Calculus Examples

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

Step 1

Find the second derivative.

Tap for more steps...

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

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

Find the second derivative.

Tap for more steps...

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

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

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$f'' (x) = 4 (3 x^{2})$

Multiply $3$ by $4$ .

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

The second derivative of $f (x)$ with respect to $x$ is $12 x^{2}$ .

$12 x^{2}$

Step 2

Set the second derivative equal to $0$ then solve the equation $12 x^{2} = 0$ .

Tap for more steps...

Set the second derivative equal to $0$ .

$12 x^{2} = 0$

Divide each term in $12 x^{2} = 0$ by $12$ and simplify.

Tap for more steps...

Divide each term in $12 x^{2} = 0$ by $12$ .

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

Simplify the left side.

Tap for more steps...

Cancel the common factor of $12$ .

Tap for more steps...

Cancel the common factor.

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

Divide $x^{2}$ by $1$ .

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

Simplify the right side.

Tap for more steps...

Divide $0$ by $12$ .

$x^{2} = 0$

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

$x = \pm \sqrt{0}$

Simplify $\pm \sqrt{0}$ .

Tap for more steps...

Rewrite $0$ as $0^{2}$ .

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

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

$x = \pm 0$

Plus or minus $0$ is $0$ .

$x = 0$

Step 3

Find the points where the second derivative is $0$ .

Tap for more steps...

Substitute $0$ in $f (x) = x^{4}$ to find the value of $y$ .

Tap for more steps...

Replace the variable $x$ with $0$ in the expression.

$f (0) = {(0)}^{4}$

Simplify the result.

Tap for more steps...

Raising $0$ to any positive power yields $0$ .

$f (0) = 0$

The final answer is $0$ .

$0$

The point found by substituting $0$ in $f (x) = x^{4}$ is $(0, 0)$ . This point can be an inflection point.

$(0, 0)$

Step 4

Split $(- \infty, \infty)$ into intervals around the points that could potentially be inflection points.

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

Step 5

Substitute a value from the interval $(- \infty, 0)$ into the second derivative to determine if it is increasing or decreasing.

Tap for more steps...

Replace the variable $x$ with $- 0.1$ in the expression.

$f'' (- 0.1) = 12 {(- 0.1)}^{2}$

Simplify the result.

Tap for more steps...

Raise $- 0.1$ to the power of $2$ .

$f'' (- 0.1) = 12 \cdot 0.01$

Multiply $12$ by $0.01$ .

$f'' (- 0.1) = 0.12$

The final answer is $0.12$ .

$0.12$

At $- 0.1$ , the second derivative is $0.12$ . Since this is positive, the second derivative is increasing on the interval $(- \infty, 0)$ .

Increasing on $(- \infty, 0)$ since $f'' (x) > 0$

Step 6

Substitute a value from the interval $(0, \infty)$ into the second derivative to determine if it is increasing or decreasing.

Tap for more steps...

Replace the variable $x$ with $0.1$ in the expression.

$f'' (0.1) = 12 {(0.1)}^{2}$

Simplify the result.

Tap for more steps...

Raise $0.1$ to the power of $2$ .

$f'' (0.1) = 12 \cdot 0.01$

Multiply $12$ by $0.01$ .

$f'' (0.1) = 0.12$

The final answer is $0.12$ .

$0.12$

At $0.1$ , the second derivative is $0.12$ . Since this is positive, the second derivative is increasing on the interval $(0, \infty)$ .

Increasing on $(0, \infty)$ since $f'' (x) > 0$

Step 7

An inflection point is a point on a curve at which the concavity changes sign from plus to minus or from minus to plus. There are no points on the graph that satisfy these requirements.

No Inflection Points