Consider the function on $\mathbb{R}^2$: \begin{equation*} V(x,y) = (x^2+y^2)^2-a(x^2+y^2)+a^2/4 \end{equation*} with $a$ real. I want to decide weather the critical points are max, min or saddle points.
We can see that the point $(0,0)$ and the points on the circumference $x^2+y^2 = \frac{a}{2}$ are critical points. If we calculate the hessian matrix we have \begin{equation*} \mathcal{H} = \begin{pmatrix} 4(3x^2+y^2)-2a & 8xy \\ 8xy & 4(x^2+3y^2)-2a \\ \end{pmatrix} \end{equation*} and \begin{equation*} \det \mathcal{H} = 16 \cdot \left[ 3 (x^2+y^2)^2-2a(x^2+y^2)+\frac{a^2}{4} \right] \end{equation*} Then $\det \mathcal{H}_{(0,0)} = 4a^2$ so the point $(0,0)$ is max if $a>0$ and min if $a<0$. And by deduction a min for $a=0$.
How can I decide the point on the circumference $x^2+y^2 = \frac{a}{2}$?
