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Finding the maximum value given an interval

  1. Nov 11, 2011 #1
    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. jcsd
  3. Nov 11, 2011 #2

    I like Serena

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    Either x or (1-x) are solutions, or otherwise you can divide away x^a and (1-x)^b.
     
  4. Nov 11, 2011 #3

    SammyS

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    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.
     
  5. Nov 11, 2011 #4
    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...
     
  6. Nov 11, 2011 #5

    I like Serena

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    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.
     
  7. Nov 11, 2011 #6
    OHH no wonder...*facepalms myself* the minus sign... and yea, the problem wanted the x and y coordinate)
    Thank you so much owo
     
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