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 MINIMA.

117

suppressing the factor si2
sin x
= cos m x cos x (m tan x - tan mv).
Now it is clear from geometrical considerations that
there is some value of mx between 0 and -, which makes
tan m x = m tan x: let this value be ma; then f' (a) = 0. Also
differentiating (1),
f2 (x) = - (m2 - 1) sin mx sin x,
which is negative when x = a, since m2 is > 1, and ma
and therefore a between 0 and -. Hence the root of the
2
equation tan m x - m tan x = 0 which multiplied by m lies
T sin me
between o and - makes - a maximum.
2 sin x
t f(, = /sin n m 2 Example 4.
Let f(x)= -(sin,
sin m j
then f' (x) = 2 i — (m cos mx sin x - sin mx cos x)
= sin mx sin M (m cos mx sin x - sin mx cos av),
sin4 07
multiplying by the factor -.
Now x a (see last example) makes f'(() = 0; also
sin mx sin x is positive when x = a, and may therefore be
suppressed so far as this value of x is concerned, therefore
f2(x) has the same value as before and is therefore negative
when = a; which indicates a maximum.
Again, v = - makes sin mx = o, and the product of the
other factors of f' () negative; therefore by (165) so far
as this value of x is concerned we may put
f'(x) = - sin mx;