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Math Help - 2 Variables Calculus Help

  1. #1
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    2 Variables Calculus Help

    Let f(x,y)=xy(1-x-y).

    Prove that max{f(x,y) | x>0 , y>0} = 1/27 .

    I want to use the statement that sais that if f is continous in a compactic set A then f gets its minimum and maximum in A.
    I'm not so sure on which compactic set I need to activate this statement...
    Hope you'll be able to help me


    Thanks in advance
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  2. #2
    MHF Contributor matheagle's Avatar
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    the only critical point is (1/3,1/3) in that region.

    Just maximize f(x,y)=xy-x^2y-xy^2

    The two partial derivatives lead to

    2x+y=1 and x+2y=1, which leads you to that one point in the first quadrant.
    Then plug in and see that f(1/3,1/3)=1/27.
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  3. #3
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    I agree that we can tell that the partial deriatives are zero at this point...And that we can find that f(1/3,1/3)=1/27 ... But how can we prove that this point is the absolute maximum in the first quadrant ? I mean... How can we prove that there is no point such as  f(x,y) > 1/27 in the first quadrant...??

    I hope my question isn't too vague... Hope you'll be able to help me


    Thanks a lot!
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  4. #4
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    First derivatives are zero, then
    df=(1/2)(f_xx*dx^2+2*f_xy*dxdy+f_yy*dy^2)
    f_xx=-2/3
    f_yy=-2/3
    f_xy=-1/3
    df=-1/3(dx^2+dxdy+dy^2)
    dx=rcos(a)
    dy=rsin(a)
    df=(-1/3)r^2(1+cos(a)sin(a))=(-1/3)r^2(1+sin(2a)/2)<0
    this point is absolute maximum.
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