# Finding the maximum value given an interval

## Homework Statement

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.?

## Homework Equations

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

## 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..

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

SammyS
Staff Emeritus
Homework Helper
Gold Member

## Homework Statement

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.?

## Homework Equations

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

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.

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...

I like Serena
Homework Helper
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.

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.

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