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

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8. If u =2a'xy - 3ax'y - ay' + Xly + wy3, it is required
to prove that when x=a and y-1 athe value of u a='.
31 4~
a minimum: when x=4a ard y -ia, the value of u = a,
a maximum: when x=-I-a and y'= -'-a, the value of u =1a4,
a maximum: and when x= a and y -a, the value of
u - a, a minimum.
9. Given ut=x4+ y4-4axy', it is required to shew that
when x=a~- and y=24a, the value of u= - a, a
minimum.
10. If 0u =y2 (a - x - y)3, it is required to prove that
when x= a and y a, the value of u=4 a6, a maximum.
11. If u 'y (x4 + 2 y3 - a)', it is required to prove
13a
that corresponding to x= - and y the value
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of u is a maxmimum.
12. Given u=y4-S-y'+18y'-8y+x'-3x2-3x, it is
required to shew that when x= 1- <2 and y =2, the value
of u=3+ 4 V a maximum; and when x=1 +V'2 and
y= 2+V', the value ofu = - 6 - 4 N/2, a minimum.
13. Given u = y' - 8y' + 18y'2 -8y -_x' + 3x2 + 3, it is
required to shew that when x = 1 + - and y = 2, the value
of u= 13~4V4\2, a maximu: and when x=1 - and
y = 2 + 2, the value of u = 4 - 4 V2, a minimum.
ao a3
14. If u=xy + - + -, then if x= a and y=a, it is
X Y
required to prove that u = 3 a 2 is a minimum.
15. If u =sin x + sin y + sin (x - y), it is required to
prove that when x = 6o0 and y = 600, the value of a = -i <p3
a maximum.
16. If u = sin x sin y sin (x + y), it is required to shew
that when x = 600 and y = 60', li = 3 V3, a maximum.
17. Find whether the function u =x2y 3(a- 2x -Y)"
admit-s of a maxinqmu or a mrni'mum, and determine its value.