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Math Help - Maximum and Minimum of multivariable calc

  1. #1
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    Maximum and Minimum of multivariable calc

    Find the absolute maximum and minimum values of f on the set D.
    f(x, y) = xy^2 + 9D = {(x, y) | x ≥ 0, y ≥ 0, x^2 + y^2 ≤ 3}
    Last edited by mr fantastic; March 17th 2009 at 08:12 PM. Reason: Clarified the equations in light of a follow-up post by the OP
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  2. #2
    MHF Contributor matheagle's Avatar
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    I think you can just use common sense here.
    The 9 does NOT affect the min and the max.
    I now see that x\ge 0 and y\ge 0,
    so the min is 9 which occurs along the x axis and the y axis.
    Next insert the constraint in the function to get the max along the circle in the first quadrant.
    f(x)=xy^2+9=x(3-x^2)+9=3x-x^3+9.
    Now maximize this wrt x and you get x=1, since we're only in the first quadrant.
    That gives y^2=2 and a max of 11.

    I also got this via LaGrange, to check my work.
    Last edited by matheagle; March 17th 2009 at 08:50 PM.
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  3. #3
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    So the minimum is when x and y are both -sqrt(3)?
    I'm a bit confused.
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  4. #4
    MHF Contributor matheagle's Avatar
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    Quote Originally Posted by acg716 View Post
    So the minimum is when x and y are both -sqrt(3)?
    I'm a bit confused.

    Nope, I was using the entire circle, my bad.
    Since and and y are nonnegative, the smallest this can be is 9 when either is zero.
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  5. #5
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    sorry i meant maximum!
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