Lagrange Multipliers: A Review

Basic Problem:

Find the point where f(x,y,z) has its maximum (or minimum) value, given that (x,y,z) satisfies g(x,y,z)=c.

To solve this problem we are going to look for the points where the level surfaces of f and the surface g(x,y,z)=c are tangent. (Why?) We know that the gradient of f and the gradient of g will be parallel whenever the level surfaces are tangent. We also know that two vectors are parallel when one is a multiple of the other. This gives us two equations that we want to solve:

So here's the recipe for using Lagrange multipliers
1. Put the constraint equation into the form g(x,y,z)=c, with c a constant.
2. Find the gradients of f and of g.
3. Solve the equations

4. Plug each of the points you get in step 3 into f and see where f has its maximum (or minimum value).

Return to Quiz 7.