Determine local max., local min., and saddle points of the following function: $4x + 4y + x^2y + xy^2$

Question

Determine local max., local min., and saddle points of the following function: $4x + 4y + x^2y + xy^2$

I know that we take the gradient vector and set it equal to zero to find the critical points. And observe the behavior of the determinant to decide whether a point is a saddle point,local max, or local min.

The gradient that I found was:$<4+2xy+y^2, 4+x^2+2xy>$. When I set them equal to zero, I'm unable to solve the system of equations.

Any help is appreciated.

Answer 1

you have $$4+2xy=-y^2$$ and $$4+2xy=-x^2$$ therefore $$-x^2=-y^2$$ or $$(x-y)(x+y)=0$$