How to interpret multiple critical points (from Lagrange multipliers) that all give a maximum value

Question

If I have 6 critical points, 3 of which give the same maximum possible value of a function f(x,y,z), subject to a constraint g=c, is there something more to say about this solution -- or we just simply note that the function attains a max at 3 different locations on the level curve g=c?

Thanks,

Answer 1

I think there is nothing more to say in such a constraint optimisation problem.

However, you can check whether the function actually does what it says it does. For instance, to show that they are maximum points, you convert the function to one-variable equation using the constraint, then differentiate once, and substitute. If the values are -ve, then you're fine.