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?

  • $\begingroup$ It should not be possible to get exactly one critical point, since that would imply that you have exactly one extreme point $\endgroup$ – Omnomnomnom Dec 20 '16 at 22:18

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