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Thread: 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:

    $\displaystyle
    f(x,y)=x^3+y^3-3x-3y
    $

    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:

    $\displaystyle
    f(x,y)=x^3+y^3-3x-3y
    $

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

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

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

    This implies that $\displaystyle x^2=1$ and $\displaystyle y^2=1$

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

    So our critical points are $\displaystyle (-1,1)$, $\displaystyle (-1,-1)$, $\displaystyle (1,-1)$, $\displaystyle (1,1)$.

    Now apply the second partials test:

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

    Recall that if:

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

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

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

    $\displaystyle D=0$, then no conclusion can be drawn.

    Try to take it from here.

    --Chris
    Last edited by Chris L T521; Sep 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 $\displaystyle (-1,-1),~(-1,1),~(1,-1),~(1,1)$. 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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