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Math Help - multivarible extremum point

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    multivarible extremum point

    Let f(x,y)=((x^(2)y^(2))/(x^(2)+y^(2))), classify the behavior of f near the critical point (0,0).
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  2. #2
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    Quote Originally Posted by bobby77 View Post
    Let f(x,y)=((x^(2)y^(2))/(x^(2)+y^(2))), classify the behavior of f near the critical point (0,0).
    I don't know how you are supposed to approach this on your course,
    but the following works:

    Along the x \, and y\, axes the function is a constant equal to zero. Along
    any other ray through the origin we may put y=\lambda x\,, when:

    f(x,\lambda x)=\frac{\lambda^2 x^2}{1+\lambda^2}\,

    which has a quadratic like mininum at x=0\,.

    A surface plot shows what this looks like (see attachment)

    (this assumes that we give the function a value of 0\, at x=y=0\,)

    RonL
    Attached Thumbnails Attached Thumbnails multivarible extremum point-gash.jpg  
    Last edited by ThePerfectHacker; February 7th 2007 at 11:33 AM.
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    Quote Originally Posted by CaptainBlank View Post
    I don't know how you are supposed to approach this on your course,
    but the following works:
    I think he wants it done through the second partials test.
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    Quote Originally Posted by ThePerfectHacker View Post
    I think he wants it done through the second partials test.
    In this case that will be inconclustve (I think, I haven't checked)

    RonL
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    Quote Originally Posted by CaptainBlack View Post
    In this case that will be inconclustve (I think, I haven't checked)

    RonL
    You mean, a saddle point.

    I agree with that, because by different views (planes) the point is both a maximum and minimum. Hence it is neither.
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    Quote Originally Posted by ThePerfectHacker View Post
    You mean, a saddle point.

    I agree with that, because by different views (planes) the point is both a maximum and minimum. Hence it is neither.
    No the determinant of the Hessian is indeterminate. The point is tippling on
    the edge of being a saddle point, but is not quite there.

    The point is a minima.

    RonL
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