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

Feb 5, 2016

The minimum is $0$ at $x = 0$, and the maximum is ${4}^{4} / {e}^{4}$ at $x = 4$

#### Explanation:

Note first that, on $\left[0 , \infty\right)$, $f$ is never negative.

Furthermore, $f \left(0\right) = 0$ so that must be the minimum.

$f ' \left(x\right) = \frac{{x}^{3} \left(4 - x\right)}{e} ^ x$ which is positive on $\left(0 , 4\right)$ and negative on $\left(4 , \infty\right)$.

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