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

122

MAXIMA AND MINIMA

when x = 0, therefore v = o may give a maximum or minimum
of y. To determine whether it does, put y = uvm as in 145, and
we have
U4 4m- 4 Aa2u +1w + 1V- = 0;
4m = m+ 1 gives m =, and 4mor + 1 <4;.'. we may
assume m = ~, and this gives U4 - 4a2l = o when w = o; and.u. u = (4a'2), we therefore have
y = (4a'2)xA+ R.
Again, 4m = 4 makes 4m or 4 > mr + 1, and must therefore be rejected.
And again, m + 1 = 4 makes m= 3, and.'. m +1 or
4 < 4m; which gives us u = when = 0, and
4 a'
4as
X3
~-. Y + R.
Now by (168), both these expressions for y shew that y = 0
is neither a maximum nor a minimum value of y.
It appears from what has been just explained, that when we
put y = Uxm for y- b = u (x - a)" if m = a and y= b be the
values we are concerned with }, then we may immediately reject
any value of m which is not an even number, or a fraction
with an even numerator and odd denominator: bearing this in
mind we may very readily obtain the maxima or minima values
of y.
y3 dy
Lastly, m= Y makes - =co, and in virtue of (1) we
1 3
have y = 38a, I= 38ar
Now assume v = 3sa + z, and y = 3a +uzm, and (1)
becomes
(38a + uzm)4 - 4a2 (38a + utz"') (38a + z) + (3a + z)' =0,