Higher Engineering Mathematics

356 DIFFERENTIAL CALCULUS

a minimum if less than at all points in the imme-
diate 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 immediate
vicinity. Figure 36.4 shows a minimum value of a
function of two variables and it is seen that the sur-
facez=f(x,y) is lower at (x,y)=(p,q) than at any
point in the immediate vicinity.

Maximum

z point

b

a

y

x

Figure 36.3

Minimum

point

y

q

p

x

z

Figure 36.4

Ifz=f(x,y) and a maximum occurs at (a,b),
the curve lying in the two planesx=aandy=b
must also have a maximum point (a,b) as shown in
Fig. 36.5. Consequently, the tangents (shown ast 1
andt 2 ) to the curves at (a,b) must be parallel toOx

andOyrespectively. This requires that

∂z

∂x

=0 and

Maximum

point

b

y

z

a

O

t 1

t 2

x

Figure 36.5

∂z

∂y

=0 at all maximum and minimum values, and

the solution of these equations gives the stationary

(or critical) points ofz.

With functions of two variables there are three

types of stationary points possible, these being a

maximum point, a minimum point, and asaddle

point. A saddle pointQis shown in Fig. 36.6 and

is such that a pointQis a maximum for curve 1 and

a minimum for curve 2.

Q

Curve 1

Curve 2

Figure 36.6

36.3 Procedure to determine maxima,

minima and saddle points for

functions of two variables

Givenz=f(x,y):

(i) determine

∂z

∂x

and

∂z

∂y

(ii) for stationary points,

∂z

∂x

=0 and

∂z

∂y

=0,