Type of extremum in Lagrange Multiplier Method Let's say I'm given that there are rectangular boxes all of which have a constant surface area, say **S**. Now, I want to find the box with either the maximum volume or the minimum volume. If I apply Lagrange Multiplier Method and solve the set of equations, I'll get a solution. But how do I know whether it is for the maximum volume or minimum volume? Is there a way to say what type of extremum we get after applying Lagrange Method, if we don't already know the type of extremum?

The minimum is obviously the degenerate case some-lenght = 0. The maximum obviously exists (why?). What can be your solution?