Calculus Examples

$- 2 x^{2} - 6 x - 2$

Step 1

Write $- 2 x^{2} - 6 x - 2$ as a function.

$f (x) = - 2 x^{2} - 6 x - 2$

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 $- 2 x^{2} - 6 x - 2$ with respect to $x$ is $\frac{d}{d x} [- 2 x^{2}] + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [- 2]$ .

$\frac{d}{d x} [- 2 x^{2}] + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [- 2]$

Step 2.1.2

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

Tap for more steps...

Step 2.1.2.1

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

$- 2 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [- 2]$

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 $n = 2$ .

$- 2 (2 x) + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [- 2]$

Step 2.1.2.3

Multiply $2$ by $- 2$ .

$- 4 x + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [- 2]$

Step 2.1.3

Evaluate $\frac{d}{d x} [- 6 x]$ .

Tap for more steps...

Step 2.1.3.1

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 x$ with respect to $x$ is $- 6 \frac{d}{d x} [x]$ .

$- 4 x - 6 \frac{d}{d x} [x] + \frac{d}{d x} [- 2]$

Step 2.1.3.2

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

$- 4 x - 6 \cdot 1 + \frac{d}{d x} [- 2]$

Step 2.1.3.3

Multiply $- 6$ by $1$ .

$- 4 x - 6 + \frac{d}{d x} [- 2]$

Step 2.1.4

Differentiate using the Constant Rule.

Tap for more steps...

Step 2.1.4.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$- 4 x - 6 + 0$

Step 2.1.4.2

Add $- 4 x - 6$ and $0$ .

$f' (x) = - 4 x - 6$

Step 2.2

Find the second derivative.

Tap for more steps...

Step 2.2.1

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

$\frac{d}{d x} [- 4 x] + \frac{d}{d x} [- 6]$

Step 2.2.2

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

Tap for more steps...

Step 2.2.2.1

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

$- 4 \frac{d}{d x} [x] + \frac{d}{d x} [- 6]$

Step 2.2.2.2

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

$- 4 \cdot 1 + \frac{d}{d x} [- 6]$

Step 2.2.2.3

Multiply $- 4$ by $1$ .

$- 4 + \frac{d}{d x} [- 6]$

Step 2.2.3

Differentiate using the Constant Rule.

Tap for more steps...

Step 2.2.3.1

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

$- 4 + 0$

Step 2.2.3.2

Add $- 4$ and $0$ .

$f'' (x) = - 4$

Step 2.3

The second derivative of $f (x)$ with respect to $x$ is $- 4$ .

$- 4$

Step 3

Set the second derivative equal to $0$ then solve the equation $- 4 = 0$ .

Tap for more steps...

Step 3.1

Set the second derivative equal to $0$ .

$- 4 = 0$

Step 3.2

Since $- 4 \neq 0$ , there are no solutions.

No solution

Step 4

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

No Inflection Points