12.8 Lagrange Multipliers
. Find the extreme values of a given function f (x, y, z) subject to the constraint g(x, y, z) = k .
Problem
Suppose f has an extreme value at a point (x0 , y0 , z0 ). Then the gradient vector
∇f (x0 , y0 , z0 ) is orthogonal to the tangent vector of every curve in the level surface
g(x, y, z) = k . Therefore two gradient vectors ∇f (x0 , y0 , z0 ) and ∇g(x0 , y0 , z0 ) must
be parallel.
Method of Lagrange Multipliers
To nd the maximum and minimum values of f (x, y, z) subject to the constraint
g(x, y, z) = k (assuming that these extreme values exist):
1. Find all values of x, y , z , and λ such that
∇f (x, y, z) = λ∇g(x, y, z)
and g(x, y, z) = k .
2. Evaluate f at all the points (x, y, z) that arise from step 1. The largest of these
values is the maximum value of f ; the smallest is the minimum value of f .
The number λ is called a Lagrange multiplier. If we rewrite the vector equation
∇f = λ∇g in terms of its components, then the equation in step 1 become
fx = λgx ,
fy = λgy ,
fz = λgz ,
g(x, y, z) = k
Example 1. A rectangular box without a lid is to be made from 12 m2 of cardboard.
Find the maximum volume of such a box.
Example 2. Find the extreme values of f (x, y) = x2 + y 2 + 4x − 4y on the disk
x2 + y 2 ≤ 9.