Question

For what values of x is `f(x)=(x^2−x)e^x` concave or convex?

Answer 1

The function is convex on `(-oo,-3)uu(0,oo)`.
The function is concave on `(-3,0)`.

Explanation:

First, find the second derivative.

First Derivative

Use product rule.

`f'(x)=(2x-1)e^x+(x^2-x)e^x`

`=>e^x(x^2+x-1)`

Second Derivative

Use product rule again.

`f''(x)=e^x(x^2+x-1)+e^x(2x+1)`

`=>e^x(x^2+3x)=xe^x(x+3)`

Create a sign chart to find when `f''(x)` is positive (convex) and negative (concave). To find the important values on the chart, set `f''(x)=0`.

`xe^x(x+3)=0`

`x=-3,0`

`color(white)(ssssssssss)-3color(white)(ssssssssssssss)0`
`larr-------------rarr`
`color(white)(sssss)+color(white)(ssssssssssss)-color(white)(ssssssssssss)+`

The function is convex on `(-oo,-3)uu(0,oo)`.
The function is concave on `(-3,0)`.

graph{e^x(x^2-x) [-10, 10, -5, 5]}