Clearly, this function has a minimum at (0,0) yet, it is not differentiable at (0,0). How then should I show that it has a minimum at (0,0)?
First partial:
1)find partial derivative of z w.r.t. x.
2)equalise that to 0 and solve.
3)similar process for z w.r.t y.
4)you will get a point(s) (x,y) which will give either max or min values.
5) fxx.fyy-(fxy)^2>0 for max or min points
6)fxx>0 implies min value otherwise max value.
fxx=partial derivative for 2 times each time wrt x
fyy=partial derivative for 2 times wrt y
fxy=partial derivative once wrt x then wrt y.