Calculus Examples

f (x) = \frac{2}{15} x^{6} - 3 x^{4}

$f (x) = \frac{2}{15} x^{6} - 3 x^{4}$

Step 1

Find the

$x$ values where the second derivative is equal to

$0$ .

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Step 1.1

Find the second derivative.

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Step 1.1.1

Find the first derivative.

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Step 1.1.1.1

By the Sum Rule, the derivative of

$\frac{2}{15} x^{6} - 3 x^{4}$ with respect to

$x$ is

$\frac{d}{d x} [\frac{2}{15} x^{6}] + \frac{d}{d x} [- 3 x^{4}]$ .

$\frac{d}{d x} [\frac{2}{15} x^{6}] + \frac{d}{d x} [- 3 x^{4}]$

Step 1.1.1.2

Evaluate

$\frac{d}{d x} [\frac{2}{15} x^{6}]$ .

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Step 1.1.1.2.1

Since

$\frac{2}{15}$ is constant with respect to

$x$ , the derivative of

$\frac{2}{15} x^{6}$ with respect to

$x$ is

$\frac{2}{15} \frac{d}{d x} [x^{6}]$ .

$\frac{2}{15} \frac{d}{d x} [x^{6}] + \frac{d}{d x} [- 3 x^{4}]$

Step 1.1.1.2.2

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 6$ .

$\frac{2}{15} (6 x^{5}) + \frac{d}{d x} [- 3 x^{4}]$

Step 1.1.1.2.3

Combine

$6$ and

$\frac{2}{15}$ .

$\frac{6 \cdot 2}{15} x^{5} + \frac{d}{d x} [- 3 x^{4}]$

Step 1.1.1.2.4

Multiply

$6$ by

$2$ .

$\frac{12}{15} x^{5} + \frac{d}{d x} [- 3 x^{4}]$

Step 1.1.1.2.5

Combine

$\frac{12}{15}$ and

$x^{5}$ .

$\frac{12 x^{5}}{15} + \frac{d}{d x} [- 3 x^{4}]$

Step 1.1.1.2.6

Cancel the common factor of

$12$ and

$15$ .

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Step 1.1.1.2.6.1

Factor

$3$ out of

$12 x^{5}$ .

$\frac{3 (4 x^{5})}{15} + \frac{d}{d x} [- 3 x^{4}]$

Step 1.1.1.2.6.2

Cancel the common factors.

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Step 1.1.1.2.6.2.1

Factor

$3$ out of

$15$ .

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

Step 1.1.1.2.6.2.2

Cancel the common factor.

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

Step 1.1.1.2.6.2.3

Rewrite the expression.

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

Step 1.1.1.3

Evaluate

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

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Step 1.1.1.3.1

Since

$- 3$ is constant with respect to

$x$ , the derivative of

$- 3 x^{4}$ with respect to

$x$ is

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

$\frac{4 x^{5}}{5} - 3 \frac{d}{d x} [x^{4}]$

Step 1.1.1.3.2

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 4$ .

$\frac{4 x^{5}}{5} - 3 (4 x^{3})$

Step 1.1.1.3.3

Multiply

$4$ by

$- 3$ .

$f' (x) = \frac{4 x^{5}}{5} - 12 x^{3}$

Step 1.1.2

Find the second derivative.

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Step 1.1.2.1

By the Sum Rule, the derivative of

$\frac{4 x^{5}}{5} - 12 x^{3}$ with respect to

$x$ is

$\frac{d}{d x} [\frac{4 x^{5}}{5}] + \frac{d}{d x} [- 12 x^{3}]$ .

$\frac{d}{d x} [\frac{4 x^{5}}{5}] + \frac{d}{d x} [- 12 x^{3}]$

Step 1.1.2.2

Evaluate

$\frac{d}{d x} [\frac{4 x^{5}}{5}]$ .

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Step 1.1.2.2.1

Since

$\frac{4}{5}$ is constant with respect to

$x$ , the derivative of

$\frac{4 x^{5}}{5}$ with respect to

$x$ is

$\frac{4}{5} \frac{d}{d x} [x^{5}]$ .

$\frac{4}{5} \frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 12 x^{3}]$

Step 1.1.2.2.2

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 5$ .

$\frac{4}{5} (5 x^{4}) + \frac{d}{d x} [- 12 x^{3}]$

Step 1.1.2.2.3

Combine

$5$ and

$\frac{4}{5}$ .

$\frac{5 \cdot 4}{5} x^{4} + \frac{d}{d x} [- 12 x^{3}]$

Step 1.1.2.2.4

Multiply

$5$ by

$4$ .

$\frac{20}{5} x^{4} + \frac{d}{d x} [- 12 x^{3}]$

Step 1.1.2.2.5

Combine

$\frac{20}{5}$ and

$x^{4}$ .

$\frac{20 x^{4}}{5} + \frac{d}{d x} [- 12 x^{3}]$

Step 1.1.2.2.6

Cancel the common factor of

$20$ and

$5$ .

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Step 1.1.2.2.6.1

Factor

$5$ out of

$20 x^{4}$ .

$\frac{5 (4 x^{4})}{5} + \frac{d}{d x} [- 12 x^{3}]$

Step 1.1.2.2.6.2

Cancel the common factors.

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Step 1.1.2.2.6.2.1

Factor

$5$ out of

$5$ .

$\frac{5 (4 x^{4})}{5 (1)} + \frac{d}{d x} [- 12 x^{3}]$

Step 1.1.2.2.6.2.2

Cancel the common factor.

$\frac{5 (4 x^{4})}{5 \cdot 1} + \frac{d}{d x} [- 12 x^{3}]$

Step 1.1.2.2.6.2.3

Rewrite the expression.

$\frac{4 x^{4}}{1} + \frac{d}{d x} [- 12 x^{3}]$

Step 1.1.2.2.6.2.4

Divide

$4 x^{4}$ by

$1$ .

$4 x^{4} + \frac{d}{d x} [- 12 x^{3}]$

Step 1.1.2.3

Evaluate

$\frac{d}{d x} [- 12 x^{3}]$ .

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Step 1.1.2.3.1

Since

$- 12$ is constant with respect to

$x$ , the derivative of

$- 12 x^{3}$ with respect to

$x$ is

$- 12 \frac{d}{d x} [x^{3}]$ .

$4 x^{4} - 12 \frac{d}{d x} [x^{3}]$

Step 1.1.2.3.2

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 3$ .

$4 x^{4} - 12 (3 x^{2})$

Step 1.1.2.3.3

Multiply

$3$ by

$- 12$ .

$f'' (x) = 4 x^{4} - 36 x^{2}$

Step 1.1.3

The second derivative of

$f (x)$ with respect to

$x$ is

$4 x^{4} - 36 x^{2}$ .

$4 x^{4} - 36 x^{2}$

Step 1.2

Set the second derivative equal to

$0$ then solve the equation

$4 x^{4} - 36 x^{2} = 0$ .

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Step 1.2.1

Set the second derivative equal to

$0$ .

$4 x^{4} - 36 x^{2} = 0$

Step 1.2.2

Factor the left side of the equation.

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Step 1.2.2.1

Rewrite

$x^{4}$ as

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

$4 {(x^{2})}^{2} - 36 x^{2} = 0$

Step 1.2.2.2

Let

$u = x^{2}$ . Substitute

$u$ for all occurrences of

$x^{2}$ .

$4 u^{2} - 36 u = 0$

Step 1.2.2.3

Factor

$4 u$ out of

$4 u^{2} - 36 u$ .

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Step 1.2.2.3.1

Factor

$4 u$ out of

$4 u^{2}$ .

$4 u (u) - 36 u = 0$

Step 1.2.2.3.2

Factor

$4 u$ out of

$- 36 u$ .

$4 u (u) + 4 u (- 9) = 0$

Step 1.2.2.3.3

Factor

$4 u$ out of

$4 u (u) + 4 u (- 9)$ .

$4 u (u - 9) = 0$

Step 1.2.2.4

Replace all occurrences of

$u$ with

$x^{2}$ .

$4 x^{2} (x^{2} - 9) = 0$

Step 1.2.3

If any individual factor on the left side of the equation is equal to

$0$ , the entire expression will be equal to

$0$ .

$x^{2} = 0$

$x^{2} - 9 = 0$

Step 1.2.4

Set

$x^{2}$ equal to

$0$ and solve for

$x$ .

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Step 1.2.4.1

Set

$x^{2}$ equal to

$0$ .

$x^{2} = 0$

Step 1.2.4.2

Solve

$x^{2} = 0$ for

$x$ .

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Step 1.2.4.2.1

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

$x = \pm \sqrt{0}$

Step 1.2.4.2.2

Simplify

$\pm \sqrt{0}$ .

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Step 1.2.4.2.2.1

Rewrite

$0$ as

$0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 1.2.4.2.2.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Step 1.2.4.2.2.3

Plus or minus

$0$ is

$0$ .

$x = 0$

Step 1.2.5

Set

$x^{2} - 9$ equal to

$0$ and solve for

$x$ .

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Step 1.2.5.1

Set

$x^{2} - 9$ equal to

$0$ .

$x^{2} - 9 = 0$

Step 1.2.5.2

Solve

$x^{2} - 9 = 0$ for

$x$ .

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Step 1.2.5.2.1

Add

$9$ to both sides of the equation.

$x^{2} = 9$

Step 1.2.5.2.2

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

$x = \pm \sqrt{9}$

Step 1.2.5.2.3

Simplify

$\pm \sqrt{9}$ .

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Step 1.2.5.2.3.1

Rewrite

$9$ as

$3^{2}$ .

$x = \pm \sqrt{3^{2}}$

Step 1.2.5.2.3.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 3$

Step 1.2.5.2.4

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

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Step 1.2.5.2.4.1

First, use the positive value of the

$\pm$ to find the first solution.

$x = 3$

Step 1.2.5.2.4.2

Next, use the negative value of the

$\pm$ to find the second solution.

$x = - 3$

Step 1.2.5.2.4.3

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

$x = 3, - 3$

Step 1.2.6

The final solution is all the values that make

$4 x^{2} (x^{2} - 9) = 0$ true.

$x = 0, 3, - 3$

Step 2

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 3

Create intervals around the

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

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

Step 4

Substitute any number from the interval

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

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Step 4.1

Replace the variable

$x$ with

$- 6$ in the expression.

$f'' (- 6) = 4 {(- 6)}^{4} - 36 {(- 6)}^{2}$

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Raise

$- 6$ to the power of

$4$ .

$f'' (- 6) = 4 \cdot 1296 - 36 {(- 6)}^{2}$

Step 4.2.1.2

Multiply

$4$ by

$1296$ .

$f'' (- 6) = 5184 - 36 {(- 6)}^{2}$

Step 4.2.1.3

Raise

$- 6$ to the power of

$2$ .

$f'' (- 6) = 5184 - 36 \cdot 36$

Step 4.2.1.4

Multiply

$- 36$ by

$36$ .

$f'' (- 6) = 5184 - 1296$

Step 4.2.2

Subtract

$1296$ from

$5184$ .

$f'' (- 6) = 3888$

Step 4.2.3

The final answer is

$3888$ .

$3888$

Step 4.3

The graph is concave up on the interval

$(- \infty, - 3)$ because

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

Concave up on

$(- \infty, - 3)$ since

$f'' (x)$ is positive

Concave up on

$(- \infty, - 3)$ since

$f'' (x)$ is positive

Step 5

Substitute any number from the interval

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

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Step 5.1

Replace the variable

$x$ with

$- 2$ in the expression.

$f'' (- 2) = 4 {(- 2)}^{4} - 36 {(- 2)}^{2}$

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

$- 2$ to the power of

$4$ .

$f'' (- 2) = 4 \cdot 16 - 36 {(- 2)}^{2}$

Step 5.2.1.2

Multiply

$4$ by

$16$ .

$f'' (- 2) = 64 - 36 {(- 2)}^{2}$

Step 5.2.1.3

Raise

$- 2$ to the power of

$2$ .

$f'' (- 2) = 64 - 36 \cdot 4$

Step 5.2.1.4

Multiply

$- 36$ by

$4$ .

$f'' (- 2) = 64 - 144$

Step 5.2.2

Subtract

$144$ from

$64$ .

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

Step 5.2.3

The final answer is

$- 80$ .

$- 80$

Step 5.3

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 6

Substitute any number from the interval

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

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Step 6.1

Replace the variable

$x$ with

$2$ in the expression.

$f'' (2) = 4 {(2)}^{4} - 36 {(2)}^{2}$

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

$2$ to the power of

$4$ .

$f'' (2) = 4 \cdot 16 - 36 {(2)}^{2}$

Step 6.2.1.2

Multiply

$4$ by

$16$ .

$f'' (2) = 64 - 36 {(2)}^{2}$

Step 6.2.1.3

Raise

$2$ to the power of

$2$ .

$f'' (2) = 64 - 36 \cdot 4$

Step 6.2.1.4

Multiply

$- 36$ by

$4$ .

$f'' (2) = 64 - 144$

Step 6.2.2

Subtract

$144$ from

$64$ .

$f'' (2) = - 80$

Step 6.2.3

The final answer is

$- 80$ .

$- 80$

Step 6.3

The graph is concave down on the interval

$(0, 3)$ because

$f'' (2)$ is negative.

Concave down on

$(0, 3)$ since

$f'' (x)$ is negative

Concave down on

$(0, 3)$ since

$f'' (x)$ is negative

Step 7

Substitute any number from the interval

$(3, \infty)$ into the second derivative and evaluate to determine the concavity.

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Step 7.1

Replace the variable

$x$ with

$6$ in the expression.

$f'' (6) = 4 {(6)}^{4} - 36 {(6)}^{2}$

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

$6$ to the power of

$4$ .

$f'' (6) = 4 \cdot 1296 - 36 {(6)}^{2}$

Step 7.2.1.2

Multiply

$4$ by

$1296$ .

$f'' (6) = 5184 - 36 {(6)}^{2}$

Step 7.2.1.3

Raise

$6$ to the power of

$2$ .

$f'' (6) = 5184 - 36 \cdot 36$

Step 7.2.1.4

Multiply

$- 36$ by

$36$ .

$f'' (6) = 5184 - 1296$

Step 7.2.2

Subtract

$1296$ from

$5184$ .

$f'' (6) = 3888$

Step 7.2.3

The final answer is

$3888$ .

$3888$

Step 7.3

The graph is concave up on the interval

$(3, \infty)$ because

$f'' (6)$ is positive.

Concave up on

$(3, \infty)$ since

$f'' (x)$ is positive

Concave up on

$(3, \infty)$ since

$f'' (x)$ is positive

Step 8

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

Concave up on

$(- \infty, - 3)$ since

$f'' (x)$ is positive

Concave down on

$(- 3, 0)$ since

$f'' (x)$ is negative

Concave down on

$(0, 3)$ since

$f'' (x)$ is negative

Concave up on

$(3, \infty)$ since

$f'' (x)$ is positive

Step 9