The problem: Find all points on the surface $z=x^3+xy^2$ at which the tangent plane is parallel to the plane $2x+2y+z=0$
So I established $f(x,y,z)=x^3+xy^2-z$ and the normal vector determined from the plane being $<2,2,1>$.
The normal vector for a tangent plane at the point $(x_0,y_0,z_0)$ on the surface is such:
$$\nabla f(x_0,y_0,z_0)=<3x_0^2+y_0^2, 2x_0y_0,-1>=k<2,2,1>$$
Anyway I try to solve this equation, I can't find any points. Is this a possible solution, or am I wrong?