Cubic degenerate saddle
= Cubic degenerate saddle
{title2=$F(x,v)=x^3+v^2$}
At the origin the <gradient> vanishes and the <Hessian matrix> is $\operatorname{diag}(0,2)$, which is <positive semidefinite>. Nevertheless, $F(t,0)=t^3$ takes both signs arbitrarily close to the origin. Thus the origin is a <saddle point of a scalar function>. This example shows why a zero <eigenvalue> prevents the ordinary <second-derivative test> from establishing a <local minimum>.