L
agrange
M
ultipliers
In Example 3 of the previous section, we maximized the volume
V
(
l
,
w
,
h
) =
lwh
subject to the constraint that
A
(
l
,
w
,
h
) = 2
hl
+ 2
hw
+
lw
= 12. We solved that problem by
substituting the constraint into the function we wished to maximize and then found
critical points.
In this section, we present a second method for dealing with these problems known as
Lagrange’s method for maximizing (or minimizing) a general function
f
(
x
,
y
) subject to a
constraint
g
(
x
,
y
) =
c
. This applies to a more general setting than the method we used in
the previous section. (Below, we present the theory for two variables, but the method
applies to functions of three or more variables, like Example 3 from the last section.)
Suppose we want to find the extreme values of
f
(
x
,
y
) subject to a constant
g
(
x
,
y
) =
c
.
That is, we want to find the extreme values of
f
(
x
,
y
) when the point (
x
,
y
) is restricted to
the level curve
g
(
x
,
y
) =
c
. Figure 1 below shows some level curves of
f
(
x
,
y
) (in blue) as
well as the constraint
g
(
x
,
y
) =
c
(in red).
1
2
3
4
5
6
7
0.2
0.4
0.6
0.8
1
x
y
0
,
0
Figure 1: Level curves of
f
(
x
,
y
) and the constraint
g
(
x
,
y
) =
c
To maximize
f
(
x
,
y
) subject to
g
(
x
,
y
) =
c
is the same as finding the largest
c
value
such that the level curve
f
(
x
,
y
) =
k
intersects
g
(
x
,
y
) =
c
. In the above figure, this happens
at the point (
x
0
,
y
0
). Notice that at this point, the curves just touch each other. That is, they
have a common tangent line. (Otherwise, the value of
k
could be increased further.)
But if the two curves have a common tangent line, this means that their gradient
vectors must be parallel. (See the colored arrows above.) This gives us the relationship
“
f
(
x
,
y
) =
l“
g
(
x
,
y
), for some scalar
l
.
1
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View Full DocumentUsing this relationship, we can describe the method of Lagrange multipliers.
Method of Lagrange Multipliers
To find the maximum and minimum values of
f
(
x
,
y
) subject to the
constraint
g
(
x
,
y
) =
c
(assuming these extreme values exist)
1. Find all values of
x
,
y
and
l
such that
“
f
(
x
,
y
) =
l“
g
(
x
,
y
)
and
g
(
x
,
y
) =
c
2. Evaluate
f
(
x
,
y
) at all of the points found in (1). The largest of these
values is the maximum value of
f
(
x
,
y
) and the smallest value is the
minimum value of
f
(
x
,
y
).
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 Spring '07
 Hohnhold
 Calculus, Optimization, lagrange multipliers

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