Question

If a and b are positive numbers, find the maximum value of f(x) = x^a(5 − x)^b on the interval 0 ≤ x ≤ 5.

Answer 1

set derivative equal to 0
use product rule
f = x^a f' = ax^(a-1)
g = (5-x)^b g' = -b(5-x)^(b-1)
\[-b x^{a} (5-x)^{b-1} + ax^{a-1}(5-x)^{b} = 0\]
\[bx = a(5-x)\]
\[x = \frac{5a}{a+b}\]
this is when max occurs

Answer 2

you sure ? cuz my assignment says there is an error for the answer you posted