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Thread: Critical points

  1. #1
    Junior Member
    Feb 2011

    Critical points

    I have to find the critical points of f(x,y)=xye^{-x^2-y^2} and decide if they're local maxima or local minima or neither of both. (This one has a name but I don't remember it.)

    Well first we have that \nabla f(x,y)=\left( \left( y-2{{x}^{2}}y \right){{e}^{-{{x}^{2}}-{{y}^{2}}}},\left( x-2x{{y}^{2}} \right){{e}^{-{{x}^{2}}-{{y}^{2}}}} \right), so we have y(1-2x^2)=0 and x(1-2y^2)=0. Critical points are

    \begin{aligned}<br />
   x=0&,y=0 \\ <br />
  x=-\frac{1}{\sqrt{2}}&,y=-\frac{1}{\sqrt{2}} \\ <br />
  x=-\frac{1}{\sqrt{2}}&,y=\frac{1}{\sqrt{2}} \\ <br />
  x=\frac{1}{\sqrt{2}}&,y=-\frac{1}{\sqrt{2}} \\ <br />
  x=\frac{1}{\sqrt{2}}&,y=\frac{1}{\sqrt{2}}. <br />

    Now the Hessian is

    H\,f(x,y)=\left[ \begin{matrix}<br />
   2xy\left( 2{{x}^{2}}-3 \right){{e}^{-{{x}^{2}}-{{y}^{2}}}} & \left( 2{{x}^{2}}-1 \right)\left( 2{{y}^{2}}-1 \right){{e}^{-{{x}^{2}}-{{y}^{2}}}}  \\<br />
   \left( 2{{x}^{2}}-1 \right)\left( 2{{y}^{2}}-1 \right){{e}^{-{{x}^{2}}-{{y}^{2}}}} & 2xy\left( 2{{y}^{2}}-3 \right){{e}^{-{{x}^{2}}-{{y}^{2}}}}  \\<br />
\end{matrix} \right].

    Now my problem is when a point is a maximum or minimum, what are the criteria?
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  2. #2
    MHF Contributor alexmahone's Avatar
    Oct 2008
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