What are all the critical points of the function $f(x,y) = (x^2 + y^2) e^{y^2 - x^2}$

Question

I know that I must find the partial derivatives with respect to $x$ and $y$, then set them equal to zero.

So:

$$f_x = 2xe^{y^2 - x^2} (1 - (x^2 + y^2))$$ $$f_y = 2ye^{y^2 - x^2} (1 + (x^2 + y^2))$$

I can't seem to figure out how I can find the critical points form here.

Answer 1

The critical points are where partial derivatives are 0, right?

Hint:

So we have a product of numbers. A product which is equal to 0 is when at least one of the factors is 0. Also we can use that the exponential function is never $0$ for any input.

Answer 2

If $a.b=0$ , then either $a =0$ or $b=0$ or $a=b=0$.

Now from $f_x=0$, we have $x=0$ or $x^2+y^2 =1$. From $f_y=0$, we have $y=0$ or $x^2+y^2 =-1$.

EDIT : critical points would be, $(0,0)$, $(1,0)$, $(0, i)$