Chapter 36

Maxima, minima and saddle

points for functions of two

variables

36.1 Functions of two independent

variables

If a relation between two real variables, x and y,
is such that when x is given,yis determined, then
y is said to be a function of x and is denoted by
y=f(x);xis called the independent variable andy
the dependent variable. Ify=f(u,v),thenyis a func-
tion of two independent variablesuandv. For example,
if, say, y=f(u,v)= 3 u^2 − 2 vthen whenu=2and
v=1,y= 3 ( 2 )^2 − 2 ( 1 )=10. This may be written as
f( 2 , 1 )=10. Similarly,ifu=1andv=4,f( 1 , 4 )=−5.

6

0

2

3

p

p 9

x

z

y

Figure 36.1

Consider a function of two variables x and y

defined by z=f(x,y)= 3 x^2 − 2 y.If(x,y)=( 0 , 0 ),

then f( 0 , 0 )=0andif(x,y)=( 2 , 1 ),thenf( 2 , 1 )=10.

Each pair of numbers, (x,y), may be represented

by a point P in the(x,y)plane of a rectangular

Cartesian co-ordinate system as shown in Fig. 36.1.

The corresponding value ofz=f(x,y)may be rep-

resented by a linePP′ drawnparalleltothez-axis.

Thus, if, for example,z= 3 x^2 − 2 y,asabove,andP

is the co-ordinate (2, 3) then the length of PP′ is

3 ( 2 )^2 − 2 ( 3 )=6. Figure 36.2 shows that when a large

number of(x,y)co-ordinates are taken for a function

z

x

o y

Figure 36.2