Multivariate Calculus - Bruce E. Shapiro

Lecture 15
Constrained Optimization:
Lagrange Multipliers
In the previous section we learned how to find the maximum and minimum points of
a function of two variables. In this section we will study how to find the maximum
and minimum points on a function subject to a constraint using a technique called
Lagrange’s Method or The Method of Langrange Multipliers.
Theorem 15.1 Lagrange’s Method. An extreme value of the function f(x, y)
subject to the constraint
g(x, y) = 0
occurs when
for some number λ.
∇f(x, y) = λ∇g(x, y)
We introduce the method along with the concept of a constraint via the following
example.
Example 15.1 Find the dimensions and area of the largest rectangular area that
can be enclosed with a fixed length of fence M.
Solution Let the sides of the rectangle have length x and y. Then the area is
given by the function f(x, y) where
. Similarly, the perimeter M is
f(x, y) = xy
M = 2x + 2y
One way to solve this problem is to solve for one of the variables, say y, as a
function of x and M, substitute the result into the equation for the area, and then
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