Finding local extrema of $f(x) = x^2 e^{-x}$

Question

I must find and identify (max or min) the local extrema of $f(x) = x^2 e^{-x}$.

This is a simple problem if it was in a calculus exam - but it's not. I'm not sure how to structure the solution for an analysis question.

Answer 1

Hint:

$$\frac{df}{dx}=2x e^{-x}+x^2(-e^{-x})=(2x-x^2)e^{-x}$$

Extrema can occur only when $df/dx=0$, and $e^{-x}$ is always positive.