# For what values of x is f(x)=(x+2)(x-3)(x+1) concave or convex?

Apr 13, 2017

The function is concave for $x \in \left(- \infty , 0\right)$
The function is convex for $x \in \left(0 , + \infty\right)$

#### Explanation:

We need

$\left(u v w\right) ' = u ' v w + u v ' w + u v w '$

We must calculate the second derivative and determine the sign.

$f \left(x\right) = \left(x + 2\right) \left(x - 3\right) \left(x + 1\right)$

$f ' \left(x\right) = \left(x - 3\right) \left(x + 1\right) + \left(x + 2\right) \left(x + 1\right) + \left(x + 2\right) \left(x - 3\right)$

$= {x}^{2} - 2 x - 3 + {x}^{2} + 3 x + 2 + {x}^{2} - x - 6$

$= 3 {x}^{2} - 7$

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

Therefore,

$f ' ' \left(x\right) = 0$ when $x = 0$

This is the point of inflexion.

We can build a chart

$\textcolor{w h i t e}{a a a a}$$I n t e r v a l$$\textcolor{w h i t e}{a a a a a a}$$\left(- \infty , 0\right)$$\textcolor{w h i t e}{a a a a a a}$$\left(0 , + \infty\right)$

$\textcolor{w h i t e}{a a a a}$$s i g n f ' ' \left(x\right)$$\textcolor{w h i t e}{a a a a a a a}$$-$$\textcolor{w h i t e}{a a a a a a a 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 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}$$\cup$