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Math Help - Critical Points

  1. #1
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    Critical Points

    Hi everyone!

    I understand critical points very well, but I am just unsure... I'm doing a practice final with no answers, and there are a lot of points attributed to this question:

    Find all critical points of f(x,y) = 3xy - x^2y - xy^2

    I found (0,0) and (1,1) - are there others? Thank you very much.
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  2. #2
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    Quote Originally Posted by Volcanicrain View Post
    Hi everyone!

    I understand critical points very well, but I am just unsure... I'm doing a practice final with no answers, and there are a lot of points attributed to this question:

    Find all critical points of f(x,y) = 3xy - x^2y - xy^2

    I found (0,0) and (1,1) - are there others? Thank you very much.

    f_x=3y-2xy-y^2=0 \Longrightarrow -y(y+2x-3)=0

    f_y=3x-x^2-2xy=0 \Longrightarrow -x(x+2y-3)=0

    Solving the above system we get: (0,0)\,,\,y+2x=3=x+2y\Longrightarrow y-x=0\Longrightarrow y=x\,\,\,and\,\,\,x+2x=3\Longrightarrow x=1

    Thus, indeed (0,0)\,,\,(1,1) are the only function's critical points.

    Tonio
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