Applied Mathematics for Business and Economics

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