105] 
DERIVATIVES AND INTEGRALS 
211 
where \—bA—aB, fx — cA —aC. Writing further £ for Xx+/x and rj for 
(d/X 2 ) {Ay - a) we obtain an equation of the form 
This transformation from {x, y) to (£, rj) amounts only to a shifting of the 
origin, keeping the axes parallel to themselves, a change of scale along each 
axis, and (if X<0) a reversal in direction of the axis of abscissae, and so a 
minimum of y, considered as a function of x, corresponds to a minimum of rj 
considered as a function of £, and vice versa; and similarly for a maximum. 
The derivative of rj with respect to £ vanishes if 
(£-?)(£-?)-£(£-.p)-£(£-?)=o, 
or if £ 2 =pq- Thus there are two roots of the derivative if p and q have the 
same sign, none if they have opposite signs. In the latter case the form of 
the graph of rj is as shown in Fig. 49 a. 
0 
Fig. 49 a. 
Fig. 49 6. 
Fig. 49 c. 
When p and q are positive the general form of the graph is as shown in 
Fig. 49 b, and it is easy to see that £=Jpq gives a maximum and £= —\Jpq 
a minimum*. 
In the particular case in which p=q the 
graph of 
is of the form shown in Fig. 49 c. 
The preceding discussion fails if X = 0, i.e. 
if a/A = blB. But in this case we have 
y - {a/A) = p/{A {Ax 2 + 2Bx+C)} 
= (mM 2 )/{(^ ~Xi)(x- x 2 )}, 
Fig. 50. 
say, and dyjdx—0 gives the single value x = ^(xi+x 2 ). On drawing a graph 
it is clear that this is a maximum or minimum according as ¡x is positive or 
negative. The graph shown in Fig. 50 corresponds to the former case. 
[A full discussion of the general function y = {ax 2 + 2bx + c)/{Ax 2 + 2Bx+C), 
will be found in Chrystal’s Algebra, vol. I. pp. 464-7 : there however only 
purely algebraical methods are used.] 
23. Show that {x - a) {x - [H)/{x - y) assumes all real values as x varies, if a 
lies between /3 and y, and otherwise assumes all values except those included 
in an interval of length 4 y/{{a ~/3) (a ~ y)]. 
* The maximum is -1 l{Jp - Jq)' 2 , the minimum -1 /(v / P +V?) 2 » which the 
latter is the greater. 
14—2