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System of equations (multivariable second derivative test)

  1. Jul 11, 2014 #1
    I am doing critical points and using the second derivative test (multivariable version)

    1. The problem statement, all variables and given/known data
    [tex]f(x,y) = (x^2+y^2)e^{x^2-y^2}[/tex]
    Issue I am having is with the system of equations to get the critical points from partial wrt x, wrt y

    3. The attempt at a solution
    [tex]f_{x} = 2xe^{x^2-y^2}+2xe^{x^2-y^2}(x^2+y^2)[/tex]
    [tex]f_{y} = 2ye^{x^2-y^2}-2ye^{x^2-y^2}(x^2+y^2)[/tex]

    It is pretty easy to see (0,0) makes both equal to zero

    I know +/- 1 = y and x = 0 is a solution as well.

    [tex](x^2+y^2)[/tex] is never negative let alone zero [over the reals]
    [tex]e^{x^2-y^2}[/tex] won't be zero, it will get infinitesimally close to zero for values of square of y > square of x

    and 2x+2x=0 only when x = 0

    and 2y-2y=0 for all real values of y

    how do I arrive at y = +/- 1 [ I can see it in the line just above but then I can use any real value for y]
     
  2. jcsd
  3. Jul 11, 2014 #2
    You should note that you have
    \begin{equation*}
    \frac{\partial f} {\partial x} = 2x e^{x^2 - y^2}(1 + x^2 + y^2)
    \end{equation*}

    and similarly for ## \frac{\partial f} {\partial y} ##.
     
  4. Jul 11, 2014 #3
    thank you
     
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