Lagrange Multipliers

Definition:

• To find the maximum and minimum values of $f(x,y,z)$ subject to the constraint $g(x,y,z)=k$
  1. Find sets of values of $x,y,z$ and $\lambda$ such that:
     • $\nabla f(x,y,z)=\lambda \nabla g(x,y,z)$
       • same tangent line in the level curves
     • $g(x,y,z)=k$
  2. Evaluate all points from 1. The largest of these values is the maximum value of $f$
• For two constraints:
  • To find the max and min values of $f(x,y,z)$ subject to the constraints $g(x,y,z)=k$ and $h(x,y,z)=l$
    • $\nabla f(x,y,z)=\lambda \nabla g(x,y,z)+\mu \nabla h(x,y,z)$
    • $g(x,y,z)=k$
    • $h(x,y,z)=l$