You have the right idea. You want to equate the partials to zero and solve the resulting system to find the critical points:
Solve this system, we find:
Now you want to use the second partials test for relative extrema:
Let be a critical point of and suppose and are continuous in a rectangular region containing .
i) If and , then is a relative minimum.
ii) If and , then is a relative maximum.
iii) If , then is not an extremum.
iv) If , then no conclusion can be drawn concerning a relative extremum.
Using this test, what conclusions can be drawn about the critical points?