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

47
39. If u y/a +x-2x; when xw= oc, u= 0,:which,
according to the criterion, is neither a maximum nor a minitmum; but since it cannot become negative, u =, may be considered a minimum.
40. If u"q+mx + a2 b + q-nx2 = o;
when x 2nb + m\/n2 + (2n- 4n) zna
when- -x =
n (m2-4n)
- mb - 2 /nb' + (m - 4n) na
u ' --— n ---, a maximtum;
m" -- n
2nb-m/nb 2 + (m2 - 4n) na2
and when x = 2 -
n (m - 4n)
-mb + 2V/nb2 + (m - 4n) na2
= ---, a minimum.
m - 4n
b 1
If m = o, = —, and u = + -- Vnb — 4n'2a2, which
2n - 2n
are respectively a maximum and a minimum.
If n = o, = oo, and u = —, which is neither a maximum
m
nor a minimum.
m a + b2 m a2" _b2
If m= 4n, x= - mb, and u=,b which
mn b -m b
is a minimum.
5
41. If uu = 2x — x2 (1 - x)2; when x = 1, u = 1, a maxi9 2869
mum: when x= -= u= --, a minimum of the first kind;
25 3125
and when x =1, u = 1, which is also a maximum of the second
kind.
42. If =cos x +cos2x+cos 3x; when =O, u=3,
a maximum; when x cos -1 - - u= 27 -\ 6 27
a maximum.