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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37. Find the maximum or minimum value of u-=x'y,
subject to the condition expressed by the equation abY = - k.
38. Required the maximum or minimum value of
= (x + a) (y + b) (z + c),
the equation of condition being abycCZ — k.
39. Determine the maximum or minimum value of
U = (a - 1) (by -1) (CZ - ),
when the equation of condition is axbycz =k.
40. It is required to find the maximum or minimum
value of u = (. when the equation of
\ oo I \x \ %j
condition is xwyzay = k.
41. What is the maximum value of the function
u = oyz-y^tc,
when the quantities x, y, z and t are subject to the condition
expressed by the equation x + 2y + 3z + 4t = a?
42. If x + y + z = w, it is required to find the value of
- = sin x sin2 y sina z, when it is a maximum or a minimum.
43. Find the maximum or minimum value of
= sin x sin y + sin x sin z + sin y sin z,
when the equation of condition is x + y + z = 45~.
44. Determine whether the function xm log y sin z, admits
of a maximum or a minimum, when the equation of condition is
x y z
- - + -=1.
a b c
45. In the equation u = a — (X2 y2), it is required to
prove by the substitution of o + h and o + k in the places of
x and y, that when x =O and y=o, the value of u =a is
a maximum.
R