I understand that for Lagrange multipliers, [itex] ∇f = λ∇g [/itex] And that you can use this to solve for extreme values. I have a set of questions because I don't understand these on a basic level. 1. How do you determine whether it is a max, min, or saddle point, especially when you only get one extreme value/critical point. 2. Why does this work? Could someone help paint a picture or better description of why you can find these critical points using Lagrange multipliers? 3. Is there a more significant purpose for Lagrange multipliers? You may use any problem where you have either [itex] f(x,y) [/itex] with the constraint [itex] g(x,y) = k [/itex] or with [itex] f(x,y,z) [/itex] with the constraint [itex] g(x,y,z) = k [/itex] Both would be preferred; The former preferred for a basic understanding, the latter for a more complex example. Any help would be appreciated, I have a quiz and test over it this week.