358 Higher Engineering Mathematics
f (x, y), and then f (x, y) calculated for each, a large
number of lines such as P P 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 function z = f (x, y) represents a surface and
not a curve.
36.2 Maxima, minima and saddle
points
Partial differentiation is used when determining stationary 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 surface z = 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 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
If z = f (x, y) and a maximum occurs at (a, b), the
curve lying in the two planes x = a and y = b must also
have a maximum point (a, b) as shown in Fig. 36.5. Consequently, the tangents (shown as t 1 and t 2 ) to the curves
at (a, b) must be parallel to Ox and Oy respectively.
This requires that
∂z
∂x
= 0 and
∂z
∂ y
= 0 at all maximum
and minimum values, and the solution of these equations
gives the stationary (or critical) points of z.
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 a saddle point. A saddle point Q is shown in Fig. 36.6 and is such that a
point Q is a maximum for curve 1 and a minimum for
curve 2.
Curve 2
Curve 1
Q
Figure 36.6