So how much I understood it, it gives us the extremum of a functional.
So we use it in isoperimetric problems too (just dealing with 2D problems).
So there as we know we have to find the curve which has max area under it. But we just extremize the functional (which we create using Lagrange multiplier) and how much I understood, this curve can be either max or min as the equation helps us to find extremum which can be any of the two.
Then how do just consider the result as the curve covering max area as it can be min too.