# How do you find interval of increasing, decreasing, concave up and down for f(x) = 2x^3-3x^2-36x-7?

Jul 23, 2018

The intervals of increasing are $x \in \left(- \infty , - 2\right) \cup \left(3 , + \infty\right)$ and the interval of decreasing is $x \in \left(- 2 , 3\right)$. Please see below for the concavities.

#### Explanation:

The function is

$f \left(x\right) = 2 {x}^{3} - 3 {x}^{2} - 36 x - 7$

To fd the interval of increasing and decreasing, calculate the first derivative

$f ' \left(x\right) = 6 {x}^{2} - 6 x - 36$

To find the critical points, let $f ' \left(x\right) = 0$

$6 {x}^{2} - 6 x - 36 = 0$

$\implies$, ${x}^{2} - x - 6 = 0$

$\implies$, $\left(x - 3\right) \left(x + 2\right) = 0$

The critical points are

$\left\{\begin{matrix}x = 3 \\ x = - 2\end{matrix}\right.$

Build a variation chart

$\textcolor{w h i t e}{a a a a}$$x$$\textcolor{w h i t e}{a a a a}$$- \infty$$\textcolor{w h i t e}{a a a a}$$- 2$$\textcolor{w h i t e}{a a a a}$$3$$\textcolor{w h i t e}{a a a a}$$+ \infty$

$\textcolor{w h i t e}{a a a a}$$f ' \left(x\right)$$\textcolor{w h i t e}{a a a a a}$$+$$\textcolor{w h i t e}{a a a a}$$-$$\textcolor{w h i t e}{a a a a}$$+$

$\textcolor{w h i t e}{a a a a}$$f \left(x\right)$$\textcolor{w h i t e}{a a a a a a}$↗$\textcolor{w h i t e}{a a a a}$↘$\textcolor{w h i t e}{a a a a}$↗

The intervals of increasing are $x \in \left(- \infty , - 2\right) \cup \left(3 , + \infty\right)$ and the interval of decreasing is $x \in \left(- 2 , 3\right)$

Calculate the second derivative

$f ' ' \left(x\right) = 12 x - 6$

The point of inflection is when $f ' ' \left(x\right) = 0$

$\implies$, $12 x - 6 = 0$

$\implies$, $x = \frac{1}{2}$

The intervals to consider are $\left(- \infty , \frac{1}{2}\right)$ and $\left(\frac{1}{2} , + \infty\right)$

Build a variation chart

$\textcolor{w h i t e}{a a a a}$$\text{ Interval }$$\textcolor{w h i t e}{a a a a}$$\left(- \infty , \frac{1}{2}\right)$$\textcolor{w h i t e}{a a a a}$$\left(\frac{1}{2} , + \infty\right)$

$\textcolor{w h i t e}{a a a a}$$\text{ sign f''(x) }$$\textcolor{w h i t e}{a a a a a a}$$\left(-\right)$$\textcolor{w h i t e}{a a a a a a a a a a}$$\left(+\right)$

$\textcolor{w h i t e}{a a a a}$$\text{ f(x) }$$\textcolor{w h i t e}{a a a a a a a a a a a}$$\cap$$\textcolor{w h i t e}{a a a a a a a a a a a a}$$\cup$

The function is concave down in the interval $\left(- \infty , \frac{1}{2}\right)$ and concave down in the interval $\left(\frac{1}{2} , + \infty\right)$.

graph{2x^3-3x^2-36x-7 [-26.64, 46.44, 1.46, 38]}