Higher Engineering Mathematics, Sixth Edition

Maxima, minima and saddle points for functions of two variables 359

36.3 Procedure to determine

maxima, minima and saddle

points for functions of two

variables

Givenz=f(x,y):

(i) determine

∂z

∂x

and

∂z

∂y

(ii) for stationary points,

∂z

∂x

=0and

∂z

∂y

=0,

(iii) solve the simultaneous equations

∂z

∂x

=0and

∂z

∂y

=0forxandy, which gives the co-ordinates

of the stationary points,

(iv) determine

∂^2 z

∂x^2

,

∂^2 z

∂y^2

and

∂^2 z

∂x∂y

(v) for each of the co-ordinates of the stationary

points,substitutevaluesofxandyinto

∂^2 z

∂x^2

,

∂^2 z

∂y^2

and

∂^2 z

∂x∂y

and evaluate each,

(vi) evaluate

(

∂^2 z

∂x∂y

) 2

for each stationary point,

(vii) substitute the values of

∂^2 z

∂x^2

,

∂^2 z

∂y^2

and

∂^2 z

∂x∂y

into the equation

=

(

∂^2 z

∂x∂y

) 2

−

(

∂^2 z

∂x^2

)(

∂^2 z

∂y^2

)

and evaluate,

(viii) (a) if> 0 then the stationary point is a

saddle point.

(b) if< 0 and

∂^2 z

∂x^2

< 0 , then the stationary

point is amaximum point,

and

(c) if< 0 and

∂^2 z

∂x^2

> 0 , then the stationary

point is aminimum point.

36.4 Worked problems on maxima,

minima and saddle points for

functions of two variables

Problem 1. Show that the function

z=(x− 1 )^2 +(y− 2 )^2 has one stationary point only

and determine its nature. Sketch the surface

represented byzand produce a contour map in the

x-yplane.

Following the above procedure:

(i)

∂z

∂x

= 2 (x− 1 )and

∂z

∂y

= 2 (y− 2 )

(ii) 2(x− 1 )= 0 ( 1 )

2 (y− 2 )= 0 ( 2 )

(iii) From equations (1) and (2),x=1andy=2, thus

the only stationary point exists at (1, 2).

(iv) Since

∂z

∂x

= 2 (x− 1 )= 2 x− 2 ,

∂^2 z

∂x^2

= 2

and since

∂z

∂y

= 2 (y− 2 )= 2 y− 4 ,

∂^2 z

∂y^2

= 2

and

∂^2 z

∂x∂y

=

∂

∂x

(

∂z

∂y

)

=

∂

∂x

( 2 y− 4 )= 0

(v)

∂^2 z

∂x^2

=

∂^2 z

∂y^2

=2and

∂^2 z

∂x∂y

= 0

(vi)

(

∂^2 z

∂x∂y

) 2

= 0

(vii) =( 0 )^2 −( 2 )( 2 )=− 4

(viii) Since<0and

∂^2 z

∂x^2

>0,the stationary point

(1, 2) is a minimum.

The surfacez=(x− 1 )^2 +(y− 2 )^2 isshowninthree

dimensions in Fig. 36.7. Looking down towards the

x-yplane from above, it is possible to produce acon-

tour map. A contour is a line on a map which gives

places having the same vertical height above a datum

line (usuallythe mean sea-level on a geographical map).