Maxima, minima and saddle points for functions of two variables 361
and substituting in equation (2) gives:
−6x + 3
1
2
x
2
2
= 0
−6x +
3
4
x
4
= 0
3x
x 3
4
− 2
= 0
from which, x = 0 or
x 3
4
− 2 =0
i.e. x 3 = 8 and x = 2
When x = 0, y = 0 and when x = 2, y = 2 from
equations (1) and (2).
Thus stationary points occur at (0, 0)
and (2, 2).
(iv)
∂ 2 z
∂x 2 = 6x,
∂ 2 z
∂ y 2 = 6y and
∂ 2 z
∂x∂ y
=
∂
∂x
∂z
∂ y
=
∂
∂x
(−6x + 3y
2
) = −6
(v) for (0, 0)
∂ 2 z
∂x 2 = 0,
∂ 2 z
∂ y 2 = 0
and
∂
2 z
∂x∂ y
= −6
for (2, 2),
∂ 2 z
∂x 2 = 12,
∂ 2 z
∂ y 2 = 12
and
∂ 2 z
∂x∂ y
= −6
(vi) for (0, 0),
∂ 2 z
∂x∂ y
2
= (−6) 2 = 36
for (2, 2),
∂ 2 z
∂x∂ y
2
= (−6) 2 = 36
(vii) (0, 0) =
∂ 2 z
∂x∂ y
2
−
∂ 2 z
∂x 2
∂ 2 z
∂ y 2
= 36 − (0)(0) = 36
(2, 2) = 36 − (12)(12) = −108
(viii) Since (0, 0) > 0 then (0, 0) is a saddle point.
Since (2, 2) < 0 and
∂ 2 z
∂x 2 > 0, then (2, 2) is a
minimum point.
Now try the following exercise
Exercise 143 Further problems on
maxima, minima and saddle points for
functions of two variables
1. Find the stationary point of the surface
f (x, y) = x 2 + y 2 and determine its nature.
Sketch the surface represented by z.
[Minimum at (0, 0)]
2. Find the maxima, minima and saddle points
for the following functions:
(a) f (x, y) = x 2 + y 2 − 2x + 4y + 8
(b) f (x, y) = x 2 − y 2 − 2x + 4y + 8
(c) f (x, y) = 2x + 2y − 2x y − 2x
2
− y
2
+ 4.
⎡
⎣
(a) Minimum at (1, −2)
(b) Saddle point at (1, 2)
(c) Maximum at (0, 1)
⎤
⎦
3. Determine the stationary values of the function f (x, y) = x 3 − 6x 2 − 8y 2 and distinguish
between them. Sketch an approximate contour
map to represent the surface f (x, y).
Maximum point at (0, 0),
saddle point at (4, 0)
4. Locate the stationary point of the function
z =12x 2 + 6x y + 15y 2 .
[Minimum at (0, 0)]
5. Find the stationary points of the surface
z = x 3 − x y + y 3 and distinguish between
them.
saddle point at (0, 0),
minimum at
1
3 ,
1
3
36.5 Further worked problems on
maxima, minima and saddle
points for functions of two
variables
Problem 3. Find the co-ordinates of the
stationary points on the surface
z = (x
2
+ y
2
)
2
− 8(x
2
− y
2
)
and distinguish between them. Sketch the
approximate contour map associated with z.
and substituting in equation (2) gives:
−6x + 3
1
2
x
2
2
= 0
−6x +
3
4
x
4
= 0
3x
x 3
4
− 2
= 0
from which, x = 0 or
x 3
4
− 2 =0
i.e. x 3 = 8 and x = 2
When x = 0, y = 0 and when x = 2, y = 2 from
equations (1) and (2).
Thus stationary points occur at (0, 0)
and (2, 2).
(iv)
∂ 2 z
∂x 2 = 6x,
∂ 2 z
∂ y 2 = 6y and
∂ 2 z
∂x∂ y
=
∂
∂x
∂z
∂ y
=
∂
∂x
(−6x + 3y
2
) = −6
(v) for (0, 0)
∂ 2 z
∂x 2 = 0,
∂ 2 z
∂ y 2 = 0
and
∂
2 z
∂x∂ y
= −6
for (2, 2),
∂ 2 z
∂x 2 = 12,
∂ 2 z
∂ y 2 = 12
and
∂ 2 z
∂x∂ y
= −6
(vi) for (0, 0),
∂ 2 z
∂x∂ y
2
= (−6) 2 = 36
for (2, 2),
∂ 2 z
∂x∂ y
2
= (−6) 2 = 36
(vii) (0, 0) =
∂ 2 z
∂x∂ y
2
−
∂ 2 z
∂x 2
∂ 2 z
∂ y 2
= 36 − (0)(0) = 36
(2, 2) = 36 − (12)(12) = −108
(viii) Since (0, 0) > 0 then (0, 0) is a saddle point.
Since (2, 2) < 0 and
∂ 2 z
∂x 2 > 0, then (2, 2) is a
minimum point.
Now try the following exercise
Exercise 143 Further problems on
maxima, minima and saddle points for
functions of two variables
1. Find the stationary point of the surface
f (x, y) = x 2 + y 2 and determine its nature.
Sketch the surface represented by z.
[Minimum at (0, 0)]
2. Find the maxima, minima and saddle points
for the following functions:
(a) f (x, y) = x 2 + y 2 − 2x + 4y + 8
(b) f (x, y) = x 2 − y 2 − 2x + 4y + 8
(c) f (x, y) = 2x + 2y − 2x y − 2x
2
− y
2
+ 4.
⎡
⎣
(a) Minimum at (1, −2)
(b) Saddle point at (1, 2)
(c) Maximum at (0, 1)
⎤
⎦
3. Determine the stationary values of the function f (x, y) = x 3 − 6x 2 − 8y 2 and distinguish
between them. Sketch an approximate contour
map to represent the surface f (x, y).
Maximum point at (0, 0),
saddle point at (4, 0)
4. Locate the stationary point of the function
z =12x 2 + 6x y + 15y 2 .
[Minimum at (0, 0)]
5. Find the stationary points of the surface
z = x 3 − x y + y 3 and distinguish between
them.
saddle point at (0, 0),
minimum at
1
3 ,
1
3
36.5 Further worked problems on
maxima, minima and saddle
points for functions of two
variables
Problem 3. Find the co-ordinates of the
stationary points on the surface
z = (x
2
+ y
2
)
2
− 8(x
2
− y
2
)
and distinguish between them. Sketch the
approximate contour map associated with z.