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Math Help - Find the critical points.

  1. #1
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    Find the critical points.

    f(x,y)=xy+lnx+2y^2

    I know fx=y+(1/x) and fy=x+4y.

    So my question is how do I find the critical points? I know it involves setting the two equations equal to 0, but I'm having a brain lapse. Please help.
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  2. #2
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    As you said, set both equations equal to 0. Then take one of the equations and solve it for a variable. For instance, you could say (1.) y=\frac{-1}{x}. Then plug this in for y in the second equation and you get 0=x-\frac{4}{x}. Solve this equation for x and you get x=\pm2. Plugging in to (1.), you get y(2)=\frac{-1}{2} and y(-2)=\frac{1}{2}.


    When the system involves partial derivatives it seems to make people hesitate, but you can essentially just treat f_{x}=0 and f_{y}=0 like you would any other set of two equations involving two independent variables. It's just the interpretation of the results that makes the system special.
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  3. #3
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    Thanks, this really cleared it up for me.
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