An elementary treatise on the differential calculus, in which the method of limits is exclusively made use of, by the Rev. M. O'Brien.

CHAPTER XI.
DETERMINATION OF THE MAXIMA AND MINIMA VALUES OF
FUNCTIONS.
Maxima 161. THE Differential Calculus may be applied with
and Minima values great success to determine the maxima and minima values
defined. of a function, i.e. those particular values which are either
greater or less than any of the neighbouring values.
If f(ix) increases when tv approaches a certain value a
(supposing iv to increase continually), and diminishes when
i, passes the value a, then f(a) must be greater than any
value of f(x) which is either a little less or a little greater
than a, and is therefore called a maximum value of f(v).
And if f(v) diminishes when tv approaches a and increases
when v passes a, f(a) must be less than any value of f (v)
which is either a little less or a little greater than a, and
is therefore called a minimum value of f(i).
How these 162. Now by Lemma XVIII. f(tv) is increasing or
values of
f(x) may diminishing according as f'(x) is positive or negative: therebeefound by fore if f(a) be a maximum, f'(') must be positive for all
means of
f (x). values of s a little less than a, and negative for all values of v
a little greater than a; i.e. f'(v) must change its sign from
+ to - when.v passes through the value a: and, conversely,
if f'(v) changes its sign from + to - when ov passes through
the value a, f(a) is a maximulm.
In like manner if f(a) be a minimum, f'(v) must change
its sign from - to + when tO passes through the value a; and
conversely iff'(v) does so change its sign, f(a) is a minimum.
Hence by Lemma XVII if f(a) be a maximum or minimum, f'(a) must be zero or infinity: but of course, sincef'(a)
may be zero or infinity without f (v) changing its sign, it does