Question

If a and b are positive numbers, find the maximum value of
f(x)=xa(1−x)b, 0≤x≤1
Your answer may depend on a and b.

Answer 1

f(x)=ab x(1-x)
f'=ab (1-2x)=0 then x=0.5
f(0)=0
f(1)=0
f(0.5)=ab (0.25)
so if ab>0 then max is f(.5)=0.25 ab
if ab <0 max f(x)=0