Okay suppose I form a lagrangian, where the constraint set $S$ is compact so therefore my objective function $f$ attains both maxima and minima.
Now Suppose lagrangian gives me just one point. Then I check the bordered hessian and see that point is a minimum.
Then in other to find the other maximum point? What should I do?
For instance consider a set constraint set $S : \ x^2+y^2≤1$
Or is it even possible that Lagrangian gives me just one point?