Question

Is `f(x)=-xe^x+x^3-x^2` concave or convex at `x=-2`?

Answer 1

Since `f''(-2)<0`, the function is concave

Explanation:

First, you would calculate the first and the second derivatives:

`f'(x)=(-1*e^x+(-x)*e^x)+3x^2-2x`

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

`f''(x)=(-e^x-e^x-xe^x)+6x-2`

`=-2e^x-xe^x+6x-2`

Then you would calculate

`f''(-2)=-2e^(-2)-(-2)e^(-2)+6(-2)-2`

`=cancel(-2e^-2)cancel(+2e^-2)-12-2`

`-14<0`

Since `f''(-2)<0`, the function is concave
graph{-xe^x+x3-x^2 [-10.035, 9.965, -7.32, 2.68]}