Lecture Note Function of Two Variables
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5 Lagrange Multipliers
5.1 Contrained Optimization Problems
In many applied problems, a function of two variables is to be optimized subject to a
restriction or constraint on the varaibles. For, example, an editor constrained to stay
within a fixed budget of $60,000, may wish to decide how to divide this money
between development and promotion in order to maximize the future sales of a new
book. If x denotes the amount of money allocated to development, y the amount
allocated to promotion, and f(xy, the corresponding number of books that will be
sold, the editor would like to maximize the sales functionf(xy, ) subject to the
budgetary constraint that. To deal with this problem, we use a
technique called the method of Lagrange multipliers.
xy+=60, 000
The Method of Lagrange Multipliers
Suppose f(xy, )( and g xy, )are functions whose first-order-partial
derivatives exist. To find the relative maximum and relative minimum of
f(xy, )subject to the constraint thatgxy K( , )= for some constant K,
introduce a new variable λ (the Greek letter lambda) and solve the following
three equations simultaneously:
() ( )
() ()
()
,,
,,
,
xx
yy
f xy g xy
f xy g xy
gxy K
λ
λ
⎧ =
⎪
⎨ =
⎪ =
⎩
The desired relative extrema will be found among the resulting points()x,y.
Example 1
Find the maximum and minimum values of the function f(xy xy, )= subject to the
constraintxy^22 += 8.
Solution
Let gxy x y(), =+^22 and use the partial derivatives
fxyx=== =yf xg,,2,and 2x gyy
to get the three Lagragnge equations
yxxy xy== + 2 λλ, 2 , and 8^22 =
The first two equations can be rewritten as
2
y
x
λ= and 2
x
y
λ=
which implies that
yx
x y
= or x^22 =y
Now substitute x^22 =y into the third equation to get
28 x^2 = or x=± 2