Applied Mathematics for Business and Economics

Lecture Note Function of Two Variables

skim milk kñúgcMnYnxnigyháaLúgerogKña. ]bmafaéføTwkedaHeKa whole milk KW

p()x=− 100 xnigTwkedaHeKaskim milk KW qy( )= 100 −y ehIysnμtfaGnuKmn_cMNay

rYmKWCxy x(), =++^2 xy y^2. etI xnigyKYrmantémøb:unμanedIm,I[R)ak;cMeNjmankMriGtibrma?

25 A dairy produces whole milk and skim milk in quantities x and y gallons,

respectively. Supose that the price of whole milk is p(xx)=20 5− and that of

skim milk isqy( )=− 42 yand assume that Cx( ,2y)= xy+ 4 is the joint-cost

function of the commodities. What should x and y be in order to maximize profit?

(Answer: ݔൌ2,ݕൌ0).

26 Find the maximum value of the functionf(xy xy, )= subject to the constraint

x+=y 1 .cUrrkGtibrmarbs;GnuKmn_f(xy, )=xyeRkamlkçxNÐRBMEdn x+=y 1.
( )
27 Find the minimum value of the function f xy x, =^2 +y^2 subject to the

constraintxy= 1. rkGb,brmarbs;GnuKn_f(xy x y, )=^2 +^2 eRkamlkçxNÐRBMEdnxy= 1.
( ,)
28 Find the minimum value of the function f xy x=^22 +− 2 y xysubject to the

constraint 222 xy+ =.

29 Find the minimum value of the function f(xy x, )=^2 −y^2 subject to the

constraint^22 = 4.

(^224)
xy+

(),8

30 Find the maximum and minimum values of the function

f xy=−x xy y+^2 subject to the constraintxy^22 + = 1.

31 A manufacturer has $8,000 to spend on the development and promotion of a new
product. It is estimated that if x thousand dollars is spent on development and y
thousand is spent on promotion, sales will be approximately f(xy,50)= x y12 32
units. How much money should the manuacturer allocate to development and
how much to promotion to maximize sales?plitkrmñak;man 8000 Ban;duløasRmab;

cMNayeTAelIkarplit nigpSBVpSayplitplfμImYy. eKrMBwgTukfaebIebIcMNay xBan;duløaeTAelI

karplit nigyBan;duløaeTAelIkarpSBVpSay enaHeKlk;dac;RbEhlf(xy,50)= x y12 32Ékta.

etIplitkrKYrEbgEckeTAelIkarplitb:unμan nigeTAelIkarpSBVpSayb:unμanedIm,IeFVI[karlk;eLIgdl;

kMritGtibrma?

32 If x thousand dollars is spent on labor and y thousand dollars is spent on

equipment, the output at a certain factory will be Qx( ,60y)= x y13 2 3units. If

$120,000 is available, how should this be allocated between labor and equipment

to generate the largest possible output?ebIeKcMNayxBan;duløaeTAelIkMlaMgBlkmμ nigy

Ban;duløaeTAelIsmÖar³ enaHcMnYnplitplecjBIeragcRkmYykMNt;edayQxy( ,60)= x y13 2 3

Ékta. ebIeKman 120000 duløa enaHetIeKKYrEbgEcky:agdUcemþcedIm,I[cMnYnplitplmankMrit

Gtibrma?