Degenerate critical point ($\nabla f(x_{\ast })=0,\det H_f(x_{\ast })=0$)

A critical point whose Hessian matrix is a singular matrix. The second-derivative test does not completely classify such a point. Higher-order terms can produce a local minimum, a local maximum, or a saddle point of a scalar function.

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.