For example with

Well I know there is a critical point at (0,0). So I calculated the second derivatives but they are all 0 here so that doesn't help.

I also tried using the Taylor expansion to show that f(x,y)>f(0,0) or not but that didn't get me anywhere.

Then I tried considering the type of critical point on x=0,y=0,y=-x etc. but again I didn't get anywhere.

Any ideas?