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

120

MAXIMA AND MINIMA

former value,f'(y) changes its sign from + to -, and throlgh
the latter from - to +: the latter therefore makes +' (y) a
minimum.
Hence f(vy) has a total minimum value obtained by
putting v = (ay2)~ and y = a; i.e. v = 2Qa and y 2a.
(For more Examples see Appendix K.)
Theequa- 171. Since f(xay) is a maximum or minimum on the
tions
d,f(xy) supposition that y is constant, it is clear that d f(z'y) must
d=0 or become 0 or c when V1 is put for Ci, and this being true
=0 or co, whatever value is assigned to y, it must be true when y= b
give us
thosevalues and therefore x = a. Hence if f(ab) be a total maximum or
of x and y
whichmay minimum, we must have
make
f(y) a df(xy) = 0 or o... (1)
total maximum or
minimum. when a and b are put for v and y.
And in exactly the same way we may shew, by treating
y as we have done v, and x as we have done y, that we
must have
dyf(xy)= 0 or o... (2)
when a and b are put for i and y.
a and b are therefore values of,v and y got from the
equations (1) and (2) taken together; and these equations
therefore give us all the values of v and y which may make
f(ivy) a total maximum or minimum.
The method of distinguishing maxima and minima of functions of two variables given by Lagrange is often troublesome;
and since it does not include the cases where any of the partial
differential coefficients of f(vy) become infinite, or where those
of the second order vanish, it must be considered as very incomplete.
How to find tt 172. When y and x are connected by an equation,
maximum
and mini- and we wish to find what values of Cv make y a maximum or
mumvalues minimum, we have only to differentiate the equation to find
of y, when