# Determine critical points of 2 variable functions without 2nd derivative test

I'm new to two variable calculus and having trouble classifying critical points for some functions which the second derivative test isn't really applicable.

E.g.

$$f(x,y) = x^2+y^2+x^2y+4$$

$$f_x=2x+2xy=0$$

$$f_y=2y+x^2=0$$

Critical Point $$(0,0)$$

But I don't know how to determine the nature of this critical point.

Express your objective function as: $$f(x,y)=x^2(1+y)+y^2+4.$$ Note that in the close neighborhood of $$(0,0)$$, the objective function is greater than or equal to $$4$$, hence $$f(0,0)=4$$ is the local minimum.
Two more critical points are $$(\sqrt 2,-1)$$ and $$(-\sqrt 2,-1)$$. For finding the nature we use the hessian matrix as following $$H=\begin{bmatrix}2+2y&2x\\2x& 2y\end{bmatrix}$$The eigenvalues of $$H$$ for $$(\pm \sqrt 2,-1)$$ are $$\lambda (\lambda+2)-8=0\\\to \lambda_1=2,\lambda_2=-4$$therefore the points $$(\pm \sqrt 2,-1)$$ are saddle points. For $$(0,0)$$ we have $$H\succeq 0$$. Hence $$(0,0)$$ is a local minimum.