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Section2.3
DifferentiableFunctionsofOneVariable
83
(Exercise2.3.3),andthereforec ¤ ¤ a andc c ¤ b. . Hence, , a a < < c c < b, , andtherefore
g
0
.c/D0,byTheorem2.3.7.From(2.3.18),f
0
.c/D.
Theproofforthecasewheref0.b/<<f0.a/canbeobtainedbyapplyingthisresult
tof.
MeanValueTheorems
Theorem2.3.10(GeneralizedMeanValueTheorem)
Iffandgarecon-
tinuousontheclosedintervalŒa;banddifferentiableontheopeninterval.a;b/;then
Œg.b/g.a/f
0
.c/DŒf.b/f.a/g
0
.c/
(2.3.20)
forsomecin.a;b/:
Proof
Thefunction
h.x/DŒg.b/g.a/f.x/Œf.b/f.a/g.x/
iscontinuousonŒa;banddifferentiableon.a;b/,and
h.a/Dh.b/Dg.b/f.a/f.b/g.a/:
Therefore,Rolle’stheoremimpliesthath
0
.c/D0forsomecin.a;b/.Since
h
0
.c/DŒg.b/g.a/f
0
.c/Œf.b/f.a/g
0
.c/;
thisimplies(2.3.20).
ThefollowingspecialcaseofTheorem2.3.10isimportantenoughtobestatedseparately.
Theorem2.3.11(MeanValueTheorem)
Iff iscontinuousontheclosed
intervalŒa;banddifferentiableontheopeninterval.a;b/;then
f
0
.c/D
f.b/f.a/
ba
forsomecin.a;b/:
Proof
ApplyTheorem2.3.10withg.x/Dx.
Theorem2.3.11impliesthatthetangenttothecurveyDf.x/at.c;f.c//isparallelto
thelineconnectingthepoints.a;f.a//and.b;f.b//onthecurve(Figure2.3.5,page84).
Consequences oftheMeanValueTheorem
Iff isdifferentiableon.a;b/andx
1
, x
2
2 .a;b/thenf iscontinuousontheclosed
intervalwithendpointsx
1
andx
2
anddifferentiableonitsinterior.Hence,themeanvalue
theoremimpliesthat
f.x
2
/f.x
1
/Df
0
.c/.x
2
x
1
/
forsomecbetweenx
1
andx
2
.(Thisistruewhetherx
1
<x
2
orx
2
<x
1
.)Thenextthree
theoremsfollowfromthis.
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84 Chapter2
DifferentialCalculusofFunctionsofOneVariable
Theorem2.3.12
Iff
0
.x/D0forallxin.a;b/;thenf isconstanton.a;b/:
Theorem2.3.13
Iff
0
existsanddoesnotchangesignon.a;b/;thenf ismonotonic
on.a;b/Wincreasing;nondecreasing;decreasing;ornonincreasingas
f
0
.x/>0; f
0
.x/0; f
0
.x/<0; or f
0
.x/0;
respectively;forallxin.a;b/:
Theorem2.3.14
If
jf
0
.x/jM; a<x<b;
then
jf.x/f.x
0
/jMjxx
0
j; x;x
0
2.a;b/:
(2.3.21)
Afunctionthatsatisfiesaninequalitylike(2.3.21)forallxandx
0
inanintervalissaid
tosatisfyaLipschitzconditionontheinterval.
y
x
b
c
a
y = f(x)
f(b)
f(c)
f(a)
Figure2.3.5
2.3Exercises
1.
Provethatafunctionf isdifferentiableatx
0
ifandonlyif
lim
x!x
0
f.x/f.x
0
/m.xx
0
/
xx
0
D0
forsomeconstantm.Inthiscase,f
0
.x
0
/Dm.
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Section2.3
DifferentiableFunctionsofOneVariable
85
2.
Prove:Iffisdefinedonaneighborhoodofx
0
,thenf isdifferentiableatx
0
ifand
onlyifthediscontinuityof
h.x/D
f.x/f.x
0
/
xx
0
atx
0
isremovable.
3.
UseLemma2.3.2toprovethatiff0.x
0
/>0,thereisaı>0suchthat
f.x/<f.x
0
/if x
0
ı<x<x
0
and f.x/>f.x
0
/if x
0
<x<x
0
Cı:
4.
Supposethatpiscontinuouson.a;canddifferentiableon.a;c/,whileqiscon-
tinuousonŒc;b/anddifferentiableon.c;b/.Let
f.x/D
(
p.x/; a<xc;
q.x/; c<x<b:
(a)
Showthat
f
0
.x/D
(
p
0
.x/; a<x<c;
q
0
.x/; c<x<b:
(b)
Underwhatadditionalconditionsonpandqdoesf
0
.c/exist? Provethat
yourstatedconditionsarenecessaryandsufficient.
5.
Findallderivativesoff.x/Dxn1jxj,wherenisapositiveinteger.
6.
Supposethatf0.0/existsandf.xCy/Df.x/f.y/forallxandy.Provethatf0
existsforallx.
7.
Supposethatc
0
.0/Daands
0
.0/Dbwherea
2
Cb
2
¤0,and
c.xCy/Dc.x/c.y/s.x/s.y/
s.xCy/Ds.x/c.y/Cc.x/s.y/
forallxandy.
(a)
Showthatcandsaredifferentiableon.1;1/,andfindcands0interms
ofc,s,a,andb.
(b)
(Forthosewhohavestudieddifferentialequations.)Findcandsexplicitly.
8. (a)
Supposethatf andgaredifferentiableatx
0
, f.x
0
/ D g.x
0
/ D 0, , and
g
0
.x
0
/¤0.WithoutusingL’Hospital’srule,showthat
lim
x!x
0
f.x/
g.x/
D
f0.x
0
/
g0.x
0
/
:
(b)
Statethecorrespondingresultsforone-sidedlimits.
9.
ProveTheorem2.3.4
(a)
.
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86 Chapter2
DifferentialCalculusofFunctionsofOneVariable
10.
ProveTheorem2.3.4
(b)
.
11.
ProveTheorem2.3.4
(d)
.
12.
Provebyinduction:Ifn1andf
.n/
.x
0
/andg
.n/
.x
0
/exist,thensodoes.fg/
.n/
.x
0
/,
and
.fg/
.n/
.x
0
/D
Xn
mD0
n
m
!
f
.m/
.x
0
/g
.nm/
.x
0
/:
H
INT
:SeeExercise1.2.19:ThisisLeibniz’srulefordifferentiatingaproduct.
13.
Whatiswrongwiththe“proof”ofthechainrulesuggestedafterExample2.3.3?
Correctit.
14.
Supposethatf iscontinuousandincreasingonŒa;b. . Letf bedifferentiableata
pointx
0
in.a;b/,withf
0
.x
0
/¤0.Ifgistheinverseoff Theorem2.2.15),show
thatg
0
.f.x
0
//D1=f
0
.x
0
/.
15. (a)
Showthatf
0
C
.a/Df
0
.aC/ifbothquantitiesexist.
(b)
Example2.3.4showsthatf
0
C
.a/mayexisteveniff
0
.aC/doesnot.Givean
examplewheref
0
.aC/existsbutf
0
C
.a/doesnot.
(c)
Completethefollowingstatementsoitbecomes atheorem, andprovethe
theorem:“Iff
0
.aC/existsandf is
ata,thenf
0
C
.a/Df
0
.aC/.”
16.
Showthatf.aC/andf.b/exist(finite)iff
0
isboundedon.a;b/. H
INT
:See
Exercise2.1.38:
17.
Supposethatf iscontinuousonŒa;b,f0
C
.a/exists,andisbetweenf0
C
.a/and
.f.b/f.a//=.ba/.Showthatf.c/f.a/D.ca/forsomecin.a;b/.
18.
Supposethatf iscontinuousonŒa;b,f
0
C
.a/<<f
0
.b/,and
.f.b/f.a//=.ba/¤:
Showthateitherf.c/f.a/D.ca/orf.c/f.b/D.cb/forsomec
in.a;b/.
19.
Let
f.x/D
sinx
x
; x¤0:
(a)
Definef.0/sothatfiscontinuousatxD0.H
INT
:UseExercise2.3.8:
(b)
Showthatif
xisalocalextremepointoff,then
jf.
x/jD.1C
x
2
/
1=2
:
H
INT
:Express sinx andcosx intermsoff.x/andf0.x/;andaddtheir
squarestoobtainausefulidentity:
(c)
Showthatjf.x/j1forallx.Forwhatvalueofxisequalityattained?
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Section2.3
DifferentiableFunctionsofOneVariable
87
20.
Letnbeapositiveintegerand
f.x/D
sinnx
nsinx
; x¤k (kDinteger):
(a)
Definef.k/sothatf iscontinuousatk.H
INT
:UseExercise2.3.8:
(b)
Showthatif
xisalocalextremepointoff,then
jf.
x/jD
1C.n
2
1/sin
2
x
1=2
:
H
INT
:Expresssinnxandcosnxintermsoff.x/andf
0
.x/;andaddtheir
squarestoobtainausefulidentity:
(c)
Showthatjf.x/j1forallx.Forwhatvaluesofxisequalityattained?
21.
Wesaythatf hasatleastnzeros,countingmultiplicities,onanintervalI I ifthere
aredistinctpointsx
1
,x
2
,...,x
p
inI suchthat
f
.j/
.x
i
/D0; 0j j n
i
1; 1ip;
andn
1
CCn
p
D n. Prove: Iff isdifferentiableandhasatleastnzeros,
countingmultiplicities,onanintervalI,thenf
0
hasatleastn1zeros,counting
multiplicities,onI.
22.
Giveanexampleofafunctionf suchthatf
0
existsonaninterval.a;b/andhasa
jumpdiscontinuityatapointx
0
in.a;b/,orshowthatthereisnosuchfunction.
23.
Letx
1
,x
2
,...,x
n
andy
1
,y
2
,...,y
n
bein.a;b/andy
i
<x
i
,1i n. Show
thatiff isdifferentiableon.a;b/,then
Xn
iD1
Œf.x
i
/f.y
i
/Df
0
.c/
Xn
iD1
.x
i
y
i
/
forsomecin.a;b/.
24.
Proveorgiveacounterexample:Iffisdifferentiableonaneighborhoodofx
0
,then
f satisfiesaLipschitzconditiononsomeneighborhoodofx
0
.
25.
Let
f
00
.x/Cp.x/f.x/D0 and g
00
.x/Cp.x/g.x/D0; a<x<b:
(a)
ShowthatW Df
0
gfg
0
isconstanton.a;b/.
(b)
Prove: IfW ¤0andf.x
1
/Df.x
2
/D0wherea<x
1
<x
2
<b,then
g.c/D0forsomecin.x
1
;x
2
/.H
INT
: Considerf=g:
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88 Chapter2
DifferentialCalculusofFunctionsofOneVariable
26.
Supposethatweextendthedefinitionofdifferentiabilitybysayingthatf isdiffer-
entiableatx
0
if
f
0
.x
0
/D lim
x!x
0
f.x/f.x
0
/
xx
0
existsintheextendedreals.Showthatif
f.x/D
(p
x;
x0;
p
x; x<0;
thenf0.0/D1.
27.
Proveorgiveacounterexample:Iff isdifferentiableatx
0
intheextendedsenseof
Exercise2.3.26,thenf iscontinuousatx
0
.
28.
Assumethatf isdifferentiableon.1;1/andx
0
isacriticalpointoff.
(a)
Leth.x/Df.x/g.x/,wheregisdifferentiableon.1;1/and
f.x
0
/g
0
.x
0
/¤0:
ShowthatthetangentlinetothecurveyDh.x/at.x
0
;h.x
0
//andthetangent
linetothecurveyDg.x/at.x
0
;g.x
0
/intersectonthex-axis.
(b)
Supposethatf.x
0
/¤0. Leth.x/Df.x/.xx
1
/,wherex
1
isarbitrary.
ShowthatthetangentlinetothecurveyDh.x/at.x
0
;h.x
0
//intersectsthe
x-axisat
xDx
1
.
(c)
Supposethatf.x
0
/¤0.Leth.x/Df.x/.xx
1
/
2
,wherex
1
¤x
0
.Show
thatthetangentlinetothecurveyDh.x/at.x
0
;h.x
0
//intersectsthex-axis
atthemidpointoftheintervalwithendpointsx
0
andx
1
.
(d)
Leth.x/D.ax
2
CbxCc/.xx
1
/,wherea¤0andb
2
4ac¤0.Let
x
0
D
b
2a
. ShowthatthetangentlinetothecurveyDh.x/at.x
0
;h.x
0
//
intersectsthex-axisat
xDx
1
.
(e)
Lethbeacubicpolynomialwithzeros˛,ˇ,and,where˛andˇaredistinct
and isreal. Letx
0
D
˛Cˇ
2
. Showthatthetangentlinetothecurve
yDh.x/at.x
0
;h.x
0
//intersectstheaxisat
xD.
2.4L’HOSPITAL’SRULE
ThemethodofTheorem2.1.4forfindinglimitsofthesum,difference,product,andquo-
tientoffunctionsbreaksdowninconnectionwithindeterminateforms. Thegeneralized
meanvaluetheorem(Theorem2.3.10)leadstoamethodforevaluatinglimitsofindetermi-
nateforms.
Theorem2.4.1(L’Hospital’sRule)
Supposethatf andgaredifferentiable
andg
0
hasnozeroson.a;b/:Let
lim
x!b
f.x/D lim
x!b
g.x/D0
(2.4.1)
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Section2.4
L’Hospital’sRule
89
or
lim
x!b
f.x/D˙1
and
lim
x!b
g.x/D˙1;
(2.4.2)
andsupposethat
lim
x!b
f
0
.x/
g0.x/
DL .finiteor˙1/:
(2.4.3)
Then
lim
x!b
f.x/
g.x/
DL:
(2.4.4)
Proof
WeprovethetheoremforfiniteLandleavethecasewhereL D˙1toyou
(Exercise2.4.1).
Supposethat>0.From(2.4.3),thereisanx
0
in.a;b/suchthat
ˇ
ˇ
ˇ
ˇ
f
0
.c/
g0.c/
L
ˇ
ˇ
ˇ
ˇ
< if x
0
<c<b:
(2.4.5)
Theorem2.3.10impliesthatifxandtareinŒx
0
;b/,thenthereisacbetweenthem,and
thereforein.x
0
;b/,suchthat
Œg.x/g.t/f
0
.c/DŒf.x/f.t/g
0
.c/:
(2.4.6)
Sinceg
0
hasnozerosin.a;b/,Theorem2.3.11impliesthat
g.x/g.t/¤0 if x;t2.a;b/:
Thismeansthatgcannothavemorethanonezeroin.a;b/. Therefore,wecanchoosex
0
sothat,inadditionto(2.4.5),ghasnozerosinŒx
0
;b/.Then(2.4.6)canberewrittenas
f.x/f.t/
g.x/g.t/
D
f0.c/
g0.c/
;
so(2.4.5)impliesthat
ˇ
ˇ
ˇ
ˇ
f.x/f.t/
g.x/g.t/
L
ˇ
ˇ
ˇ
ˇ
< if x;t2Œx
0
;b/:
(2.4.7)
If(2.4.1)holds,letxbefixedinŒx
0
;b/,andconsiderthefunction
G.t/D
f.x/f.t/
g.x/g.t/
L:
From(2.4.1),
lim
t!b
f.t/D lim
t!b
g.t/D0;
so
lim
t!b
G.t/D
f.x/
g.x/
L:
(2.4.8)
90 Chapter2
DifferentialCalculusofFunctionsofOneVariable
Since
jG.t/j< if x
0
<t<b;
becauseof(2.4.7),(2.4.8)impliesthat
ˇ
ˇ
ˇ
ˇ
f.x/
g.x/
L
ˇ
ˇ
ˇ
ˇ
:
Thisholdsforallxin.x
0
;b/,whichimplies(2.4.4).
Theproofunderassumption(2.4.2)ismorecomplicated.Againchoosex
0
sothat(2.4.5)
holdsandghasnozerosinŒx
0
;b/.LettingtDx
0
in(2.4.7),weseethat
ˇ
ˇ
ˇ
ˇ
f.x/f.x
0
/
g.x/g.x
0
/
L
ˇ
ˇ
ˇ
ˇ
< if x
0
x<b:
(2.4.9)
Sincelim
x!b
f.x/D˙1,wecanchoosex
1
>x
0
sothatf.x/¤0andf.x/¤f.x
0
/
ifx
1
<x<b.Thenthefunction
u.x/D
1g.x
0
/=g.x/
1f.x
0
/=f.x/
isdefinedandnonzeroifx
1
<x<b,and
lim
x!b
u.x/D1;
(2.4.10)
becauseof(2.4.2).
Since
f.x/f.x
0
/
g.x/g.x
0
/
D
f.x/
g.x/
1f.x
0
/=f.x/
1g.x
0
/=g.x/
D
f.x/
g.x/u.x/
;
(2.4.9)impliesthat
ˇ
ˇ
ˇ
ˇ
f.x/
g.x/u.x/
L
ˇ
ˇ
ˇ
ˇ
< if x
1
<x<b;
whichcanberewrittenas
ˇ
ˇ
ˇ
ˇ
f.x/
g.x/
Lu.x/
ˇ
ˇ
ˇ
ˇ
<ju.x/j if x
1
<x<b:
(2.4.11)
Fromthisandthetriangleinequality,
ˇ
ˇ
ˇ
ˇ
f.x/
g.x/
L
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
f.x/
g.x/
Lu.x/
ˇ
ˇ
ˇ
ˇ
CjLu.x/Ljju.x/jCjLjju.x/1j: (2.4.12)
Becauseof(2.4.10),thereisapointx
2
in.x
1
;b/suchthat
ju.x/1j< andtherefore ju.x/j<1C if x
2
<x<b:
This,(2.4.11),and(2.4.12)implythat
ˇ
ˇ
ˇ
ˇ
f.x/
g.x/
L
ˇ
ˇ
ˇ
ˇ
<.1C/CjLj if x
2
<x<b;
Section2.4
L’Hospital’sRule
91
whichproves(2.4.4)underassumption(2.4.2).
Theorem2.4.1andtheproofgivenhere remainvalidifb D 1and“x ! b”is
replacedby“x !1”throughout.Onlyminorchangesintheproofarerequiredtoshow
thatsimilartheoremsarevalidforlimitsfromtheright,limitsat1,andordinary(two-
sided)limits.Wewilltaketheseasgiven.
TheIndeterminateForms
0=0
and
1=1
Wesaythatf=gisoftheform0=0asx!bif
lim
x!b
f.x/D lim
x!b
g.x/D0;
oroftheform1=1asx!bif
lim
x!b
f.x/D˙1
and
lim
x!b
g.x/D˙1:
Thecorrespondingdefinitionsforx !bCandx!˙1aresimilar.Iff=gisofoneof
theseformsasx!bandasx!bC,thenwesaythatitisofthatformasx!b.
Example2.4.1
Theratiosinx=xisoftheform0=0asx!0,andL’Hospital’srule
yields
lim
x!0
sinx
x
D lim
x!0
cosx
1
D1:
Example2.4.2
Theratioe
x
=xisoftheform1=1asx!1,andL’Hospital’s
ruleyields
lim
x!1
e
x
x
D lim
x!1
e
x
1
D1:
Example2.4.3
UsingL’Hospital’srulemayleadtoanotherindeterminateform;thus,
lim
x!1
e
x
x2
D lim
x!1
e
x
2x
ifthelimitontherightexistsintheextendedreals.ApplyingL’Hospital’sruleagainyields
lim
x!1
e
x
2x
D lim
x!1
e
x
2
D1:
Therefore,
lim
x!1
e
x
x2
D1:
Moregenerally,
lim
x!1
e
x
x˛
D1
foranyrealnumber˛(Exercise2.4.33).
92 Chapter2
DifferentialCalculusofFunctionsofOneVariable
Example2.4.4
SometimesitpaystocombineL’Hospital’srulewithothermanipula-
tions.Forexample,
lim
x!0
44cosx2sin
2
x
x4
D lim
x!0
4sinx4sinxcosx
4x3
D
lim
x!0
sinx
x

lim
x!0
1cosx
x2
D
lim
x!0
sinx
x

lim
x!0
sinx
2x
D
1
2
lim
x!0
sinx
x
2
D
1
2
.1/
2
D
1
2
(Example2.4.1):
Asanotherexample,L’Hospital’sruleyields
lim
x!0
e
x
2
log.1Cx/
x
D lim
x!0
2xe
x
2
log.1Cx/Ce
x
2
.1Cx/
1
1
D1:
However,itisbettertoremovethe“determinate”partoftheratiobeforeusingL’Hospital’s
rule:
lim
x!0
e
x
2
log.1Cx/
x
D
lim
x!0
e
x
2

lim
x!0
log.1Cx/
x
D.1/lim
x!0
log.1Cx/
x
D lim
x!0
1=.1Cx/
1
D1:
InusingL’Hospital’sruleweusuallywrite,forexample,
lim
x!b
f.x/
g.x/
D lim
x!b
f
0
.x/
g0.x/
(2.4.13)
andthentrytofindthelimitontheright.Thisisconvenient,buttechnicallyincorrect,since
(2.4.13)istrueonlyifthelimitontherightexistsintheextendedreals.Itmayhappenthat
thelimitontheleftexistsbuttheoneontherightdoesnot.Inthiscase,(2.4.13)isincorrect.
Example2.4.5
If
f.x/Dxx
2
sin
1
x
and g.x/Dsinx;
then
f
0
.x/D12xsin
1
x
Ccos
1
x
and g
0
.x/Dcosx:
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