Calculus Examples

f (x) = x^{4} - 6

$f (x) = x^{4} - 6$

Step 1

Find the

x

$x$ values where the second derivative is equal to

0

$0$ .

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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

x^{4} - 6

$x^{4} - 6$ with respect to

x

$x$ is

\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 6]

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

\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 6]

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

Step 1.1.1.2

Differentiate using the Power Rule which states that

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

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

n x^{n - 1}

$n x^{n - 1}$ where

n = 4

$n = 4$ .

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

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

Step 1.1.1.3

Since

- 6

$- 6$ is constant with respect to

x

$x$ , the derivative of

- 6

$- 6$ with respect to

x

$x$ is

0

$0$ .

4 x^{3} + 0

$4 x^{3} + 0$

Step 1.1.1.4

Add

4 x^{3}

$4 x^{3}$ and

0

$0$ .

$f' (x) = 4 x^{3}$

Step 1.1.2

Find the second derivative.

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Step 1.1.2.1

Since

$4$ is constant with respect to

$x$ , the derivative of

$4 x^{3}$ with respect to

$x$ is

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

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

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

$n = 3$ .

$4 (3 x^{2})$

Step 1.1.2.3

Multiply

$3$ by

$4$ .

$f'' (x) = 12 x^{2}$

Step 1.1.3

The second derivative of

$f (x)$ with respect to

$x$ is

$12 x^{2}$ .

$12 x^{2}$

Step 1.2

Set the second derivative equal to

$0$ then solve the equation

$12 x^{2} = 0$ .

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Step 1.2.1

Set the second derivative equal to

$0$ .

$12 x^{2} = 0$

Step 1.2.2

Divide each term in

$12 x^{2} = 0$ by

$12$ and simplify.

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Step 1.2.2.1

Divide each term in

$12 x^{2} = 0$ by

$12$ .

$\frac{12 x^{2}}{12} = \frac{0}{12}$

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Cancel the common factor of

$12$ .

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Step 1.2.2.2.1.1

Cancel the common factor.

$\frac{12 x^{2}}{12} = \frac{0}{12}$

Step 1.2.2.2.1.2

Divide

$x^{2}$ by

$1$ .

$x^{2} = \frac{0}{12}$

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Divide

$0$ by

$12$ .

$x^{2} = 0$

Step 1.2.3

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

Simplify

$\pm \sqrt{0}$ .

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Step 1.2.4.1

Rewrite

$0$ as

$0^{2}$ .

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

Step 1.2.4.2

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

$x = \pm 0$

Step 1.2.4.3

Plus or minus

$0$ is

$0$ .

$x = 0$

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.

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

Step 4

Substitute any number from the interval

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

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Step 4.1

Replace the variable

$x$ with

$- 2$ in the expression.

$f'' (- 2) = 12 {(- 2)}^{2}$

Step 4.2

Simplify the result.

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Step 4.2.1

Raise

$- 2$ to the power of

$2$ .

$f'' (- 2) = 12 \cdot 4$

Step 4.2.2

Multiply

$12$ by

$4$ .

$f'' (- 2) = 48$

Step 4.2.3

The final answer is

$48$ .

$48$

Step 4.3

The graph is concave up on the interval

$(- \infty, 0)$ because

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

Concave up on

$(- \infty, 0)$ since

$f'' (x)$ is positive

Concave up on

$(- \infty, 0)$ since

$f'' (x)$ is positive

Step 5

Substitute any number from the interval

$(0, \infty)$ 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.

$f'' (2) = 12 {(2)}^{2}$

Step 5.2

Simplify the result.

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Step 5.2.1

Raise

$2$ to the power of

$2$ .

$f'' (2) = 12 \cdot 4$

Step 5.2.2

Multiply

$12$ by

$4$ .

$f'' (2) = 48$

Step 5.2.3

The final answer is

$48$ .

$48$

Step 5.3

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 6

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

Concave up on

$(- \infty, 0)$ since

$f'' (x)$ is positive

Concave up on

$(0, \infty)$ since

$f'' (x)$ is positive

Step 7

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