Let phi : R^2 --> R be a function given by
phi(x,y) = x^3 + xy + y^3 +1
For which points p =(0,0) , p=(1/3,1/3), p =(-1/3,-1/3) is the subset
phi^-1 ( phi(p)) an imbedded sub-manifold of R^2.
Please justify all your steps and claims. And try to solve the problem in a manner similar to the book mentioned above.