G
Differential calculus
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,xandy,
is such that whenxis given,yis determined, then
yis said to be a function ofxand is denoted by
y=f(x);xis called the independent variable andy
the dependent variable. Ify=f(u,v), thenyis a
function of two independent variablesuandv.For
example, if, say,y=f(u,v)= 3 u^2 − 2 vthen when
u=2 andv=1,y=3(2)^2 −2(1)=10. This may be
written asf(2, 1)=10. Similarly, ifu=1 andv=4,
f(1, 4)=−5.
Consider a function of two variablesxandy
defined byz=f(x,y)= 3 x^2 − 2 y.If(x,y)=(0, 0),
thenf(0, 0)=0 and if (x,y)=(2, 1), thenf(2, 1)=10.
Each pair of numbers, (x,y), may be represented
by a pointPin 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′drawn parallel to thez-axis.
Thus, if, for example,z= 3 x^2 − 2 y, as above, and
Pis the co-ordinate (2, 3) then the length ofPP′
xyz60 3(^2) p
p′
Figure 36.1
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 functionf(x,y), and thenf(x,y) calculated for
each, a large number of lines such as PP′ can
be constructed, and in the limit when all points
in the (x,y) plane are considered, a surface is
seen to result as shown in Fig. 36.2. Thus the
functionz=f(x,y) represents a surface and not
a curve.
y
x
z
o
Figure 36.2
36.2 Maxima, minima and saddle
points
Partial differentiation is used when determining sta-
tionary points for functions of two variables. A
functionf(x,y) is said to be a maximum at a point
(x,y) if the value of the function there is greater
than at all points in the immediate vicinity, and is