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Math Help - Critical points of functions of two variables

  1. #1
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    Critical points of functions of two variables

    Show that
    <br />
\Delta=f_{xx}f_{yy}-(f_{xy})^2<br />
    is zero at the origin. Then classify this critical point by visualizing the surface
    <br />
f(x, y) = x^3 + y^3<br />

    I have proved the first condition, but I am not sure how to proceed with the second part to classify this critical point by visualizing the surface.

    Any help will be greatly appreciated!
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  2. #2
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    You should at least be able to visualise what the cross-sections of the surface along the coordinate axes look like. On the x-axis, for example, where y=0, f(x,y) is equal to x^3, and you surely know what the graph of that curve looks like, with a point of inflection at the origin.
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    From what I am understanding the critical point then should be a saddle point, am I right?
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  4. #4
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    Quote Originally Posted by hasanbalkan View Post
    From what I am understanding the critical point then should be a saddle point, am I right?
    No, because there is no direction in which it is either a local maximum or a local minimum. It is neither a max nor a min nor a saddle point. It is a point of inflection. In fact, the function f(x,y) is identically zero along the line y=x. It is positive above that line and negative below it.
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  5. #5
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    Thank you a lot for the reply. I did some research and according to the results I found this critical point is called a stationary point of inflection also known as a saddle point.

    Saddle Point -- from Wolfram MathWorld

    Once again, great help on this forum...
    Last edited by hasanbalkan; November 14th 2008 at 06:57 PM. Reason: typo
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