The Chemistry Maths Book, Second Edition

256 Chapter 9Functions of several variables

The optimization problem in this example has been simplified by using the constraint

to eliminate one of the variables. In general, however, such a simplification is either

difficult or impossible, particularly for functions of more than two variables or when

there are several constraint relations. A general procedure for solving many of the

constrained optimization problems in the physical sciences is the method of

Lagrange multipliers.

3

We consider a function of three variables,f(x, y, z), and the constraint relation

g(x, y, z) 1 = 1 contant. By the method of Lagrangian multipliers there exists a number λ

such that

(9.11)

at the stationary points. These three equations and the constraint relation are

sufficient to determine the values of the stationary points and of the corresponding

values of the Lagrange multiplierλ.

EXAMPLE 9.7Use the method of Lagrange multipliers to find the extremum value

of the functionf 1 = 13 x

2

1 − 12 y

2

subject to the constraintx 1 + 1 y 1 = 12 (Example 9.6).

We haveg 1 = 1 x 1 + 1 yandf 1 − 1 λg 1 = 13 x

2

1 − 12 y

2

1 − 1 λ(x 1 + 1 y), so that the three equations to be

solved are

The solution isx 1 = 1 −4,y 1 = 1 6,λ 1 = 1 − 24 , andf(−4, 6) 1 = 1 − 24 as in Example 9.6.

0 Exercise 27(ii), 28(ii)

In the general case, the optimization is that of a function of nvariables,

f(x

1

, x

2

, x

3

,1=, x

n

) (9.12a)

subject tom(< 1 n)constraint relations

g

k

(x

1

,x

2

,x

3

,1=,x

n

) 1 = 1 a

k

, k 1 = 1 1, 2, 3,1=, m (9.12b)

∂

∂

−=−=,

∂

∂

−=−−=,

x

fgx

y

( λλ) 60 (fg yλ λ) 40 ggxy=+= 2

∂

∂

−

∂

∂

=,

∂

∂

−

∂

∂

=,

∂

∂

−

∂

∂

=

f

x

g

x

f

y

g

y

f

z

g

z

λλλ 000

3

Joseph-Louis Lagrange (1736–1813), born in Turin, made important contributions to many branches

of mathematics, and has been called the greatest mathematician of the 18th century (‘Lagrange is the lofty

pyramid of the mathematical sciences’, Napoleon Bonaparte). His greatest achievement was the development of

the calculus of variations, and in his Mécanique analytiqueof 1788 he extended the mechanics of Newton

and Euler. He emphasized that problems in mechanics can generally be solved by reducing them to differential

equations. Lagrange invented the name ‘derived function’ (hence ‘derivative’) and notationf′(x)for the derivative

off(x).