Regarding use of Lagrangian as a maximization tool

[Asd1995]

Does lagrangian method give you only the critical points (and we have to use other ways to see if there is a maxima or minima) or does it maximize f(X) in the expression f(x) plus lambda*constraint=L(x)?

[eco aspirant]

Yes lagrangian gives us the critical points (possible candidates for max/min) . To check that those points are indeed a maximum/ minimum you need to apply second order conditions just like we do in case of single variable optimization.

[Asd1995]

But if function is quasi concave are we assured that the points are maxima? Similarly if it is quasi convex will it be a minima?

[eco aspirant]

I think no because the sufficient condition for a maximum to exist requires that the lagrangian function is concave and quasi concavity may or may not imply a concave lagrangian function

[Asd1995]

but if utility is quasi concave then indifference curves are well behaved and convex so solution would be given by the tangency condition?

if utility is quasi convex then ICs are concave and solution will be at the corner

do these make sense?

[Econ17]

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[Asd1995]

Now I'm confused. If utility is quasi concave wouldn't lagrangian critical points always be maxima points (because I'm maximising utility with convex ICs, only critical point is tangency)