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DERIVATIVES AND INTEGRALS 
209 
2. Show that the polynomials 
2 xP + 3# 2 — 12# + 7, 3# 4 + 8# 3 — 6# 2 - 24# +19, 
are positive for #>1. 
3. Show that # — sin# is an increasing function for all values of #, and 
that tan x — x increases as x varies from to ¿77. For what values of a is 
ax- sinx a steadily increasing or decreasing function of #? 
4. Show that tan x — x also increases from x = \tt to #=|tt, from x=\n 
to x = %n, and so on, and deduce that there is one and only one root of the 
equation tan #=# in each of these intervals (cf. Ex. xvni. 5). 
5. Deduce from Ex. 3 that sin x - x < 0 if x > 0, from this that 
cos#— 1 +^# 2 >0, and from this that sin# — # + $# 3 >0. And, generally, 
prove that if 
0G* 
C' 2ot =cos#- 1 + 2 f 
£2m + l = sin#-# + g-| 
^2 m +1 
/-I)»» 
k ; (2m + l)!’ 
and #>0, then (7 m ^0 and S 2m +1 < 0 according as m is odd or even. 
6. Show that # sin #+cos x + ^ cos 2 # increases as x increases from 
0 tO ijTT. 
7. If /(#) and /"(#) are continuous and have the same sign at every 
point of an interval {a, b), this interval can include at most one root of 
/(#) = 0. 
8. The functions u, v and their derivatives u', v' are continuous 
throughout a certain interval of values of #, and uv —u'v never vanishes 
at any point of the interval. Show that between any two roots of u = 0 
occurs one of v—0, and conversely. 
[If v does not vanish between two roots of u=0, say a and /3, the 
function u/v is continuous throughout the interval (a, /3) and vanishes at its 
extremities. Hence {u/v)' = {u'v-uv')lv 2 must vanish between a and ¡3, which 
contradicts our hypotheses.] 
9. Verify the preceding theorem when u = cos#, v = sin#. 
10. Show how to determine as completely as possible the multiple roots 
of P (#)=0, where P{x) is a polynomial, with their degrees of multiplicity, 
by means of the elementary algebraical operations. 
[If Hi is the highest common factor of P and P\ II 2 the highest common 
factor of H\ and P”, H 3 that of H 2 and P"\ and so on, then the roots of 
Hi¡H9=0 are the double roots of P=0, the roots of H 2 /H 3 =0 the treble roots, 
and so on. But it may not be possible to complete the solution of Hi/H 2 = 0, 
H 2 /H 3 =0, etc. Thus if P(#) = (#- 1) 3 (# 5 — #— 7) 2 , ^T 1 /i/ 2 = # 6 —# —7 and 
//2/7/3=# - 1 ; and we cannot solve the first equation.] 
11. Show that # 6 — 10# 2 +15# — 6=0 has a treble root, and find it. 
12. If (/>(#) is a polynomial and /(#) = (#- a) r cf> (#), then 
/'(«)=/"(«)= ••• =/( r - 1 )(a) = 0, f( r ) (a)=r\<j> (a).