Min max using partial derivatives

[StephenDoty]

(x^2 + y^2)*e^(y^2 - x^2)
I am having trouble finding the critical points.

Fx=2xe^(y^2-x^2)(1-x^2-y^2)=0
Fy=2ye^(y^2 - x^2)(1+x^2 +y^2)=0

or
0=x(1-x^2-y^2)
0=y(1+x^2+y^2)

now, finding all the roots is giving me trouble. x and y obviously = 0, but I am unsure how to move forward to get the rest of the zeros, therefore giving me the critical points. Any help would be appreciated. Thanks.

Stephen

 

[Mark44]

F_x = 0 iff x = 0 or x^2 + y^2 = 1
That is, F_x = 0 when x = 0 or when (x, y) is a point on the unit circle centered at (0, 0).

F_y = 0 iff y = 0 (1 + x^2 + y^2 >= 1 for all real (x, y) )
So both partials are 0 simultaneously for (0, 0), (1, 0), or (-1, 0).