Using Lagrange Multipliers Better?

Question

My question is that "Can we use lagrange multipliers to solve any problem where we need to find local/global minima/maxima?" Also, "Is it much easier to use lagrange multipliers, and if so what cases would it be better to do the conventional method of finding $f_x,f_y$ to find critical points, and then finding max/min?"

Answer 1

The method of Lagrange's multipliers is a theorem with a few assumptions. Check it out on your favorite advanced calculus book. But, what is even more important, this method works for constrained critical points and should not be seen as an alternative to solving $\nabla f=0$.

Answer 2

The Lagrange multipliers method is a procedure to find the maximum and minimum of functions of multiple variables subject to restrictions, so it is not a universal method since it works with some conditions or assumptions