Cubic degenerate saddle ($F(x,v)=x^3+v^2$)

At the origin the gradient vanishes and the Hessian matrix is $\mathrm{diag}(0,2)$, 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.