A local minimum of a quasi-convex function is a global minimum. At a critical point that is not a local minimum and where the function is $C^2$, the Hessian matrix must be singular.

A local minimum of a quasi-convex function is a global minimum. At a critical point where the function is $C^2$, the Hessian matrix is positive semidefinite; if such a critical point is not a local minimum, the Hessian matrix must be singular there. An example of this is the function $f(x) = x^3$.