Calculus Examples

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

Find the second derivative.

Tap for more steps...

Find the first derivative.

Tap for more steps...

By the Sum Rule, the derivative of $x^{4} - 6$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 6]$ .

$f' (x) = \frac{d}{d x} (x^{4}) + \frac{d}{d x} (- 6)$

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

$f' (x) = 4 x^{3} + \frac{d}{d x} (- 6)$

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6$ with respect to $x$ is $0$ .

$f' (x) = 4 x^{3} + 0$

Add $4 x^{3}$ and $0$ to get $4 x^{3}$ .

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

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 $4$ to get $12$ .

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

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

$12 x^{2}$

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

Tap for more steps...

Rewrite $12 x^{2}$ as a set of linear factors.

$12 (x^{2}) = 0$

Divide each term 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 of the equation by cancelling the common factors.

Tap for more steps...

Multiply $12$ by $x^{2}$ to get $12 x^{2}$ .

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

Reduce the expression by cancelling the common factors.

Tap for more steps...

Cancel the common factor.

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

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

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

Divide $0$ by $12$ to get $0$ .

$x^{2} = 0$

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

$x = \pm \sqrt{0}$

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

Tap for more steps...

Simplify the right side of the equation.

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.

$x = \pm 0$

$\pm 0$ is equal to $0$ .

$x = 0$

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

Tap for more steps...

Substitute $0$ in $f (x) = x^{4} - 6$ to find the value of $y$ .

Tap for more steps...

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

$f (0) = {(0)}^{4} - 6$

Simplify the result.

Tap for more steps...

Simplify each term.

Tap for more steps...

Remove parentheses around $0$ .

$f (0) = 0^{4} - 6$

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

$f (0) = 0 - 6$

Subtract $6$ from $0$ to get $- 6$ .

$f (0) = - 6$

The final answer is $- 6$ .

$- 6$

The point found by substituting $0$ in $f (x) = x^{4} - 6$ is $(0, - 6)$ . This point can be an inflection point.

$(0, - 6)$

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

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

There are no inflection points and the interval to test the concavity on is $(- \infty, \infty)$ .

$(- \infty, \infty)$

The graph is concave up because the second derivative is positive.

The graph is concave up

Enter YOUR Problem