I have the following question: of course we know that in Lagrange multiplication method we can use bordered Hessian to check if some points are local extremum. But do this method, this bordered Hessian, says something about the fact that the minimum/maximum does not exist? I mean if the special properties of this Hessian are not satisfied then do we know that extremum does not exists or this is only one way implication?
More about on https://en.wikipedia.org/wiki/Hessian_matrix#Bordered_Hessian
Thank you!