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

116

MAXIMA AND MINIMA.

Now 8iv - 5a =0 when iv = -, and changes its sign
5a
from - to + when v passes through the value - hence
X = 5 a gives a minimum value of f(iv), vii.
(i)' ( 5a - (a
8 8 8i/
Or thus f'Qev)= 2v - 5sa;.f'~ (iv) =2;.~. f2 (+) = positive,
88
which indicates a minimum by (167).
Example 2. Let f(IV)
(i- - a)''
iv2
then f'(v)= - (- 2iv-;3a)
(iv- a)F
— (2,v + 3a), suppressing the factor -
(V - a)"'
3a
SNow - (2iv + 3a) is zero when v=, and changes
its sign from + to - when x passes through that value:
hence x = - gives a maximum value of f(v).
2
Or thus, f'(v) = - (2ix + 3a)
f ). = negative,
which indicates a maximum.
sin miv:
Example 3. Let f(s) =, m being an integer,
then ) = 2~
sin x
=mcos mx sin x - sin mx cosi ()
sin 2 X
m cos mv sin x - sin mx cos x...()