LESSON 13
Lagrange Multipliers
READ:
Section 15.8.
NOTES:
At the end of the last lesson, we looked at the problem of ﬁnding the maximum value of
f
(
x,y
) =
x
2
+
y
for points (
x,y
) on the circle
x
2
+
y
2
= 1. The idea used there was to parameterize the circle as
x
= cos
t
,
y
= sin
t
, 0
≤
t
≤
2
π
. Then those component functions were plugged into
f
for
x
and
y
. The result was a
function of just one variable (
t
). Now we can determine the max and min of
f
along the boundary in the
oldfashioned calculus I way. In this lesson, we will look at another way to determine the max and min of
f
along the boundary.
To help visualize the problem, imagine we are walking about on the surface given by equation
f
(
x,y
) =
x
2
+
y
,
but we can’t walk just anywhere on the surface: we are constrained to the part of the surface directly above
the circle
x
2
+
y
2
= 1 in the
x
,
y
plane. The question we are trying to answer is: under that constraint, what
is the highest point we will reach and what is the lowest point we will reach as we go all the way around
above the circle. The question is usually posed as: what are the maximum and minimum values of
x
2
+
y
subject to the constraint
x
2
+
y
2
= 1?
Lagrange devised a method for solving such problems. The method can be explained geometrically. Suppose
the maximum value on the surface
z
=
f
(
x,y
) directly above the constraint curve
g
(
x,y
) = 0 is
M
. (In our
example above, the constraint curve would be given by equation
g
(
x,y
) =
x
2
+
y
2

1 = 0.) Imagine the level
curve for the surface produced by slicing through the surface at
z
=
M
. Now down in the
x
,
y
plane, the
constraint curve and the level curve will intersect, since there is a spot on the surface with
z
=
M
somewhere
above the constraint curve. Call the point of intersection (
x
0
,y
0
). If there is any justice in life, the level
curve and the constraint curve ought to be tangent to each other at the point where they intersect, since
if the constraint curve crossed over the level curve, then that would mean there would be points above the
constraint curve that were higher than
M
, which we assumed was the maximum value above the constraint
curve. So it seems reasonable that the level curve and the constraint curve are tangent to each other at the