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Math Help - Finding extreme point in a multivariable function

  1. #1
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    Finding extreme point in a multivariable function

    Hello,
    As far as I know, critical points are the point in which the gradient equals zero or not defined. In the following function Z=X(1+Y)^{1/2}+Y(1+X)^{1/2}, the gradient is not defined for all X\leq-1 or Y\leq-1. How do I prove that only (-1,-1) is an extreme point (maximum)?

    Thanks in advance,
    Michael
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  2. #2
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    Re: Finding extreme point in a multivariable function

    Are you sure (-1, -1) is the maximum of z? For a function that is continuous on a closed interval we know that the function will achieve its extrema at some f(a) and f(b) with a and b contained in the closed interval, and f'(a)=f'(b)=0. How can you get the interval on which z is continuous to be closed?

    If you are still struggling, a good first step would be to reconsider your statement that z(-1, -1) does not exist.
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  3. #3
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    Re: Finding extreme point in a multivariable function

    Quote Originally Posted by letepsilonbenegative View Post
    Are you sure (-1, -1) is the maximum of z? For a function that is continuous on a closed interval we know that the function will achieve its extrema at some f(a) and f(b) with a and b contained in the closed interval, and f'(a)=f'(b)=0. How can you get the interval on which z is continuous to be closed?

    If you are still struggling, a good first step would be to reconsider your statement that z(-1, -1) does not exist.
    I did not say that z(-1,-1) does bot exist, it does. In addition, it is a maximum point because z(-1,-1)=0, and the points in the small neighborhood are either negative or not defined.
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  4. #4
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    Re: Finding extreme point in a multivariable function

    The function itself is not defined for x< -1 or y< -1 so there can't be a max or min there.
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