Calculus Examples

$\frac{1}{10} x^{5} + 2 x^{4} + 9 x^{3}$

Step 1

Write

$\frac{1}{10} x^{5} + 2 x^{4} + 9 x^{3}$ as a function.

$f (x) = \frac{1}{10} x^{5} + 2 x^{4} + 9 x^{3}$

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

$\frac{1}{10} x^{5} + 2 x^{4} + 9 x^{3}$ with respect to

$x$ is

$\frac{d}{d x} [\frac{1}{10} x^{5}] + \frac{d}{d x} [2 x^{4}] + \frac{d}{d x} [9 x^{3}]$ .

$\frac{d}{d x} [\frac{1}{10} x^{5}] + \frac{d}{d x} [2 x^{4}] + \frac{d}{d x} [9 x^{3}]$

Step 2.1.1.2

Evaluate

$\frac{d}{d x} [\frac{1}{10} x^{5}]$ .

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Step 2.1.1.2.1

Since

$\frac{1}{10}$ is constant with respect to

$x$ , the derivative of

$\frac{1}{10} x^{5}$ with respect to

$x$ is

$\frac{1}{10} \frac{d}{d x} [x^{5}]$ .

$\frac{1}{10} \frac{d}{d x} [x^{5}] + \frac{d}{d x} [2 x^{4}] + \frac{d}{d x} [9 x^{3}]$

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

$n = 5$ .

$\frac{1}{10} (5 x^{4}) + \frac{d}{d x} [2 x^{4}] + \frac{d}{d x} [9 x^{3}]$

Step 2.1.1.2.3

Combine

$5$ and

$\frac{1}{10}$ .

$\frac{5}{10} x^{4} + \frac{d}{d x} [2 x^{4}] + \frac{d}{d x} [9 x^{3}]$

Step 2.1.1.2.4

Combine

$\frac{5}{10}$ and

$x^{4}$ .

$\frac{5 x^{4}}{10} + \frac{d}{d x} [2 x^{4}] + \frac{d}{d x} [9 x^{3}]$

Step 2.1.1.2.5

Cancel the common factor of

$5$ and

$10$ .

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Step 2.1.1.2.5.1

Factor

$5$ out of

$5 x^{4}$ .

$\frac{5 (x^{4})}{10} + \frac{d}{d x} [2 x^{4}] + \frac{d}{d x} [9 x^{3}]$

Step 2.1.1.2.5.2

Cancel the common factors.

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Step 2.1.1.2.5.2.1

Factor

$5$ out of

$10$ .

$\frac{5 x^{4}}{5 \cdot 2} + \frac{d}{d x} [2 x^{4}] + \frac{d}{d x} [9 x^{3}]$

Step 2.1.1.2.5.2.2

Cancel the common factor.

$\frac{5 x^{4}}{5 \cdot 2} + \frac{d}{d x} [2 x^{4}] + \frac{d}{d x} [9 x^{3}]$

Step 2.1.1.2.5.2.3

Rewrite the expression.

$\frac{x^{4}}{2} + \frac{d}{d x} [2 x^{4}] + \frac{d}{d x} [9 x^{3}]$

Step 2.1.1.3

Evaluate

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

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Step 2.1.1.3.1

Since

$2$ is constant with respect to

$x$ , the derivative of

$2 x^{4}$ with respect to

$x$ is

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

$\frac{x^{4}}{2} + 2 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [9 x^{3}]$

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

$n = 4$ .

$\frac{x^{4}}{2} + 2 (4 x^{3}) + \frac{d}{d x} [9 x^{3}]$

Step 2.1.1.3.3

Multiply

$4$ by

$2$ .

$\frac{x^{4}}{2} + 8 x^{3} + \frac{d}{d x} [9 x^{3}]$

Step 2.1.1.4

Evaluate

$\frac{d}{d x} [9 x^{3}]$ .

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Step 2.1.1.4.1

Since

$9$ is constant with respect to

$x$ , the derivative of

$9 x^{3}$ with respect to

$x$ is

$9 \frac{d}{d x} [x^{3}]$ .

$\frac{x^{4}}{2} + 8 x^{3} + 9 \frac{d}{d x} [x^{3}]$

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

$n = 3$ .

$\frac{x^{4}}{2} + 8 x^{3} + 9 (3 x^{2})$

Step 2.1.1.4.3

Multiply

$3$ by

$9$ .

$f' (x) = \frac{x^{4}}{2} + 8 x^{3} + 27 x^{2}$

Step 2.1.2

Find the second derivative.

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Step 2.1.2.1

By the Sum Rule, the derivative of

$\frac{x^{4}}{2} + 8 x^{3} + 27 x^{2}$ with respect to

$x$ is

$\frac{d}{d x} [\frac{x^{4}}{2}] + \frac{d}{d x} [8 x^{3}] + \frac{d}{d x} [27 x^{2}]$ .

$\frac{d}{d x} [\frac{x^{4}}{2}] + \frac{d}{d x} [8 x^{3}] + \frac{d}{d x} [27 x^{2}]$

Step 2.1.2.2

Evaluate

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

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Step 2.1.2.2.1

Since

$\frac{1}{2}$ is constant with respect to

$x$ , the derivative of

$\frac{x^{4}}{2}$ with respect to

$x$ is

$\frac{1}{2} \frac{d}{d x} [x^{4}]$ .

$\frac{1}{2} \frac{d}{d x} [x^{4}] + \frac{d}{d x} [8 x^{3}] + \frac{d}{d x} [27 x^{2}]$

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

$n = 4$ .

$\frac{1}{2} (4 x^{3}) + \frac{d}{d x} [8 x^{3}] + \frac{d}{d x} [27 x^{2}]$

Step 2.1.2.2.3

Combine

$4$ and

$\frac{1}{2}$ .

$\frac{4}{2} x^{3} + \frac{d}{d x} [8 x^{3}] + \frac{d}{d x} [27 x^{2}]$

Step 2.1.2.2.4

Combine

$\frac{4}{2}$ and

$x^{3}$ .

$\frac{4 x^{3}}{2} + \frac{d}{d x} [8 x^{3}] + \frac{d}{d x} [27 x^{2}]$

Step 2.1.2.2.5

Cancel the common factor of

$4$ and

$2$ .

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Step 2.1.2.2.5.1

Factor

$2$ out of

$4 x^{3}$ .

$\frac{2 (2 x^{3})}{2} + \frac{d}{d x} [8 x^{3}] + \frac{d}{d x} [27 x^{2}]$

Step 2.1.2.2.5.2

Cancel the common factors.

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Step 2.1.2.2.5.2.1

Factor

$2$ out of

$2$ .

$\frac{2 (2 x^{3})}{2 (1)} + \frac{d}{d x} [8 x^{3}] + \frac{d}{d x} [27 x^{2}]$

Step 2.1.2.2.5.2.2

Cancel the common factor.

$\frac{2 (2 x^{3})}{2 \cdot 1} + \frac{d}{d x} [8 x^{3}] + \frac{d}{d x} [27 x^{2}]$

Step 2.1.2.2.5.2.3

Rewrite the expression.

$\frac{2 x^{3}}{1} + \frac{d}{d x} [8 x^{3}] + \frac{d}{d x} [27 x^{2}]$

Step 2.1.2.2.5.2.4

Divide

$2 x^{3}$ by

$1$ .

$2 x^{3} + \frac{d}{d x} [8 x^{3}] + \frac{d}{d x} [27 x^{2}]$

Step 2.1.2.3

Evaluate

$\frac{d}{d x} [8 x^{3}]$ .

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Step 2.1.2.3.1

Since

$8$ is constant with respect to

$x$ , the derivative of

$8 x^{3}$ with respect to

$x$ is

$8 \frac{d}{d x} [x^{3}]$ .

$2 x^{3} + 8 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [27 x^{2}]$

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

$n = 3$ .

$2 x^{3} + 8 (3 x^{2}) + \frac{d}{d x} [27 x^{2}]$

Step 2.1.2.3.3

Multiply

$3$ by

$8$ .

$2 x^{3} + 24 x^{2} + \frac{d}{d x} [27 x^{2}]$

Step 2.1.2.4

Evaluate

$\frac{d}{d x} [27 x^{2}]$ .

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Step 2.1.2.4.1

Since

$27$ is constant with respect to

$x$ , the derivative of

$27 x^{2}$ with respect to

$x$ is

$27 \frac{d}{d x} [x^{2}]$ .

$2 x^{3} + 24 x^{2} + 27 \frac{d}{d x} [x^{2}]$

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

$n = 2$ .

$2 x^{3} + 24 x^{2} + 27 (2 x)$

Step 2.1.2.4.3

Multiply

$2$ by

$27$ .

$f'' (x) = 2 x^{3} + 24 x^{2} + 54 x$

Step 2.1.3

The second derivative of

$f (x)$ with respect to

$x$ is

$2 x^{3} + 24 x^{2} + 54 x$ .

$2 x^{3} + 24 x^{2} + 54 x$

Step 2.2

Set the second derivative equal to

$0$ then solve the equation

$2 x^{3} + 24 x^{2} + 54 x = 0$ .

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Step 2.2.1

Set the second derivative equal to

$0$ .

$2 x^{3} + 24 x^{2} + 54 x = 0$

Step 2.2.2

Factor the left side of the equation.

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Step 2.2.2.1

Factor

$2 x$ out of

$2 x^{3} + 24 x^{2} + 54 x$ .

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Step 2.2.2.1.1

Factor

$2 x$ out of

$2 x^{3}$ .

$2 x (x^{2}) + 24 x^{2} + 54 x = 0$

Step 2.2.2.1.2

Factor

$2 x$ out of

$24 x^{2}$ .

$2 x (x^{2}) + 2 x (12 x) + 54 x = 0$

Step 2.2.2.1.3

Factor

$2 x$ out of

$54 x$ .

$2 x (x^{2}) + 2 x (12 x) + 2 x (27) = 0$

Step 2.2.2.1.4

Factor

$2 x$ out of

$2 x (x^{2}) + 2 x (12 x)$ .

$2 x (x^{2} + 12 x) + 2 x (27) = 0$

Step 2.2.2.1.5

Factor

$2 x$ out of

$2 x (x^{2} + 12 x) + 2 x (27)$ .

$2 x (x^{2} + 12 x + 27) = 0$

Step 2.2.2.2

Factor.

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Step 2.2.2.2.1

Factor

$x^{2} + 12 x + 27$ using the AC method.

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Step 2.2.2.2.1.1

Consider the form

$x^{2} + b x + c$ . Find a pair of integers whose product is

$c$ and whose sum is

$b$ . In this case, whose product is

$27$ and whose sum is

$12$ .

$3, 9$

Step 2.2.2.2.1.2

Write the factored form using these integers.

$2 x ((x + 3) (x + 9)) = 0$

Step 2.2.2.2.2

Remove unnecessary parentheses.

$2 x (x + 3) (x + 9) = 0$

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

$0$ .

$x = 0$

$x + 3 = 0$

$x + 9 = 0$

Step 2.2.4

Set

$x$ equal to

$0$ .

$x = 0$

Step 2.2.5

Set

$x + 3$ equal to

$0$ and solve for

$x$ .

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Step 2.2.5.1

Set

$x + 3$ equal to

$0$ .

$x + 3 = 0$

Step 2.2.5.2

Subtract

$3$ from both sides of the equation.

$x = - 3$

Step 2.2.6

Set

$x + 9$ equal to

$0$ and solve for

$x$ .

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Step 2.2.6.1

Set

$x + 9$ equal to

$0$ .

$x + 9 = 0$

Step 2.2.6.2

Subtract

$9$ from both sides of the equation.

$x = - 9$

Step 2.2.7

The final solution is all the values that make

$2 x (x + 3) (x + 9) = 0$ true.

$x = 0, - 3, - 9$

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.

$(- \infty, - 9) \cup (- 9, - 3) \cup (- 3, 0) \cup (0, \infty)$

Step 5

Substitute any number from the interval

$(- \infty, - 9)$ into the second derivative and evaluate to determine the concavity.

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Step 5.1

Replace the variable

$x$ with

$- 12$ in the expression.

$f'' (- 12) = 2 {(- 12)}^{3} + 24 {(- 12)}^{2} + 54 (- 12)$

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

$- 12$ to the power of

$3$ .

$f'' (- 12) = 2 \cdot - 1728 + 24 {(- 12)}^{2} + 54 (- 12)$

Step 5.2.1.2

Multiply

$2$ by

$- 1728$ .

$f'' (- 12) = - 3456 + 24 {(- 12)}^{2} + 54 (- 12)$

Step 5.2.1.3

Raise

$- 12$ to the power of

$2$ .

$f'' (- 12) = - 3456 + 24 \cdot 144 + 54 (- 12)$

Step 5.2.1.4

Multiply

$24$ by

$144$ .

$f'' (- 12) = - 3456 + 3456 + 54 (- 12)$

Step 5.2.1.5

Multiply

$54$ by

$- 12$ .

$f'' (- 12) = - 3456 + 3456 - 648$

Step 5.2.2

Simplify by adding and subtracting.

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Step 5.2.2.1

Add

$- 3456$ and

$3456$ .

$f'' (- 12) = 0 - 648$

Step 5.2.2.2

Subtract

$648$ from

$0$ .

$f'' (- 12) = - 648$

Step 5.2.3

The final answer is

$- 648$ .

$- 648$

Step 5.3

The graph is concave down on the interval

$(- \infty, - 9)$ because

$f'' (- 12)$ is negative.

Concave down on

$(- \infty, - 9)$ since

$f'' (x)$ is negative

Concave down on

$(- \infty, - 9)$ since

$f'' (x)$ is negative

Step 6

Substitute any number from the interval

$(- 9, - 3)$ into the second derivative and evaluate to determine the concavity.

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Step 6.1

Replace the variable

$x$ with

$- 6$ in the expression.

$f'' (- 6) = 2 {(- 6)}^{3} + 24 {(- 6)}^{2} + 54 (- 6)$

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

$- 6$ to the power of

$3$ .

$f'' (- 6) = 2 \cdot - 216 + 24 {(- 6)}^{2} + 54 (- 6)$

Step 6.2.1.2

Multiply

$2$ by

$- 216$ .

$f'' (- 6) = - 432 + 24 {(- 6)}^{2} + 54 (- 6)$

Step 6.2.1.3

Raise

$- 6$ to the power of

$2$ .

$f'' (- 6) = - 432 + 24 \cdot 36 + 54 (- 6)$

Step 6.2.1.4

Multiply

$24$ by

$36$ .

$f'' (- 6) = - 432 + 864 + 54 (- 6)$

Step 6.2.1.5

Multiply

$54$ by

$- 6$ .

$f'' (- 6) = - 432 + 864 - 324$

Step 6.2.2

Simplify by adding and subtracting.

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Step 6.2.2.1

Add

$- 432$ and

$864$ .

$f'' (- 6) = 432 - 324$

Step 6.2.2.2

Subtract

$324$ from

$432$ .

$f'' (- 6) = 108$

Step 6.2.3

The final answer is

$108$ .

$108$

Step 6.3

The graph is concave up on the interval

$(- 9, - 3)$ because

$f'' (- 6)$ is positive.

Concave up on

$(- 9, - 3)$ since

$f'' (x)$ is positive

Concave up on

$(- 9, - 3)$ since

$f'' (x)$ is positive

Step 7

Substitute any number from the interval

$(- 3, 0)$ into the second derivative and evaluate to determine the concavity.

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Step 7.1

Replace the variable

$x$ with

$- 2$ in the expression.

$f'' (- 2) = 2 {(- 2)}^{3} + 24 {(- 2)}^{2} + 54 (- 2)$

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

$- 2$ to the power of

$3$ .

$f'' (- 2) = 2 \cdot - 8 + 24 {(- 2)}^{2} + 54 (- 2)$

Step 7.2.1.2

Multiply

$2$ by

$- 8$ .

$f'' (- 2) = - 16 + 24 {(- 2)}^{2} + 54 (- 2)$

Step 7.2.1.3

Raise

$- 2$ to the power of

$2$ .

$f'' (- 2) = - 16 + 24 \cdot 4 + 54 (- 2)$

Step 7.2.1.4

Multiply

$24$ by

$4$ .

$f'' (- 2) = - 16 + 96 + 54 (- 2)$

Step 7.2.1.5

Multiply

$54$ by

$- 2$ .

$f'' (- 2) = - 16 + 96 - 108$

Step 7.2.2

Simplify by adding and subtracting.

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Step 7.2.2.1

Add

$- 16$ and

$96$ .

$f'' (- 2) = 80 - 108$

Step 7.2.2.2

Subtract

$108$ from

$80$ .

$f'' (- 2) = - 28$

Step 7.2.3

The final answer is

$- 28$ .

$- 28$

Step 7.3

The graph is concave down on the interval

$(- 3, 0)$ because

$f'' (- 2)$ is negative.

Concave down on

$(- 3, 0)$ since

$f'' (x)$ is negative

Concave down on

$(- 3, 0)$ since

$f'' (x)$ is negative

Step 8

Substitute any number from the interval

$(0, \infty)$ into the second derivative and evaluate to determine the concavity.

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Step 8.1

Replace the variable

$x$ with

$2$ in the expression.

$f'' (2) = 2 {(2)}^{3} + 24 {(2)}^{2} + 54 (2)$

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

Multiply

$2$ by

${(2)}^{3}$ by adding the exponents.

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Step 8.2.1.1.1

Multiply

$2$ by

${(2)}^{3}$ .

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Step 8.2.1.1.1.1

Raise

$2$ to the power of

$1$ .

$f'' (2) = 2 {(2)}^{3} + 24 {(2)}^{2} + 54 (2)$

Step 8.2.1.1.1.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (2) = 2^{1 + 3} + 24 {(2)}^{2} + 54 (2)$

Step 8.2.1.1.2

Add

$1$ and

$3$ .

$f'' (2) = 2^{4} + 24 {(2)}^{2} + 54 (2)$

Step 8.2.1.2

Raise

$2$ to the power of

$4$ .

$f'' (2) = 16 + 24 {(2)}^{2} + 54 (2)$

Step 8.2.1.3

Raise

$2$ to the power of

$2$ .

$f'' (2) = 16 + 24 \cdot 4 + 54 (2)$

Step 8.2.1.4

Multiply

$24$ by

$4$ .

$f'' (2) = 16 + 96 + 54 (2)$

Step 8.2.1.5

Multiply

$54$ by

$2$ .

$f'' (2) = 16 + 96 + 108$

Step 8.2.2

Simplify by adding numbers.

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Step 8.2.2.1

Add

$16$ and

$96$ .

$f'' (2) = 112 + 108$

Step 8.2.2.2

Add

$112$ and

$108$ .

$f'' (2) = 220$

Step 8.2.3

The final answer is

$220$ .

$220$

Step 8.3

The graph is concave up on the interval

$(0, \infty)$ because

$f'' (2)$ is positive.

Concave up on

$(0, \infty)$ since

$f'' (x)$ is positive

Concave up on

$(0, \infty)$ 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

$(- \infty, - 9)$ since

$f'' (x)$ is negative

Concave up on

$(- 9, - 3)$ since

$f'' (x)$ is positive

Concave down on

$(- 3, 0)$ since

$f'' (x)$ is negative

Concave up on

$(0, \infty)$ since

$f'' (x)$ is positive

Step 10