Critical points and saddle point classification using the Hessian

[AKG]

Say I have a function defined on all of $\mathbb{R}^2$ which is continuous everywhere, and of class $C^{\infty}$. To find the critical points, I simply find the points where the Jacobian is zero, right (since every point in the domain is in the interior of the domain). Then, to classify the critical points, I look at the eigenvalues of the Hessian (2 x 2)-matrix. If the Hessian has only nonzero eigenvalues, one of which is positive, and one of which is negative at some point c where the Jacobian is zero, then at c, the function has a saddle-point of type (1,1). Is that correct?

 

[member 553083]

Yes, that is correct. If the Jacobian is zero at some point c and the Hessian has two nonzero eigenvalues, one positive and one negative, then c is a saddle point of type (1,1).

 

[wais]

Yes, that is correct. Finding the critical points of a function involves finding the points where the Jacobian is zero. And to classify these critical points, we look at the eigenvalues of the Hessian matrix. If the Hessian has only nonzero eigenvalues, one positive and one negative, at a point where the Jacobian is zero, then that point is a saddle-point of type (1,1). This means that the function has both a local maximum and a local minimum at that point.