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Math Help - maximum values

  1. #1
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    maximum values

    I am trying to find the set of points for which the maximum value of the curvature

    K=\fract{3} { (x^{-2}+y^{-2}+z^{-2})^{-2} }

    occurs for the function xyz=1

    I have equated the gradients and subject to the restriction of the function xyz=1 i only get one solution the point (1,1,1), since when i solve the equations i get x^3=y^3=z^3

    but i am not sure if i have done this correctly.Anyway any help would be appreciated.
    Last edited by CaptainBlack; September 28th 2010 at 05:59 AM.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by ulysses123 View Post
    I am trying to find the set of points for which the maximum value of the curvature

    K=\fract{3} { (x^{-2}+y^{-2}+z^{-2})^{-2} }

    occurs for the function xyz=1

    I have equated the gradients and subject to the restriction of the function xyz=1 i only get one solution the point (1,1,1), since when i solve the equations i get x^3=y^3=z^3

    but i am not sure if i have done this correctly.Anyway any help would be appreciated.
    There is something wrong with your LaTeX you have a \frac{}{} which does not render.

    Other than that the AM-GM inequality will (probably - since the question has some ambiguity due to the LaTeX problem) give your result.

    CB
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  3. #3
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    Yes i noticed that, but since this is the correct equation i didn't alter it.
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  4. #4
    Grand Panjandrum
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    Quote Originally Posted by CaptainBlack View Post
    There is something wrong with your LaTeX you have a \frac{}{} which does not render.

    Other than that the AM-GM inequality will (probably - since the question has some ambiguity due to the LaTeX problem) give your result.

    CB
    The maximum of  K occurs when: x^{-2}+y^{-2}+z^{-2} is at a minimum.

    The AM-GM inequality tells us that:

    x^{-2}+y^{-2}+z^{-2}\ge 3 \root 3 \of {x^{-2}y^{-2}z^{-2}} =3

    with equality only when x^{-2}=y^{-2}=z^{-2} , or x=y=z=1.

    So the maximum of  Z occurs at x=y=z=1

    CB
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