Calculus Examples

$f (x) = 4 x^{3} - 48 x$

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 $4 x^{3} - 48 x$ with respect to $x$ is $\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- 48 x]$ .

$\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- 48 x]$

Step 1.1.1.2

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

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Step 1.1.1.2.1

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

$4 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 48 x]$

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 $n = 3$ .

$4 (3 x^{2}) + \frac{d}{d x} [- 48 x]$

Step 1.1.1.2.3

Multiply $3$ by $4$ .

$12 x^{2} + \frac{d}{d x} [- 48 x]$

Step 1.1.1.3

Evaluate $\frac{d}{d x} [- 48 x]$ .

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Step 1.1.1.3.1

Since $- 48$ is constant with respect to $x$ , the derivative of $- 48 x$ with respect to $x$ is $- 48 \frac{d}{d x} [x]$ .

$12 x^{2} - 48 \frac{d}{d x} [x]$

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 $n = 1$ .

$12 x^{2} - 48 \cdot 1$

Step 1.1.1.3.3

Multiply $- 48$ by $1$ .

$f' (x) = 12 x^{2} - 48$

Step 1.1.2

Find the second derivative.

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Step 1.1.2.1

By the Sum Rule, the derivative of $12 x^{2} - 48$ with respect to $x$ is $\frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [- 48]$ .

$\frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [- 48]$

Step 1.1.2.2

Evaluate $\frac{d}{d x} [12 x^{2}]$ .

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Step 1.1.2.2.1

Since $12$ is constant with respect to $x$ , the derivative of $12 x^{2}$ with respect to $x$ is $12 \frac{d}{d x} [x^{2}]$ .

$12 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 48]$

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 $n = 2$ .

$12 (2 x) + \frac{d}{d x} [- 48]$

Step 1.1.2.2.3

Multiply $2$ by $12$ .

$24 x + \frac{d}{d x} [- 48]$

Step 1.1.2.3

Differentiate using the Constant Rule.

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Step 1.1.2.3.1

Since $- 48$ is constant with respect to $x$ , the derivative of $- 48$ with respect to $x$ is $0$ .

$24 x + 0$

Step 1.1.2.3.2

Add $24 x$ and $0$ .

$f'' (x) = 24 x$

Step 1.1.3

The second derivative of $f (x)$ with respect to $x$ is $24 x$ .

$24 x$

Step 1.2

Set the second derivative equal to $0$ then solve the equation $24 x = 0$ .

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Step 1.2.1

Set the second derivative equal to $0$ .

$24 x = 0$

Step 1.2.2

Divide each term in $24 x = 0$ by $24$ and simplify.

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Step 1.2.2.1

Divide each term in $24 x = 0$ by $24$ .

$\frac{24 x}{24} = \frac{0}{24}$

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Cancel the common factor of $24$ .

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Step 1.2.2.2.1.1

Cancel the common factor.

$\frac{24 x}{24} = \frac{0}{24}$

Step 1.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{24}$

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Divide $0$ by $24$ .

$x = 0$

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.

$f'' (- 2) = 24 (- 2)$

Step 4.2

Simplify the result.

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Step 4.2.1

Multiply $24$ by $- 2$ .

$f'' (- 2) = - 48$

Step 4.2.2

The final answer is $- 48$ .

$- 48$

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

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.

$f'' (2) = 24 (2)$

Step 5.2

Simplify the result.

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Step 5.2.1

Multiply $24$ by $2$ .

$f'' (2) = 48$

Step 5.2.2

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

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