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[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.