Finding critical numbers of a product involving an exponential function

Ok, so the original function is and the problem is to find the derivative with respect to x, respect to y, respect to xx, respect to yy, and respect to xy/yx and find the critical numbers as well as any relative extrema or saddle point if applicable.

I found that the derivative with respect to x is , the derivative with respect to y is , the derivative with respect to xx is , the derivative with respect to yy is and finally the derivative with respect to xy is .

Now, I tried setting from equal to from but that didn't work. I tried doing substitution for and but that ends up being 1=4 which obviously isn't true nor is it a CN...so how would I find the CN's for this problem? I'm sure I'm overlooking something simple, however, I can't find out what that simple thing is.