= Parabolic quartic critical-point classification
For $F(x,y)=(y-cx^2)^2+\alpha x^2-\beta x^4$ with $\beta>0$, its stationary points are the origin and, when $\alpha>0$, $x=\pm\sqrt{\alpha/(2\beta)}$, $y=cx^2$. The origin is a strict local minimum for $\alpha>0$ and a <saddle point of a scalar function> for $\alpha<0$. At $\alpha=0$ its <Hessian matrix> is singular, but the vertical line gives positive values and the parabola $y=cx^2$ negative values, so it remains a degenerate saddle. The other stationary points have Hessian determinant $-8\alpha$ and are saddles.
Back to article page
