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

43

than n', provided m- n2 and ma' - b' have the same sign;
and when
b n ma2 -2 nb- (m a -b2) (m- n)
0 = - - - - --
mm m -n m
a minimum or a maximum on the same hypothesis: also,
when m-n2 and ma - b2 have different signs, the values
of x and u become impossible, and consequently by (115)
there can be neither a maximum nor a minimum.
a a
If m=2, n=1 and b=o; when =-, = --
-a 3a
a minimum; and when x = -_/, - = a maximum.
5/V,/ 2 x
10. If u = 24x'A - 135x4 + 2803 - 270x2 + 120x + 180;
when x-=, 2u=1993, a maximum; when x=2, u7=188,
a minimum: and when x =1, t=199, is neither.
11. If u = (x - a)"; when x = a, u =, which is a minimum if n be even; but neither a maximum nor a minimum
if n be odd, as appears from (112) and (113).
12. If u=x2 (a-.)3; when x=0, u=0, a minimum;
when x = a, u =, which is neither a maximum nor a mini2a 108a5
mum; and when x =, u =, a maximum.
5 3125
13. If u = (x - 1)4 (x + 2)3; when x= -2, = 0, which
5
is neither a max inm nor a minimum; when = -,
112"."7
i24. 9'3
u = a maxinmum; and when = 1, u = 0, a minimum.
7
14. If u=xm (a- x)"; when x=0, u=0, which is a
minimum when m is even, but neither a maximum nor a
minimum when m is odd; when x = a, u = 0, which is a
minimum when n is even, but neither a maximum nor a. Cb qa m?z am + n
ma mmnfar
minimum when n is odd; when x= —, = ---m + n (m + n)+)m
which is a maximum: as appears from (114).