Local minimum with eigenvalues $\lambda_1 = 1$ and $\lambda_2 = 3$

Local maximum with eigenvalues $\lambda_1 = 1$ and $\lambda_2 = 3$

Saddle point with eigenvalues $\lambda_1 = 1$ and $\lambda_2 = -3$

Degenerate critical point where the Hessian is singular