# For what values of x is f(x)=x^3e^x concave or convex?

Jul 17, 2018

The function is concave for $x \in \left(- \infty , - 3 - \sqrt{3}\right) \cup \left(- 3 + \sqrt{3} , 0\right)$ and convex for $x \in \left(- 3 - \sqrt{3} , - 3 + \sqrt{3}\right) \cup \left(0 , + \infty\right)$

#### Explanation:

Calculate the first and second derivative by the product rule

$\left(u v\right) ' = u ' v - u v '$

$f \left(x\right) = {x}^{3} {e}^{x}$

$f ' \left(x\right) = 3 {x}^{2} {e}^{x} + {x}^{3} {e}^{x} = \left({x}^{3} + 3 {x}^{2}\right) {e}^{x}$

$f ' ' \left(x\right) = \left(3 {x}^{2} + 6 x\right) {e}^{x} + \left({x}^{3} + 3 {x}^{2}\right) {e}^{x}$

$= x \left({x}^{2} + 6 x + 6\right) {e}^{x}$

The points of inflections are when, $f ' ' \left(x\right) = 0$

$x \left({x}^{2} + 6 x + 6\right) {e}^{x}$, ${e}^{x} > 0$

$\implies$, $x \left({x}^{2} + 6 x + 6\right) = 0$

$\implies$, $\left\{\begin{matrix}x = 0 \\ {x}^{2} + 6 x + 6 = 0\end{matrix}\right.$

$\implies$, $x = \frac{- 6 \pm \sqrt{36 - 24}}{2} = - 3 \pm \sqrt{3}$

There are $3$ points of inflections

Therefore,

There are $4$ intervals to consider are

${I}_{1} = \left(- \infty , - 3 - \sqrt{3}\right)$ and ${I}_{2} = \left(- 3 - \sqrt{3} , - 3 + \sqrt{3}\right)$ and ${I}_{3} = \left(- 3 + \sqrt{3} , 0\right)$ and ${I}_{4} = \left(0 , + \infty\right)$

Let's consider 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 a a}$${I}_{1}$$\textcolor{w h i t e}{a a a a a}$${I}_{2}$$\textcolor{w h i t e}{a a a a}$${I}_{3}$$\textcolor{w h i t e}{a a a a}$${I}_{4}$

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

$\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}$$\cap$$\textcolor{w h i t e}{a a a a}$$\cup$$\textcolor{w h i t e}{a a a a}$$\cap$$\textcolor{w h i t e}{a a a a}$$\cup$

The function is concave for $x \in \left(- \infty , - 3 - \sqrt{3}\right) \cup \left(- 3 + \sqrt{3} , 0\right)$ and convex for $x \in \left(- 3 - \sqrt{3} , - 3 + \sqrt{3}\right) \cup \left(0 , + \infty\right)$

graph{x^3e^x [-10, 10, -5, 5]}