Calculus Examples

$f (x) = x^{3} - 12 x + 3$

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

Differentiate.

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Step 1.1.1.1.1

By the Sum Rule, the derivative of

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

$x$ is

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

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

Step 1.1.1.1.2

Differentiate using the Power Rule which states that

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

$n x^{n - 1}$ where

$n = 3$ .

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

Step 1.1.1.2

Evaluate

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

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Step 1.1.1.2.1

Since

$- 12$ is constant with respect to

$x$ , the derivative of

$- 12 x$ with respect to

$x$ is

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

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

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

$n = 1$ .

$3 x^{2} - 12 \cdot 1 + \frac{d}{d x} [3]$

Step 1.1.1.2.3

Multiply

$- 12$ by

$1$ .

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

Step 1.1.1.3

Differentiate using the Constant Rule.

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Step 1.1.1.3.1

Since

$3$ is constant with respect to

$x$ , the derivative of

$3$ with respect to

$x$ is

$0$ .

$3 x^{2} - 12 + 0$

Step 1.1.1.3.2

Add

$3 x^{2} - 12$ and

$0$ .

$f' (x) = 3 x^{2} - 12$

Step 1.1.2

Find the second derivative.

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Step 1.1.2.1

By the Sum Rule, the derivative of

$3 x^{2} - 12$ with respect to

$x$ is

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

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

Step 1.1.2.2

Evaluate

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

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Step 1.1.2.2.1

Since

$3$ is constant with respect to

$x$ , the derivative of

$3 x^{2}$ with respect to

$x$ is

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

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

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

$n = 2$ .

$3 (2 x) + \frac{d}{d x} [- 12]$

Step 1.1.2.2.3

Multiply

$2$ by

$3$ .

$6 x + \frac{d}{d x} [- 12]$

Step 1.1.2.3

Differentiate using the Constant Rule.

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Step 1.1.2.3.1

Since

$- 12$ is constant with respect to

$x$ , the derivative of

$- 12$ with respect to

$x$ is

$0$ .

$6 x + 0$

Step 1.1.2.3.2

Add

$6 x$ and

$0$ .

$f'' (x) = 6 x$

Step 1.1.3

The second derivative of

$f (x)$ with respect to

$x$ is

$6 x$ .

$6 x$

Step 1.2

Set the second derivative equal to

$0$ then solve the equation

$6 x = 0$ .

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Step 1.2.1

Set the second derivative equal to

$0$ .

$6 x = 0$

Step 1.2.2

Divide each term in

$6 x = 0$ by

$6$ and simplify.

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Step 1.2.2.1

Divide each term in

$6 x = 0$ by

$6$ .

$\frac{6 x}{6} = \frac{0}{6}$

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Cancel the common factor of

$6$ .

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Step 1.2.2.2.1.1

Cancel the common factor.

$\frac{6 x}{6} = \frac{0}{6}$

Step 1.2.2.2.1.2

Divide

$x$ by

$1$ .

$x = \frac{0}{6}$

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Divide

$0$ by

$6$ .

$x = 0$

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.

$f'' (- 2) = 6 (- 2)$

Step 4.2

Simplify the result.

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Step 4.2.1

Multiply

$6$ by

$- 2$ .

$f'' (- 2) = - 12$

Step 4.2.2

The final answer is

$- 12$ .

$- 12$

Step 4.3

The graph is concave down on the interval

$(- \infty, 0)$ because

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

Concave down on

$(- \infty, 0)$ since

$f'' (x)$ is negative

Concave down on

$(- \infty, 0)$ since

$f'' (x)$ is negative

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.

$f'' (2) = 6 (2)$

Step 5.2

Simplify the result.

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Step 5.2.1

Multiply

$6$ by

$2$ .

$f'' (2) = 12$

Step 5.2.2

The final answer is

$12$ .

$12$

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 down on

$(- \infty, 0)$ since

$f'' (x)$ is negative

Concave up on

$(0, \infty)$ since

$f'' (x)$ is positive

Step 7