The Chemistry Maths Book, Second Edition

9.4 Stationary points 255

The determination of the nature of these points is summarized in the following table.

Table 9.1

xyf

xx

f

yy

f

xy

f

xx

f

yy

1 − 1 f

2

xy

nature

2 0 12 24 0 > 10 minimum

− 20 − 12 − 24 0 > 10 maximum

4 − 816 < 10 saddle point

− 48 − 16 < 10 saddle point

0 Exercises 24–26

Optimization with constraints

The finding of the maxima and minima (extremum values) of a function is called

optimization. In Example 9.4 the variables xand yare independent variables, with no

constraints on their values. In many applications in the physical sciences, however,

the optimization may be subject to one or more constraints; we have constrained

optimization. These constraints usually take the form of one or more relations

amongst the variables.

EXAMPLE 9.6Find the extremum value of the function

f(x, y) 1 = 13 x

2

1 − 12 y

2

subject to the constraintx 1 + 1 y 1 = 12.

In the absence of the constraint, the function has a saddle point atx 1 = 1 y 1 = 10 , and no

maxima or minima. The constraint is a relation between the variables xand ythat

reduces the number of independent variables to 1. In the present case, the search for

a stationary value is restricted to the liney 1 = 121 − 1 x. Thus, substitutingy 1 = 121 − 1 xin the

function gives

f(x, y) 1 = 1 F(x) 1 = 13 x

2

1 − 1 2(2 1 − 1 x)

2

1 = 1 x

2

1 + 18 x 1 − 18

Then, for a stationary value,

F′(x) 1 = 12 x 1 + 181 = 1 0, F′′(x) 1 = 12

so that F(x)has minimum value forx 1 = 1 − 4. The extremum point off(x,y)is therefore

a minimum point at (x, y) 1 = 1 (−4, 6), and the extremum value of the function is

f(−4, 6) 1 = 1 − 24.

0 Exercises 27(i), 28(i)

−

4

3

−

2

3

4

3

2

3