210 
DERIVATIVES AND INTEGRALS 
[VI 
13. Determine the maxima and minima (if any) of (x — l) 2 (x + 2), x 3 — Zx, 
2X 3 — Zx 2 — 36a' + 10, 4a -3 — 1 8x 2 -f 27# — 7, 3# 4 —4# 3 + l, a 4 — ISa^+S. In each 
case sketch the form of the graph of the function. 
[Consider the last function, for example. Here ft (x) = 5x 2 (x 2 — 9), which 
vanishes for #=0, x — +3. It is easy to see that x— — 3 gives a maximum, 
x= +3 a minimum, while x=0 gives neither, as ft(x) is negative on both 
sides of # = 0.] 
14. Discuss the maxima and minima of #(#-1), x 2 (x-l) 2 , x 3 (x-l) 3 , 
x A (x— l) 4 . Sketch the graphs of the functions. 
15. Discuss similarly the function (x - a) (x - b) 2 (x - c) 3 , distinguishing 
the different forms of the graph which correspond to different hypotheses as 
to the relative magnitudes of a, b, c. 
16. Discuss similarly the function (x — a) m (x — b) n , where m and n are 
any positive integers, considering the different cases which occur according as 
m and n are odd or even. 
17. Show that (ax+b)/(cx+d), whatever values a, b, c, d may have, has 
no maxima or minima. Draw a graph of the function. 
18. Discuss the maxima and minima of {(1 — x)/{l+x)} 2 , 2x/{l+x 2 ), 
2x/{l-x 2 ), {l-x 2 )/{l+x 2 ), {\+x 2 )/{\ —jfi). 
19. Discuss the maxima and minima of the function 
?/ = (ax 2 + 2 bx + c)/(Ax 2 + 2 Bx + c), 
when the denominator has complex roots. 
[We may suppose a and A positive. The derivative vanishes if 
(ax + b) (Bx -\-c)-(Ax-\-B)(bx+c)—0 (1). 
This equation must have real roots. For if not the derivative would 
always have the same sign, and this is impossible, since y is continuous for all 
values of x, and y-*-a/A as x-^+oo or x-» — oo. 
It is easy to verify that the curve cuts the line y = ajA in one and only 
one point, and that it lies above this line for large positive values of x, and 
below it for large negative values, or vice versa, according as bja < BjA. 
Thus the algebraically greater root of (1) gives a maximum if b/a>B/A, a 
minimum in the contrary case.] 
20. The maximum and minimum values themselves are the values of X 
for which ax 2 + 2bx + c-\(Ax 2 + 2Bx + C) is a perfect square. [This is the 
condition that y = X should touch the curve.] 
21. In general the maxima and minima of R (x) = P (x)/Q (x) are among 
the values of X obtained by expressing the condition that P(x) - XQ(x) = 0 
should have a pair of equal roots. 
22. If Ax 2 + 2Bx+ C=0 has real roots it is convenient to proceed as 
follows. We have 
v-(ajA) — (\x + ft)¡{A (Ax 2 + 2Bx + 0)}