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Thread: Critical points

  1. #1
    Junior Member
    Feb 2011

    Critical points

    I have to find the critical points of $\displaystyle f(x,y)=xye^{-x^2-y^2}$ 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 $\displaystyle \nabla f(x,y)=\left( \left( y-2{{x}^{2}}y \right){{e}^{-{{x}^{2}}-{{y}^{2}}}},\left( x-2x{{y}^{2}} \right){{e}^{-{{x}^{2}}-{{y}^{2}}}} \right),$ so we have $\displaystyle y(1-2x^2)=0$ and $\displaystyle x(1-2y^2)=0.$ Critical points are

    $\displaystyle \begin{aligned}
    x=0&,y=0 \\
    x=-\frac{1}{\sqrt{2}}&,y=-\frac{1}{\sqrt{2}} \\
    x=-\frac{1}{\sqrt{2}}&,y=\frac{1}{\sqrt{2}} \\
    x=\frac{1}{\sqrt{2}}&,y=-\frac{1}{\sqrt{2}} \\

    Now the Hessian is

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

    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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