360 Higher Engineering Mathematics
z
1
2
1
o
x
y
Figure 36.7
A contour map for z =(x − 1) 2 + (y − 2) 2 is shown in
Fig. 36.8. The values of z are shown on the map and these
give an indication of the rise and fall to a stationary point.
Problem 2. Find the stationary points of the
surface f (x, y) = x 3 − 6x y + y 3 and determine their
nature.
Let z = f (x, y) = x 3 − 6x y + y 3
Following the procedure:
(i)
∂z
∂x
= 3x 2 − 6y and
∂z
∂ y
=−6x + 3y 2
(ii) for stationary points, 3x 2 − 6y = 0
( 1 )
and
−6x + 3y 2 = 0
( 2 )
(iii) from equation (1), 3x
2
= 6y
and
y =
3x 2
6
=
1
2
x 2
y
z 5 1
z 5 4
z 5 9
z 5 16
x
2
1
1
2
Figure 36.8
z
1
2
1
o
x
y
Figure 36.7
A contour map for z =(x − 1) 2 + (y − 2) 2 is shown in
Fig. 36.8. The values of z are shown on the map and these
give an indication of the rise and fall to a stationary point.
Problem 2. Find the stationary points of the
surface f (x, y) = x 3 − 6x y + y 3 and determine their
nature.
Let z = f (x, y) = x 3 − 6x y + y 3
Following the procedure:
(i)
∂z
∂x
= 3x 2 − 6y and
∂z
∂ y
=−6x + 3y 2
(ii) for stationary points, 3x 2 − 6y = 0
( 1 )
and
−6x + 3y 2 = 0
( 2 )
(iii) from equation (1), 3x
2
= 6y
and
y =
3x 2
6
=
1
2
x 2
y
z 5 1
z 5 4
z 5 9
z 5 16
x
2
1
1
2
Figure 36.8