Calculus Examples

$12 x^{2} - 100$

Step 1

Write $12 x^{2} - 100$ as a function.

$f (x) = 12 x^{2} - 100$

Step 2

Find the second derivative.

Tap for more steps...

Step 2.1

Find the first derivative.

Tap for more steps...

Step 2.1.1

By the Sum Rule, the derivative of $12 x^{2} - 100$ with respect to $x$ is $\frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [- 100]$ .

$\frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [- 100]$

Step 2.1.2

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

Tap for more steps...

Step 2.1.2.1

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

$12 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 100]$

Step 2.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$12 (2 x) + \frac{d}{d x} [- 100]$

Step 2.1.2.3

Multiply $2$ by $12$ .

$24 x + \frac{d}{d x} [- 100]$

Step 2.1.3

Differentiate using the Constant Rule.

Tap for more steps...

Step 2.1.3.1

Since $- 100$ is constant with respect to $x$ , the derivative of $- 100$ with respect to $x$ is $0$ .

$24 x + 0$

Step 2.1.3.2

Add $24 x$ and $0$ .

$f' (x) = 24 x$

Step 2.2

Find the second derivative.

Tap for more steps...

Step 2.2.1

Since $24$ is constant with respect to $x$ , the derivative of $24 x$ with respect to $x$ is $24 \frac{d}{d x} [x]$ .

$24 \frac{d}{d x} [x]$

Step 2.2.2

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

$24 \cdot 1$

Step 2.2.3

Multiply $24$ by $1$ .

$f'' (x) = 24$

Step 2.3

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

$24$

Step 3

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

Tap for more steps...

Step 3.1

Set the second derivative equal to $0$ .

$24 = 0$

Step 3.2

Since $24 \neq 0$ , there are no solutions.

No solution

Step 4

No values found that can make the second derivative equal to $0$ .

No Inflection Points