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Math Help - maxima, minima, saddle point problem

  1. #1
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    maxima, minima, saddle point problem

    Find and classify all local maxima, minima and saddle points for the surface:

     <br />
f(x,y)=x^3+y^3-3x-3y<br />

    last question for my assignment
    thanks!
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by Dr Zoidburg View Post
    Find and classify all local maxima, minima and saddle points for the surface:

     <br />
f(x,y)=x^3+y^3-3x-3y<br />

    last question for my assignment
    thanks!
    Find where \frac{\partial f}{\partial x}=0 and \frac{\partial f}{\partial y}=0

    \frac{\partial f}{\partial x}=3x^2-3
    \frac{\partial f}{\partial y}=3y^2-3

    So we see that we have critical points when 3x^2-3=0 and 3y^2-3=0

    This implies that x^2=1 and y^2=1

    So we see that x=\pm1 and y=\pm1

    So our critical points are (-1,1), (-1,-1), (1,-1), (1,1).

    Now apply the second partials test:

    D=f_{xx}(x_0,y_0)f_{yy}(x_0,y_0)-f_{xy}^2(x_0,y_0)

    Recall that if:

    D>0~\text{and}~f_{xx}(x_0,y_0)>0, then f has a relative minimum at (x_0,y_0)

    D>0~\text{and}~f_{xx}(x_0,y_0)><0, then f has a relative maximum at (x_0,y_0)

    D<0, then f has a saddle point at (x_0,y_0)

    D=0, then no conclusion can be drawn.

    Try to take it from here.

    --Chris
    Last edited by Chris L T521; September 13th 2008 at 11:21 AM. Reason: deadly mistake XD
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    thanks for that!
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  4. #4
    Rhymes with Orange Chris L T521's Avatar
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    I checked over my work and noticed a slight error...the critical points are really (-1,-1),~(-1,1),~(1,-1),~(1,1). There shouldn't be any zeros in the critical points. My mistake [shouldn't be math when you're dead tired ].

    Here's the graph:



    --Chris
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