Did I do this right? (Local max/min/partial derivatives)

Let

1) Find polynomial equations for the critical points of f.

I just took both partials and found:

It's asking for polynomial equations, so do I just factor the exponent e out and write the equations left?

2) Find all the second order partial derivatives of f.

As it should be, I found .

3) It can be shown that the critical points of f are (0,0), (1,1) and (-1,-1). Classify these points.

By plugging the points into , I got that (0,0) was a minimum, (1,1) was a maximum, and (-1,-1) was a saddle point.

You probably are wondering what I need help with; I really just wanted reassurance on the first part and wanted to make sure I was getting the concept right. I have a test coming up soon, and I wanted to make sure these review problems were getting done correctly so that I was prepared.