Finding the maximum value given an interval

1. Nov 11, 2011

PandaherO

1. The problem statement, all variables and given/known data

If a and b are positive numbers, find the maximum value of f(x) = x^a(1 - x)^b on the interval 0 ≤ x ≤ 1.?

2. Relevant equations

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

3. The attempt at a solution

I know to set f'(x) = 0 but i'm not sure how i'm supposed to tidy up the f'(x) above..

2. Nov 11, 2011

I like Serena

Either x or (1-x) are solutions, or otherwise you can divide away x^a and (1-x)^b.

3. Nov 11, 2011

SammyS

Staff Emeritus
Set f '(x) equal to zero.

xa b(1-x)(b-1) + (1-x)b ax(a-1) = 0

Now factor out (x(a-1) (1-x)(b-1)).

There is one solution in addition to the two mentioned by I like Serena.

4. Nov 11, 2011

PandaherO

the question also gave a hint that my maximum value may depend on a and b

so (x^(a-1) (1-x)^(b-1)) (a(1-x)+bx) =0

then do I set (a(1-x)+bx)=0?
and solve for x?

i got x=a/(a-b) and it isn't the correct answer...

5. Nov 11, 2011

I like Serena

You made a typo with a minus sign, giving you the wrong result.

Furthermore, if I read your problem correctly, it asks for the maximum value, not the x-coordinate of the maximum value.

You have 3 solutions: x=0, x=1, and this one.
You should substitute those and pick the greatest for the maximum value.

6. Nov 11, 2011

PandaherO

OHH no wonder...*facepalms myself* the minus sign... and yea, the problem wanted the x and y coordinate)
Thank you so much owo