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

CHAP. VII.
MISCELLANEOUS EXAMPLES.
1. IF w=a2 -2,ax+b2; when x =a, u will be a minimum,.
whose value is b' - a.
2. If u-X2+ ax +b'; when x= '-a, u=Vb.-+a'
will be a minimum.
3. If9x-X2 ~24x + 16; when x=29,~ u=36, a
maximum; when x = 4, u =32, a minimum.
4. If U=x'-_18X2+ 96x-20; when x=4, u=140, amaximum; when x = 8, u = 108, a minimum.
5. Ifu /- 28ax+ 8ax-9q6a'x~.+48a4; when
x=a, u=iia4, a minimum; when x=2a, U=i6a4, a
maximum; and when x =4a, u =-1i6a a mninimum.
6. If u =12x'- 45&4 +40x3; when x=,u=7, a
maximum; when x = 2, u =-16, a minimum; -and when
x = 0, u = 0 is neither a maximum nor a minimum.
7. i-f u=&6-6ax'+9ax'x+a6; -when x=o,u=a6, a
minimum; when x = 2 a, u = 17 a6,- a Maximum; and when
x=3 a, u =a, a minimum.
8. If u=10x6-12x'+1.5x4-2O'o0+2o; when x=0,
u =20, which is neither a maximum nor a minimum,; and
when x =1, u =13, a minimum.
9. If u= a-2b~ A-x when
6 + n /m a2-b' nb~ + (ma'-b2) (in- n2)
m, m in-n' u =Mi
a maximum, or a minimum, according as m is less or greater