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

114

MAXIMA AND MINIMA.

We may therefore suppress all factors of f'(v) which do
not become 0 or co when w = a, retaining only the signs they
have when x = a. This is a very important simplification.
(qu) is a 166. If (p (u) always increases when u increases, it is a
maxlmun
or mini- maximum or minimum whenever u, is a maximum or minimum
mum when-) ()
ever ui is, d~ 0 (~u) d7u
if v (u) respectively; for )= ' (j) -: and since ( (u) always
always d
increases increases when u increases, Op' (u) is always positive, and
when u increases: if d) (u) du
not the re- therefore has always the same sign as -, and.'.
verse is the dx d '
case. (u) and u become maxima or minima at the same time.
If ( (u) diminishes when x increases the reverse is the case.
We often find this consideration of use in practice, inasmuch
as it may in many cases be much easier to find the maxima and
minima of 0 (u) than of u.
We may 167. If f'(a) = 0, then by Lemma XX, Cor. 4, f '(a)
determine
whether has in general the same sign as f (a) (v - a) for all values of
af(x) hw taken sufficiently near a: therefore if f(a) be positive f(x)
changes its changes its sign from - to + when x passes through the value
sign when x
passes a, which indicates a maximum; and if f2(a) be negative the
through a,
by the con- change is fiom + to -, which indicates a maximum. If howsideration ever f2(a) = 0, let f"(a) be the first differential coefficient
second or of f(x), which does not vanish when = a; then f'(x), (by
higher differential Lemma XX, Cor. 4), has the same sign as f"(a) (m - a)"-'
coefficient. for all values of v sufficiently near a. If, therefore, n be odd,
f'(x) does not change its sign when x passes through the
value n; but if n be even it does, and the change is from
- to + or from + to - according as ff (a) is positive or
negative. Of course we here suppose that none of the differential coefficients are infinite.
Simple inspection is These considerations will enable us to determine whether
often the
best way f '(x) changes its sign when x passes through the value a, and
and some- if so, whether the change is from - to + or from + to -
only way of But this is often more easily seen by simple inspection; and
making out
whether indeed when f'(a) or any of the higher differential coefficients
f' (x) ~
changes its are Infinite, which often occurs, simple inspection is the only
sign and method we can resort to.
how.