Investigate local maxima and minima

Question

Question: $x^2 + 4xy + 4y^2 + x^3 +2x^2y+y^4 $ critical point $(0,0)$, $F_{xx}*F_{yy} - (F_{xy})^2$ = $0$ now test failed to conclude any thing. But solution says it has minimum at the origin. Please guide me how to proceed. And lastly, I am unable to find its other critical point. Please help.

Answer 1

Note that

$$ p(x,y) = x^2 + 4 x y + 4 y^2 + x^3 + 2 x^2 y + x^4=\left(\begin{array}{c}y\\ x\\ x^2\end{array}\right)^{\dagger}\left(\begin{array}{ccc}4& 0 & 0\\ 4 & 1 & 0\\ 2 & 1 & 1\end{array}\right)\left(\begin{array}{c}y\\ x\\ x^2\end{array}\right) $$

Note also that matrix $M$

$$ M = \left(\begin{array}{ccc}4& 0 & 0\\ 4 & 1 & 0\\ 2 & 1 & 1\end{array}\right) $$

has eigenvalues $\{4,1,1\}$ then it is positive definite hence $p(x,y)$ has a minimum at the origin.