The Chemistry Maths Book, Second Edition

116 Chapter 4Differentiation

This is a maximum or minimum, depending on the sign of the fourth derivative. For

example, the functiony 1 = 1 (x 1 − 1 1)

4

satisfies (4.24), with a minimum atx 1 = 11.

EXAMPLE 4.23Find the stationary points of the quarticy 1 = 13 x

4

1 − 14 x

3

1 + 11.

For the stationary values,

= 1 0 whenx 1 = 1 0orx 1 = 1 1.

To determine the kinds of stationary points,

To determine the kind of stationary point atx 1 = 10 ,

whenx 1 = 1 0, a point of inflection

The function therefore has a single turning point, a minimum, at x 1 = 11 , when

y 1 = 10 , and a point of inflection at x 1 = 10 , when y 1 = 11. In fact, the function can be

factorized,

y 1 = 1 (x 1 − 1 1)

2

(3x

2

1 + 12 x 1 + 1 1)

and has a double rootx 1 = 11 , and two complex roots.

0 Exercise 83

EXAMPLE 4.24In Hückel molecular orbital theory, the possible values of the

orbital energies of the πelectrons of ethene (C

2

H

4

) are given by the stationary values

of the quantity

ε 1 = 1 α 1 + 12 c(1 1 − 1 c

2

)

122

β

where αand βare constant ‘Hückel parameters’, and cis a variable. For the stationary

values of ε,

d

dc

ccc

ε

=− ββ− − =

−

21 2 1 0

212 2 2 12

() ()

dy

dx

x

3

=−≠72 24 0

dy

dx

xx

x

2

36 24

00

01

=−

==

=,

when

when a minimumm











dy

dx

=−= −12 12 12xxxx 1

322

()

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Figure 4.10