Section 4.8 - Lagrange Multipliers

Section Objectives

  1. Use Lagrange multipliers to solve a constrained optimization problem.



A constrained optimization problem is the problem of finding the absolute minimum or maximum value of a function subject to a constraint. For example,

Find the minimum value of subject to the condition that .


The Method of Lagrange Multipliers is a technique for solving constrained optimization problems. The method follows from the idea that the gradient vector at a point is normal to the level curve (or surface) through the point. The method applies to functions of any number of variables and with any number of constraints, but we will focus on single constraints.



The method of Lagrange multipliers

To solve the constrained optimization problem

  1. Assume (or verify) that the maximum and minimum values exist.
  2. Find all that satisfy the system of equations
  1. Evaluate at the points found in step 2.
  2. The least value is the minimum and the greatest value is the maximum.


Examples