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?