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

MAXIMA AND) M INIMA.

113

no llow that f (a) must be a maximum or minimum whenever f'(a) is zero or infinity.
163. Hence to determine the maxima and minima of Rile for
determinfix), we must determine what values of x make f'(x) zero or ing maxima
infinity, and try whether f'(x) changes its sign when x passes m ia.
through each of these values; those values which give a change
from + to - make f(x) a maximum; those which give a
change from - to + make f(x) a minimum; and those which
do not give a change must be rejected.
164. In applying this method to any example, we may Factors of
suppress any factor of f'(x) which we are sure is always f-(Spay
positive, or may introduce any such factor; since we are not pressed
". - *' ~under
concerned with the actual magnitude of f'(x) but only its certain
circumsign. This will often considerably simplify our operations, stanes
as will appear.
If therefore we find f'(x) in the form 0 (x). \f (x), and
if p (x) be always a positive quantity, then we may put
f'(x) =, (x) simply.
If P/ (x) be always a negative quantity, then f'(x) will
hlave the same sign as -, (x), and therefore we may put
f'(x) -- ().
165. If it be our object simply to examine whether f'(x)
changes its sign, and how, when x passes through a certain
value a, then if <p (a) be neither zero nor infinity, we may
suppose f'(x) = + (x) if ( (a) be positive, and f'() = -fi(x)
if p (a) be negative.
For it is clear that if p (a) be neither zero nor infinity,
then for all values of x near a 0((x) has the same sign as
0 (a), and therefore 0 (). (x) or f'(x) the same sign
as p (a) (v), i.e. f'(x) has the same sign as + 4, (x) or
- xP (x) according as the sign of 5 (a) is + or - for all values
of x near a, and therefore in examining whether f'(x) changes
its sign, and how, when x passes through a, we may put
f'(x) = + \ (x) or - \ (x) according as ) (a) is positive
or negative.
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