360 Higher Engineering Mathematics

z

(^12)
1
o
x
y
Figure 36.7
A contour map forz=(x− 1 )^2 +(y− 2 )^2 is shown in
Fig.36.8.Thevaluesofzareshown on themap and these
giveanindicationoftheriseandfall toastationarypoint.
Problem 2. Find the stationary points of the
surfacef(x,y)=x^3 − 6 xy+y^3 and determine their
nature.
Letz=f(x,y)=x^3 − 6 xy+y^3
Following the procedure:
(i)
∂z
∂x
= 3 x^2 − 6 yand
∂z
∂y
=− 6 x+ 3 y^2
(ii) for stationary points, 3x^2 − 6 y=0(1)
and − 6 x+ 3 y^2 =0(2)
(iii) from equation (1), 3x^2 = 6 y
and y=
3 x^2
6

1
2
x^2
y
z 51 z 54 z 59 z 516
(^21) x
1
2
Figure 36.8