Question

What are the absolute extrema of `f(x)=(x^4) / (e^x) in[0,oo]`?

Answer 1

The minimum is `0` at `x=0`, and the maximum is `4^4/e^4` at `x=4`

Explanation:

Note first that, on `[0,oo)`, `f` is never negative.

Furthermore, `f(0)=0` so that must be the minimum.

`f'(x) = (x^3(4-x))/e^x` which is positive on `(0,4)` and negative on `(4,oo)`.

We conclude that `f(4)` is a relative maximum. Since the function has no other critical points in the domain, this relative maximum is also the absolute maximum.