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

1 Answer
Feb 5, 2016

Answer:

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.