This preview shows pages 1–2. Sign up to view the full content.
The Lagrange Multiplier Method
© Arizona State University, Department of Mathematics and Statistics
1
Objective:
At the end of this lesson, you should be able to find the absolute maximum and minimum using
the direct substitution method and the Lagrangian method.
Background
So far, we have optimized a function subject to some constraint using the
direct substitution method.
This
works well as long as the
constraint
can be solved for one of the variables involved.
Example
1.
Find the maximum and minimum values of
( )
,
45
f x y
xy
=
under the constraint
22
2
3
5
10
x
xy
y
++=
It would be difficult to solve this problem using the direct substitution method. However, the Lagrange
method is much easier. This will be shown later in this lecture.
2.
Find the minimum value of
( )
,
34
f xy
x
y
=
+
under
the constraint 4
5
20
xy
+=
.
Now this constraint can be easily solved for either
x
or
y
and the direct substitution method works fine.
Solve the constraint 4
5
20
to get
4
4
5
x
y
=
−
. Then
substitute. The result is to the right. Then set
( )
278
128
'0
25
5
fx
x
=
−=
. Solve to get
2.302
x
=
,
Back substitute to find
( )
4 2.302
4
2.158
5
y
=
=
.
Evaluate to find the minimum value:
( ) ( ) ( )
2.302,2.158
3 2.302
4 2.158
34.525
f
=+=
.
State that the minimum value of the function is 34.525 at the point
( )
2.302,2.158 .
Finish the problem!
The Lagrange Theorem
Suppose you are given the function
f
(
x
,
y
) subject to the constraining function
g
(
x
,
y
)
=
c
. Suppose both
functions have continuous partial derivatives on a domain
A
in the
xy
plane. Suppose that the point
x
0
,
y
0
(
)
is an interior extreme point the domain under the constraint
g
(
x
,
y
)
=
c
and
g
x
x
0
,
y
0
(
)
≠
0
and g
y
x
0
,
y
0
(
)
≠
0 (one them could be zero). Then there is a unique constant
λ
(lambda to
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 06/16/2011 for the course MAT 211 taught by Professor Seal during the Summer '08 term at ASU.
 Summer '08
 SEAL
 Substitution Method

Click to edit the document details