Maxima, minima and saddle points for functions of two variables 363
yxz^50z (^5128)
z^59
h
g c
d
22
2
222 a
S
f b e
j
i
24
4
3 3
Figure 36.9
Hence thez=0 contour passes throughthe co-ordinates
(2, 0.97) and (2,−0.97) shown as acanddin Fig. 36.9.
Similarly, for thez= 9 contour, wheny=0,
9 =(x^2 + 02 )^2 − 8 (x^2 − 02 )
i.e. 9 =x^4 − 8 x^2
i.e. x^4 − 8 x^2 − 9 = 0
Hence(x^2 − 9 )(x^2 + 1 )=0.
from which,x=±3 or complex roots.
Thus thez=9 contourpasses through(3, 0) and (−3, 0),
shown aseandfin Fig. 36.9.
Ifz=9andx= 0 , 9 =y^4 + 8 y^2
i.e. y^4 + 8 y^2 − 9 = 0
i.e.(y^2 + 9 )(y^2 − 1 )= 0
from which,y=±1 or complex roots.
Thus thez=9 contour also passes through (0, 1) and
(0,−1), shown asgandhin Fig. 36.9.
When, say,x=4andy=0,
z=( 42 )^2 − 8 ( 42 )=128.
whenz=128 andx= 0 , 128 =y^4 + 8 y^2
i.e. y^4 + 8 y^2 − 128 = 0
i.e.(y^2 + 16 )(y^2 − 8 )= 0
from which,y=±
√
8 or complex roots.
Thus thez=128 contour passes through (0, 2.83) and
(0,−2.83), shown asiandjin Fig. 36.9.
In a similar manner many other points may be calculated
with the resulting approximate contour map shown in
Fig. 36.9. It is seen that two ‘hollows’ occur at the min-
imum points, and a ‘cross-over’ occurs at the saddle
pointS, which is typical of such contour maps.
Problem 4. Show that the function
f(x,y)=x^3 − 3 x^2 − 4 y^2 + 2
has one saddle point and one maximum point.
Determine the maximum value.