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Math Help - trouble finding critical points on a function, R^2 -> R

  1. #1
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    trouble finding critical points on a function, R^2 -> R

    Q: Find the local maximum and minimum values and saddle point(s) of the function
    f(x, y) = (x^2 + y^2)e^{y^2 - x^2}

    I've found the the function's gradiant vector:

    \triangledown{f} = [-2xe^{y^2-x^2}(x^2+y^2-1), 2ye^{y^2-x^2}(x^2+y^2+1)]

    but I'm having trouble manipulating these equations to find where \triangledown{f}=\vec{0}. BOB says these points are (0, 0) and (\pm{1}, 0), but I'm not sure how to go about proving these are in fact the only points.
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  2. #2
    MHF Contributor

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    Re: trouble finding critical points on a function, R^2 -> R

    An exponential is never 0 so you can divide \nabla f= \vec{0} by e^{y^2- x^2} to get \nabla f= [-2x(x^2+ y^2- 1), 2y(x^2+ y^2+ 1]= \vec{0}.

    Since a vector is 0 if and only if each component is 0, we must have -2x(x^2+ y^2- 1)= 0 and 2y(x^2+ y^2- 1)= 0. And since ab= 0 if and only if either a= 0 or b= 0 we must have x=0, or x^2+ y^2- 1= 0 and y= 0 and x^2+ y^2- 1= 0. What are the solutions to those equations? ( (0, 0) and (\pm1, 0) are solutions but there are two other solutions also.)
    Thanks from sgcb
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