Don't know how to solve

Question

If a and b are positive numbers, find the maximum value of f(x)=x^a(7−x)^b on the interval 0≤x≤7

Answer 1

f(x) = x^a(7-x)^b

on the interval from 0 to 7

a,b>0

f'(x) = x^a(b)(7-x)^(b-1) + (7-x)^b(a)(x^(a-1)

set=0, solve for x = 7a/(a+b)

check the end points, x=0,7

f(0)=0=f(7)

check the local extremum:

f(a,b) = f(0,0)=7(0)/(0+0) = 0/0 = Undefined

no real max value of f

my best guess

Answer 2

desmos.com/calculator/nnqqj9q8i1