Higher Engineering Mathematics, Sixth Edition

Maxima, minima and saddle points for functions of two variables 361

and substituting in equation (2) gives:

− 6 x+ 3

(

1

2

x^2

) 2

= 0

− 6 x+

3

4

x^4 = 0

3 x

(

x^3

4

− 2

)

= 0

from which,x=0or

x^3

4

− 2 = 0

i.e.x^3 =8andx= 2

Whenx=0,y=0andwhenx=2,y=2 from

equations (1) and (2).

Thus stationary points occur at (0, 0)

and (2, 2).

(iv)

∂^2 z

∂x^2

= 6 x,

∂^2 z

∂y^2

= 6 yand

∂^2 z

∂x∂y

=

∂

∂x

(

∂z

∂y

)

=

∂

∂x

(− 6 x+ 3 y^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 )>0then(0, 0) is a saddle point.

Since( 2 , 2 )<0and

∂^2 z

∂x^2

>0, then(2, 2) is a

minimum point.

Now try the following exercise

Exercise 143 Further problemson

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 byz.
   [Minimum at (0, 0)]


2. Find the maxima, minima and saddle points
   for the following functions:
   (a) f(x,y)=x^2 +y^2 − 2 x+ 4 y+ 8
   (b) f(x,y)=x^2 −y^2 − 2 x+ 4 y+ 8
   (c) f(x,y)= 2 x⎡+ 2 y− 2 xy− 2 x^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 func-
   tionf(x,y)=x^3 − 6 x^2 − 8 y^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= 12 x^2 + 6 xy+ 15 y^2.
   [Minimum at (0, 0)]


5. Find the stationary points of the surface
   z=x^3 −xy+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 withz.