Higher Engineering Mathematics, Sixth Edition

358 Higher Engineering Mathematics

f(x,y),andthenf(x,y)calculated for each, a large

number of lines such asPP′can be constructed, and in

the limit when all points in the(x,y)plane are consid-

ered, 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.

36.2 Maxima, minima and saddle

points

Partial differentiationis used when determiningstation-

ary points for functions of two variables. A function

f(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 a minimum if less than at

all points in the immediate vicinity. Figure 36.3 shows

geometrically a maximum value of a function of two

variables and it is seen that the surfacez=f(x,y)is

higher at(x,y)=(a,b)than at any point in the imme-

diate vicinity. Figure 36.4 shows a minimum value of a

function of two variables and it is seen that the surface

z=f(x,y)is lower at(x,y)=(p,q)than at any point

in the immediate vicinity.

z

b

Maximum

point

y

x

a

Figure 36.3

Ifz=f(x,y)and a maximum occurs at(a,b),the

curve lying in the two planesx=aandy=bmust also

haveamaximumpoint(a,b)asshowninFig.36.5.Con-

sequently, the tangents (shown ast 1 andt 2 ) to the curves

at(a,b)must be parallel toOxandOyrespectively.

This requires that

∂z

∂x

=0and

∂z

∂y

=0 at all maximum

andminimumvalues,andthesolutionoftheseequations

gives the stationary (or critical) points ofz.

Minimum

point

z

x

p

q

y

Figure 36.4

Maximum

point

z

t 1

t 2

O b

a

x

y

Figure 36.5

With functions of two variables there are three types

of stationary points possible, these being a maximum

point, a minimum point, and asaddle point.Asad-

dle pointQis shown in Fig. 36.6 and is such that a

pointQis a maximum for curve 1 and a minimum for

curve 2.

Curve 2

Curve 1

Q

Figure 36.6