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If a and b are positive numbers, find the maximum value of f(x)=x^a*(1-x)^b, 0 less than or equal to x less than or equal to 1.

Your answer may depend on a and b. What is the maximum value? ...?

Verified Answer :

f '(x) = ax^(a-1)(1 - x)^b - bx^a (1 - x)^(b-1) = x^(a-1)(1 - x)^(b-1) (a(1 - x) - bx). 


If a < 1 or b < 1, this may not be defined at one or both of the end points. f '(x) = 0 if the right most term is zero. 


a - ax - bx = 0 ==> x = a/(a + b). 


Note that this number is both well define and is between 0 and 1. 


f(0) = 0 

f(1) = 0, 

f(a/(a+b)) = a^ab^b/(a + b)^(a + b) <--- the maximum value.


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