}$

Step 3.1.2.1.2

Multiply $\frac{3}{10}$ by $0$ .

$f (0) = 0 - 13 {(0)}^{3}$

Step 3.1.2.1.3

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

$f (0) = 0 - 13 \cdot 0$

Step 3.1.2.1.4

Multiply $- 13$ by $0$ .

$f (0) = 0 + 0$

Step 3.1.2.2

Add $0$ and $0$ .

$f (0) = 0$

Step 3.1.2.3

The final answer is $0$ .

$0$

Step 3.2

The point found by substituting $0$ in $f (x) = \frac{3}{10} x^{5} - 13 x^{3}$ is $(0, 0)$ . This point can be an inflection point.

$(0, 0)$

Step 3.3

Substitute $\sqrt{13}$ in $f (x) = \frac{3}{10} x^{5} - 13 x^{3}$ to find the value of $y$ .

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Step 3.3.1

Replace the variable $x$ with $\sqrt{13}$ in the expression.

$f (\sqrt{13}) = \frac{3}{10} \cdot {(\sqrt{13})}^{5} - 13 {(\sqrt{13})}^{3}$

Step 3.3.2

Simplify the result.

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Step 3.3.2.1

Simplify each term.

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Step 3.3.2.1.1

Rewrite ${\sqrt{13}}^{5}$ as $\sqrt{13^{5}}$ .

$f (\sqrt{13}) = \frac{3}{10} \cdot \sqrt{13^{5}} - 13 {(\sqrt{13})}^{3}$

Step 3.3.2.1.2

Raise $13$ to the power of $5$ .

$f (\sqrt{13}) = \frac{3}{10} \cdot \sqrt{371293} - 13 {(\sqrt{13})}^{3}$

Step 3.3.2.1.3

Rewrite $371293$ as $169^{2} \cdot 13$ .

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Step 3.3.2.1.3.1

Factor $28561$ out of $371293$ .

$f (\sqrt{13}) = \frac{3}{10} \cdot \sqrt{28561 (13)} - 13 {(\sqrt{13})}^{3}$

Step 3.3.2.1.3.2

Rewrite $28561$ as $169^{2}$ .

$f (\sqrt{13}) = \frac{3}{10} \cdot \sqrt{169^{2} \cdot 13} - 13 {(\sqrt{13})}^{3}$

Step 3.3.2.1.4

Pull terms out from under the radical.

$f (\sqrt{13}) = \frac{3}{10} \cdot (169 \sqrt{13}) - 13 {(\sqrt{13})}^{3}$

Step 3.3.2.1.5

Multiply $\frac{3}{10} (169 \sqrt{13})$ .

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Step 3.3.2.1.5.1

Combine $169$ and $\frac{3}{10}$ .

$f (\sqrt{13}) = \frac{169 \cdot 3}{10} \cdot \sqrt{13} - 13 {(\sqrt{13})}^{3}$

Step 3.3.2.1.5.2

Multiply $169$ by $3$ .

$f (\sqrt{13}) = \frac{507}{10} \cdot \sqrt{13} - 13 {(\sqrt{13})}^{3}$

Step 3.3.2.1.5.3

Combine $\frac{507}{10}$ and $\sqrt{13}$ .

$f (\sqrt{13}) = \frac{507 \sqrt{13}}{10} - 13 {(\sqrt{13})}^{3}$

Step 3.3.2.1.6

Rewrite ${\sqrt{13}}^{3}$ as $\sqrt{13^{3}}$ .

$f (\sqrt{13}) = \frac{507 \sqrt{13}}{10} - 13 \sqrt{13^{3}}$

Step 3.3.2.1.7

Raise $13$ to the power of $3$ .

$f (\sqrt{13}) = \frac{507 \sqrt{13}}{10} - 13 \sqrt{2197}$

Step 3.3.2.1.8

Rewrite $2197$ as $13^{2} \cdot 13$ .

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Step 3.3.2.1.8.1

Factor $169$ out of $2197$ .

$f (\sqrt{13}) = \frac{507 \sqrt{13}}{10} - 13 \sqrt{169 (13)}$

Step 3.3.2.1.8.2

Rewrite $169$ as $13^{2}$ .

$f (\sqrt{13}) = \frac{507 \sqrt{13}}{10} - 13 \sqrt{13^{2} \cdot 13}$

Step 3.3.2.1.9

Pull terms out from under the radical.

$f (\sqrt{13}) = \frac{507 \sqrt{13}}{10} - 13 (13 \sqrt{13})$

Step 3.3.2.1.10

Multiply $13$ by $- 13$ .

$f (\sqrt{13}) = \frac{507 \sqrt{13}}{10} - 169 \sqrt{13}$

Step 3.3.2.2

To write $- 169 \sqrt{13}$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

$f (\sqrt{13}) = \frac{507 \sqrt{13}}{10} - 169 \sqrt{13} \cdot \frac{10}{10}$

Step 3.3.2.3

Combine fractions.

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Step 3.3.2.3.1

Combine $- 169 \sqrt{13}$ and $\frac{10}{10}$ .

$f (\sqrt{13}) = \frac{507 \sqrt{13}}{10} + \frac{- 169 \sqrt{13} \cdot 10}{10}$

Step 3.3.2.3.2

Combine the numerators over the common denominator.

$f (\sqrt{13}) = \frac{507 \sqrt{13} - 169 \sqrt{13} \cdot 10}{10}$

Step 3.3.2.4

Simplify the numerator.

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Step 3.3.2.4.1

Multiply $10$ by $- 169$ .

$f (\sqrt{13}) = \frac{507 \sqrt{13} - 1690 \sqrt{13}}{10}$

Step 3.3.2.4.2

Subtract $1690 \sqrt{13}$ from $507 \sqrt{13}$ .

$f (\sqrt{13}) = \frac{- 1183 \sqrt{13}}{10}$

Step 3.3.2.5

Move the negative in front of the fraction.

$f (\sqrt{13}) = - \frac{1183 \sqrt{13}}{10}$

Step 3.3.2.6

The final answer is $- \frac{1183 \sqrt{13}}{10}$ .

$- \frac{1183 \sqrt{13}}{10}$

Step 3.4

The point found by substituting $\sqrt{13}$ in $f (x) = \frac{3}{10} x^{5} - 13 x^{3}$ is $(\sqrt{13}, - \frac{1183 \sqrt{13}}{10})$ . This point can be an inflection point.

$(\sqrt{13}, - \frac{1183 \sqrt{13}}{10})$

Step 3.5

Substitute $- \sqrt{13}$ in $f (x) = \frac{3}{10} x^{5} - 13 x^{3}$ to find the value of $y$ .

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Step 3.5.1

Replace the variable $x$ with $- \sqrt{13}$ in the expression.

$f (- \sqrt{13}) = \frac{3}{10} \cdot {(- \sqrt{13})}^{5} - 13 {(- \sqrt{13})}^{3}$

Step 3.5.2

Simplify the result.

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Step 3.5.2.1

Simplify each term.

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Step 3.5.2.1.1

Apply the product rule to $- \sqrt{13}$ .

$f (- \sqrt{13}) = \frac{3}{10} \cdot ({(- 1)}^{5} {\sqrt{13}}^{5}) - 13 {(- \sqrt{13})}^{3}$

Step 3.5.2.1.2

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

$f (- \sqrt{13}) = \frac{3}{10} \cdot (- {\sqrt{13}}^{5}) - 13 {(- \sqrt{13})}^{3}$

Step 3.5.2.1.3

Rewrite ${\sqrt{13}}^{5}$ as $\sqrt{13^{5}}$ .

$f (- \sqrt{13}) = \frac{3}{10} \cdot (- \sqrt{13^{5}}) - 13 {(- \sqrt{13})}^{3}$

Step 3.5.2.1.4

Raise $13$ to the power of $5$ .

$f (- \sqrt{13}) = \frac{3}{10} \cdot (- \sqrt{371293}) - 13 {(- \sqrt{13})}^{3}$

Step 3.5.2.1.5

Rewrite $371293$ as $169^{2} \cdot 13$ .

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Step 3.5.2.1.5.1

Factor $28561$ out of $371293$ .

$f (- \sqrt{13}) = \frac{3}{10} \cdot (- \sqrt{28561 (13)}) - 13 {(- \sqrt{13})}^{3}$

Step 3.5.2.1.5.2

Rewrite $28561$ as $169^{2}$ .

$f (- \sqrt{13}) = \frac{3}{10} \cdot (- \sqrt{169^{2} \cdot 13}) - 13 {(- \sqrt{13})}^{3}$

Step 3.5.2.1.6

Pull terms out from under the radical.

$f (- \sqrt{13}) = \frac{3}{10} \cdot (- (169 \sqrt{13})) - 13 {(- \sqrt{13})}^{3}$

Step 3.5.2.1.7

Multiply $169$ by $- 1$ .

$f (- \sqrt{13}) = \frac{3}{10} \cdot (- 169 \sqrt{13}) - 13 {(- \sqrt{13})}^{3}$

Step 3.5.2.1.8

Multiply $\frac{3}{10} (- 169 \sqrt{13})$ .

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Step 3.5.2.1.8.1

Combine $- 169$ and $\frac{3}{10}$ .

$f (- \sqrt{13}) = \frac{- 169 \cdot 3}{10} \cdot \sqrt{13} - 13 {(- \sqrt{13})}^{3}$

Step 3.5.2.1.8.2

Multiply $- 169$ by $3$ .

$f (- \sqrt{13}) = \frac{- 507}{10} \cdot \sqrt{13} - 13 {(- \sqrt{13})}^{3}$

Step 3.5.2.1.8.3

Combine $\frac{- 507}{10}$ and $\sqrt{13}$ .

$f (- \sqrt{13}) = \frac{- 507 \sqrt{13}}{10} - 13 {(- \sqrt{13})}^{3}$

Step 3.5.2.1.9

Move the negative in front of the fraction.

$f (- \sqrt{13}) = - \frac{507 \sqrt{13}}{10} - 13 {(- \sqrt{13})}^{3}$

Step 3.5.2.1.10

Apply the product rule to $- \sqrt{13}$ .

$f (- \sqrt{13}) = - \frac{507 \sqrt{13}}{10} - 13 ({(- 1)}^{3} {\sqrt{13}}^{3})$

Step 3.5.2.1.11

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

$f (- \sqrt{13}) = - \frac{507 \sqrt{13}}{10} - 13 (- {\sqrt{13}}^{3})$

Step 3.5.2.1.12

Rewrite ${\sqrt{13}}^{3}$ as $\sqrt{13^{3}}$ .

$f (- \sqrt{13}) = - \frac{507 \sqrt{13}}{10} - 13 (- \sqrt{13^{3}})$

Step 3.5.2.1.13

Raise $13$ to the power of $3$ .

$f (- \sqrt{13}) = - \frac{507 \sqrt{13}}{10} - 13 (- \sqrt{2197})$

Step 3.5.2.1.14

Rewrite $2197$ as $13^{2} \cdot 13$ .

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Step 3.5.2.1.14.1

Factor $169$ out of $2197$ .

$f (- \sqrt{13}) = - \frac{507 \sqrt{13}}{10} - 13 (- \sqrt{169 (13)})$

Step 3.5.2.1.14.2

Rewrite $169$ as $13^{2}$ .

$f (- \sqrt{13}) = - \frac{507 \sqrt{13}}{10} - 13 (- \sqrt{13^{2} \cdot 13})$

Step 3.5.2.1.15

Pull terms out from under the radical.

$f (- \sqrt{13}) = - \frac{507 \sqrt{13}}{10} - 13 (- (13 \sqrt{13}))$

Step 3.5.2.1.16

Multiply $13$ by $- 1$ .

$f (- \sqrt{13}) = - \frac{507 \sqrt{13}}{10} - 13 (- 13 \sqrt{13})$

Step 3.5.2.1.17

Multiply $- 13$ by $- 13$ .

$f (- \sqrt{13}) = - \frac{507 \sqrt{13}}{10} + 169 \sqrt{13}$

Step 3.5.2.2

To write $169 \sqrt{13}$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

$f (- \sqrt{13}) = - \frac{507 \sqrt{13}}{10} + 169 \sqrt{13} \cdot \frac{10}{10}$

Step 3.5.2.3

Combine fractions.

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Step 3.5.2.3.1

Combine $169 \sqrt{13}$ and $\frac{10}{10}$ .

$f (- \sqrt{13}) = - \frac{507 \sqrt{13}}{10} + \frac{169 \sqrt{13} \cdot 10}{10}$

Step 3.5.2.3.2

Combine the numerators over the common denominator.

$f (- \sqrt{13}) = \frac{- 507 \sqrt{13} + 169 \sqrt{13} \cdot 10}{10}$

Step 3.5.2.4

Simplify the numerator.

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Step 3.5.2.4.1

Multiply $10$ by $169$ .

$f (- \sqrt{13}) = \frac{- 507 \sqrt{13} + 1690 \sqrt{13}}{10}$

Step 3.5.2.4.2

Add $- 507 \sqrt{13}$ and $1690 \sqrt{13}$ .

$f (- \sqrt{13}) = \frac{1183 \sqrt{13}}{10}$

Step 3.5.2.5

The final answer is $\frac{1183 \sqrt{13}}{10}$ .

$\frac{1183 \sqrt{13}}{10}$

Step 3.6

The point found by substituting $- \sqrt{13}$ in $f (x) = \frac{3}{10} x^{5} - 13 x^{3}$ is $(- \sqrt{13}, \frac{1183 \sqrt{13}}{10})$ . This point can be an inflection point.

$(- \sqrt{13}, \frac{1183 \sqrt{13}}{10})$

Step 3.7

Determine the points that could be inflection points.

$(0, 0), (\sqrt{13}, - \frac{1183 \sqrt{13}}{10}), (- \sqrt{13}, \frac{1183 \sqrt{13}}{10})$

Step 4

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

$(- \infty, - \sqrt{13}) \cup (- \sqrt{13}, 0) \cup (0, \sqrt{13}) \cup (\sqrt{13}, \infty)$

Step 5

Substitute a value from the interval $(- \infty, - \sqrt{13})$ into the second derivative to determine if it is increasing or decreasing.

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Step 5.1

Replace the variable $x$ with $- 3.70555127$ in the expression.

$f'' (- 3.70555127) = 6 {(- 3.70555127)}^{3} - 78 \cdot - 3.70555127$

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 $- 3.70555127$ to the power of $3$ .

$f'' (- 3.70555127) = 6 \cdot - 50.88133311 - 78 \cdot - 3.70555127$

Step 5.2.1.2

Multiply $6$ by $- 50.88133311$ .

$f'' (- 3.70555127) = - 305.28799871 - 78 \cdot - 3.70555127$

Step 5.2.1.3

Multiply $- 78$ by $- 3.70555127$ .

$f'' (- 3.70555127) = - 305.28799871 + 289.03299948$

Step 5.2.2

Add $- 305.28799871$ and $289.03299948$ .

$f'' (- 3.70555127) = - 16.25499922$

Step 5.2.3

The final answer is $- 16.25499922$ .

$- 16.25499922$

Step 5.3

At $- 3.70555127$ , the second derivative is $- 16.25499922$ . Since this is negative, the second derivative is decreasing on the interval $(- \infty, - \sqrt{13})$

Decreasing on $(- \infty, - \sqrt{13})$ since $f'' (x) < 0$

Step 6

Substitute a value from the interval $(- \sqrt{13}, 0)$ into the second derivative to determine if it is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $- 1.80277563$ in the expression.

$f'' (- 1.80277563) = 6 {(- 1.80277563)}^{3} - 78 \cdot - 1.80277563$

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 $- 1.80277563$ to the power of $3$ .

$f'' (- 1.80277563) = 6 \cdot - 5.85902082 - 78 \cdot - 1.80277563$

Step 6.2.1.2

Multiply $6$ by $- 5.85902082$ .

$f'' (- 1.80277563) = - 35.15412493 - 78 \cdot - 1.80277563$

Step 6.2.1.3

Multiply $- 78$ by $- 1.80277563$ .

$f'' (- 1.80277563) = - 35.15412493 + 140.61649974$

Step 6.2.2

Add $- 35.15412493$ and $140.61649974$ .

$f'' (- 1.80277563) = 105.4623748$

Step 6.2.3

The final answer is $105.4623748$ .

$105.4623748$

Step 6.3

At $- 1.80277563$ , the second derivative is $105.4623748$ . Since this is positive, the second derivative is increasing on the interval $(- \sqrt{13}, 0)$ .

Increasing on $(- \sqrt{13}, 0)$ since $f'' (x) > 0$

Step 7

Substitute a value from the interval $(0, \sqrt{13})$ into the second derivative to determine if it is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $1.80277563$ in the expression.

$f'' (1.80277563) = 6 {(1.80277563)}^{3} - 78 \cdot 1.80277563$

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 $1.80277563$ to the power of $3$ .

$f'' (1.80277563) = 6 \cdot 5.85902082 - 78 \cdot 1.80277563$

Step 7.2.1.2

Multiply $6$ by $5.85902082$ .

$f'' (1.80277563) = 35.15412493 - 78 \cdot 1.80277563$

Step 7.2.1.3

Multiply $- 78$ by $1.80277563$ .

$f'' (1.80277563) = 35.15412493 - 140.61649974$

Step 7.2.2

Subtract $140.61649974$ from $35.15412493$ .

$f'' (1.80277563) = - 105.4623748$

Step 7.2.3

The final answer is $- 105.4623748$ .

$- 105.4623748$

Step 7.3

At $1.80277563$ , the second derivative is $- 105.4623748$ . Since this is negative, the second derivative is decreasing on the interval $(0, \sqrt{13})$

Decreasing on $(0, \sqrt{13})$ since $f'' (x) < 0$

Step 8

Substitute a value from the interval $(\sqrt{13}, \infty)$ into the second derivative to determine if it is increasing or decreasing.

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Step 8.1

Replace the variable $x$ with $3.70555127$ in the expression.

$f'' (3.70555127) = 6 {(3.70555127)}^{3} - 78 \cdot 3.70555127$

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 $3.70555127$ to the power of $3$ .

$f'' (3.70555127) = 6 \cdot 50.88133311 - 78 \cdot 3.70555127$

Step 8.2.1.2

Multiply $6$ by $50.88133311$ .

$f'' (3.70555127) = 305.28799871 - 78 \cdot 3.70555127$

Step 8.2.1.3

Multiply $- 78$ by $3.70555127$ .

$f'' (3.70555127) = 305.28799871 - 289.03299948$

Step 8.2.2

Subtract $289.03299948$ from $305.28799871$ .

$f'' (3.70555127) = 16.25499922$

Step 8.2.3

The final answer is $16.25499922$ .

$16.25499922$

Step 8.3

At $3.70555127$ , the second derivative is $16.25499922$ . Since this is positive, the second derivative is increasing on the interval $(\sqrt{13}, \infty)$ .

Increasing on $(\sqrt{13}, \infty)$ since $f'' (x) > 0$

Step 9

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 $(- \sqrt{13}, \frac{1183 \sqrt{13}}{10}), (0, 0), (\sqrt{13}, - \frac{1183 \sqrt{13}}{10})$ .

$(- \sqrt{13}, \frac{1183 \sqrt{13}}{10}), (0, 0), (\sqrt{13}, - \frac{1183 \sqrt{13}}{10})$

Step 10