Applied Mathematics for Business and Economics

Lecture Note Function of Two Variables

Ifx= 2 , it follows from the equation x^2 =y^2 thatyy=2 or =−^2. Similarly, if ,

it follows that. Hence, the four points at which the constrained

extrema can occur are

x=− 2

yy==2 or 2

()

−

2, 2 ,(2, 2 ,−−)( 2, 2 , a) nd 2, 2(−−). Since

ff(2, 2)( )=−−=2, 2 4 and f f(2, 2−=− =−) ( 2, 2) 4

it follows that when xy^22 += 8 , the maximum value of f(xy, )is 4, which occurs at

the points (2, and the minimum value isെ4which occurs at(2,െ2ሻ 

and ሺെ2, 2ሻ.

2 an)d ( )−−2, 2

Example2
An editor has been allocated $60,000 to spend on the development and promotion of a
new book. It is estimated that if x thousand dollars is spent on development and y
thousand on promotion, approximately f(xy,20)= x y^32 copies of the book will be

sold. How much money should the editor allocate to development and how much to
promotion in order to maximize sales?

Solution:

The goal is to maximize the functionf(xy,20)= x y^32 subject to the constraint

gxy(),6= 0

, 3

, wheregxy x y(), =+. The corresponding Lagrange equations are

()

()

()

()

1/ 2

3/2

30 , 1

20 , 2

,6 0

xx

yy

fg xy

fg x

gxy K x y

λλ

λλ

⎧ ==⎧

⎪ ⎪

⎨⎨=⇒ =

⎪⎪=+=

⎩ ⎩

From (1) and (2) you get
30 12 20 32
2
3

x yx

y x

=

=

Substituting this expression into the (3) you get
35
60 or 60
22

xx+ ==x

From which it follows that
xy=36 and = 24

That is, to maximize sales, the editor should spend $36,000 on development and $

24,000 on promotion. If this is done, approximatelyf(36, 24)=103, 680copies of the

book will be sold.

Example 3
A consummer has $600 to spend on two commodities, the first of which costs $20 per
unit and the second $30 per unit. Suppose that the utility derived by the consumer
from x units of the first commodity and y units of the second commodity is given by

the Cobb-Douglas utility functionUxy( ,10)= x y0.6 0.4. How many units of each

commodity should the consumer buy to maximize utility? (A utility function