Local maxima/minima of a Multivariable function

Question

I'm given the function $f(x,y)$ $=$ $x^3-3x+y^4-2y^2$ and asked to find the critical points and determine whether they are a max, min, or saddle. I have a question on finding the critical points: I set $f_x$ and $f_y$ equal to 0, and get x=0,1,-1 and y=0,1,-1. I am just wondering how I know which x corresponds with which y in determining the critical points, I know how to test them in later steps. Any help is appreciated.

Answer 1

Since the partial derivatives are $$f_x(x,y)=3x^2-3=3(x^2-1)\quad \text{and}\quad f_y(x,y)=4y^3-4y=4y(y^2-1)$$ it follows that the points $(-1,-1),\;(-1,0),\;(-1,1),\;(1,-1),\;(1,0)\;\text{and}\;(1,1)$ are all critical points.