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5.2 Theﬁrst t derivativetest
97
Exercises
Inproblems1–12,ﬁndalllocalmaximumandminimumpoints(x,y)bythemethodofthissection.
1. y=x
2
−x
2. y=2+3x−x
3
3. y=x
3
−9x
2
+24x
4. y=x
4
−2x
2
+3
5. y=3x
4
−4x
3
6. y=(x
2
−1)/x
7. y=3x
2
−(1/x
2
)
8. y=cos(2x)−x
9. f(x)=
x−1 x<2
x
2
x≥2
10. f(x)=
8
<
:
x−3 x<3
x
3
3≤x≤5
1/x
x>5
11. f(x)=x
2
−98x+4
12. f(x)=
−2
x=0
1/x
2
x=0
13. Recallthatforanyrealnumber r xthereisauniqueintegernsuchthatn≤x<n+1,and
the greatest integer functionis givenby x = n, as shownin ﬁgure 3.1. Where e are the
criticalvaluesofthegreatestintegerfunction?Whicharelocalmaximaandwhicharelocal
minima?
14. Explainwhythefunctionf(x)=1/xhasnolocalmaximaorminima.
16. Showthatacubicpolynomialcanhaveatmosttwocriticalpoints. . Giveexamplestoshow
thatacubicpolynomialcanhavezero,one,ortwocriticalpoints.
17. Explore e the familyoffunctions f(x)=x
3
+cx+1where cisa constant. . Howmanyand
18. Wegeneralizetheprecedingtwoquestions. . Letnbeapositiveintegerandletf beapoly-
nomialofdegreen. Howmanycriticalpointscanfhave?(Hint: RecalltheFundamental
TheoremofAlgebra,whichsaysthatapolynomialofdegreenhasatmostnroots.)
The method of the previoussection for deciding whether there e is a a localmaximumor
minimumat a criticalvalue isnot alwaysconvenient. . We e can insteaduse information
toﬁndthecriticalvalues,thereisoftenrelativelylittleextraworkinvolvedinthismethod.
Howcanthederivativetelluswhether thereisamaximum,minimum,orneitherat
apoint? Supposethatf(a)=0. . Ifthereisalocalmaximumwhenx=a,thefunction
mustbelowernearx=athanitisrightatx=a.Ifthederivativeexistsnearx=a,this
meansf
(x)>0whenxisnearaandx<a,becausethefunctionmust“slopeup”just
totheleftofa. Similarly,f(x)<0whenxisnearaandx>a,becausef f slopesdown
fromthelocalmaximumaswemovetothe right. . Usingthesamereasoning,ifthereis
alocalminimumatx=a,thederivativeoff mustbenegativejusttotheleftofaand
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98
Chapter5 CurveSketching
positivejusttotheright.Ifthederivativeexistsnearabutdoesnotchangefrompositive
tonegativeornegativetopositive,thatis,itispostiveonbothsidesornegativeonboth
sides,thenthereisneitheramaximumnorminimumwhenx=a. Seetheﬁrstgraphin
ﬁgure5.1andthegraphinﬁgure5.2forexamples.
EXAMPLE 5.4
Findalllocalmaximumandminimumpointsforf(x)=sinx+cosx
usingtheﬁrstderivativetest. Thederivativeisf
(x)=cosx−sinxandfromexample5.3
thecriticalvaluesweneedtoconsiderareπ/4and5π/4.
Thegraphsofsinxandcosxareshowninﬁgure5.4.Justtotheleftofπ/4thecosine
islargerthanthesine,sof(x)ispositive;justtotherightthecosineissmallerthanthe
sine,sof
(x)isnegative. Thismeansthereisalocalmaximumatπ/4. Justtotheleft
of5π/4thecosineissmallerthanthesine,andtotherightthecosineislargerthanthe
sine. Thismeansthatthederivativef(x)isnegativetotheleftandpositivetotheright,
sof hasalocalminimumat5π/4.
π
4
4
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Figure 5.4
Thesineandcosine.
Exercises
In1–13,ﬁndallcriticalpointsandidentifythemaslocalmaximumpoints,localminimumpoints,
orneither.
1. y=x
2
−x
2. y=2+3x−x
3
3. y=x
3
−9x
2
+24x
4. y=x
4
−2x
2
+3
5. y=3x
4
−4x
3
6. y=(x
2
−1)/x
7. y=3x
2
−(1/x
2
)
8. y=cos(2x)−x
9. f(x)=(5−x)/(x+2)
10. f(x)=|x
2
−121|
11. f(x)=x
3
/(x+1)
12. f(x)=
x
2
sin(1/x) x=0
0
x=0
13. f(x)=sin
2
x
14. Findthemaximaandminimaoff(x)=secx.
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5.3 Thesecondderivativetest
99
15. Letf(θ)=cos
2
(θ)−2sin(θ). Findtheintervalswherefisincreasingandtheintervalswhere
f isdecreasinginsideof[0,2π]. . Usethis s informationtoclassify thecriticalpointsoff f as
eitherlocalmaximums,localminimums,orneither.
16. Letr>0.Findthelocalmaximaandminimaofthefunctionf(x)=
r−xonitsdomain
(−r,r).Sketchthecurveandexplainwhytheresultisunsurprising.
17. Letf(x)=ax
2
+bx+c witha=0. . Show w thatf f has s exactlyonecriticalpointusingthe
ﬁrstderivativetest. Giveconditionsonaandbwhichguaranteethatthecriticalpointwill
beamaximum. Itispossibletoseethiswithoutusingcalculusatall;explain.
The basis s of f the e ﬁrst derivative test t is s that t if the e derivative e changesfrom m positive e to
negativeatapointatwhichthederivativeiszerothenthereisalocalmaximumat the
point, and similarly for r a localminimum. . If f f
changesfrompositive to negative it is
decreasing;thismeansthatthederivativeoff
,f

,mightbenegative,andifinfact f

isnegativethen f
isdeﬁnitelydecreasing,so there isalocalmaximumatthe pointin
question. Note e wellthat f
mightchangefrompositiveto negative while f

iszero,in
whichcase f

givesusno information about thecriticalvalue. . Similarly, , if f
changes
fromnegativetopositivethereisalocalminimumat thepoint,andf
isincreasing. If
f

>0atthepoint,thistellsusthatf
isincreasing,andsothereisalocalminimum.
EXAMPLE 5.5
Consider again f(x) =sinx+cosx,withf
(x)=cosx−sinx and
f

(x)=−sinx−cosx. Sincef

(π/4)=−
2/2−
2/2=−
2<0,weknowthereisa
localmaximumatπ/4. Sincef(5π/4)=−−
2/2−−
2/2=
2>0,thereisalocal
minimumat5π/4.
Whenitworks,thesecondderivativetestisoftentheeasiestwaytoidentifylocalmax-
imumandminimumpoints. Sometimesthetestfails,andsometimesthesecondderivative
isquitediﬃculttoevaluate;insuchcaseswemustfallbackononeoftheprevioustests.
EXAMPLE 5.6
Letf(x) =x
4
. The e derivativesare f
(x) =4x
3
and f

(x)=12x
2
.
Zeroistheonlycriticalvalue,butf

(0)=0,sothesecondderivativetesttellsusnothing.
However,f(x)ispositiveeverywhereexceptatzero,soclearlyf(x)hasalocalminimum
atzero. Ontheotherhand,f(x)=−x
4
alsohaszeroasitsonlycriticalvalue,andthe
secondderivativeisagainzero,but−x
4
hasalocalmaximumatzero.
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100
Chapter5 Curve e Sketching
Exercises
Findalllocalmaximumandminimumpointsbythesecondderivativetest.
1. y=x
2
−x
2. y=2+3x−x
3
3. y=x
3
−9x
2
+24x
4. y=x
4
−2x
2
+3
5. y=3x
4
−4x
3
6. y=(x
2
−1)/x
7. y=3x
2
−(1/x
2
)
8. y=cos(2x)−x
9. y=4x+
1−x
10. y=(x+1)/
p
5x+35
11. y=x
5
−x
12. y=6x+sin3x
13. y=x+1/x
14. y=x
2
+1/x
15. y=(x+5)
1/4
16. y=tan
2
x
17. y=cos
2
x−sin
2
x
18. y=sin
3
x
Weknowthatthesignofthederivativetellswhetherafunctionisincreasingordecreasing.
Forexample,anytimethatf(x)>0,thatmeansthatf(x)isincreasing. Thesignofthe
secondderivativef

(x)tellsuswhetherf
isincreasingordecreasing;wehaveseenthat
iff
iszeroandincreasingat apointthenthereisalocalminimumat thepoint,andif
f iszeroanddecreasingatapointthenthereisalocalmaximumatthepoint. Thus,we

.
Wecangetinformationfromthe signoff

evenwhenf
isnotzero. Supposethat
f(a)>0. Thismeansthatnearx=af isincreasing. . Iff(a)>0,thismeansthatf
slopesupandisgettingsteeper;iff
(a)<0,thismeansthatfslopesdownandisgetting
lesssteep. Thetwosituationsareshowninﬁgure5.5. Acurvethatisshapedlikethisis
calledconcaveup.
a
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Figure5.5
f

(a)>0: f
(a)positiveandincreasing,f
(a)negativeandincreasing.
Nowsupposethatf

(a)<0.Thismeansthatnearx=af
isdecreasing.Iff
(a)>0,
thismeansthatf slopesupandisgettinglesssteep;iff
(a)<0,thismeansthatfslopes
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5.4 Concavity y andinﬂectionpoints
101
a
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Figure 5.6
f

(a)<0: f
(a)positiveanddecreasing,f
(a)negativeanddecreasing.
downandisgettingsteeper. Thetwosituationsareshowninﬁgure5.6. Acurvethatis
shapedlikethisiscalledconcavedown.
Ifweare tryingtounderstandtheshape of thegraphofafunction,knowingwhere
itisconcaveupandconcavedownhelpsustogetamoreaccuratepicture. Ofparticular
interestarepointsatwhichtheconcavitychangesfromuptodownordowntoup;such
pointsarecalledinﬂectionpoints. Iftheconcavitychangesfromuptodownatx=a,f

changesfrompositivetotheleftofatonegativetotherightofa,andusuallyf

(a)=0.
Wecanidentifysuchpointsbyﬁrstﬁndingwheref(x)iszeroandthencheckingtosee
whetherf

(x) doesin factgofrompositiveto negative ornegativeto positive atthese
points.Notethatitispossiblethatf

(a)=0buttheconcavityisthesameonbothsides;
f(x)=x
4
isanexample.
EXAMPLE5.7
Describetheconcavityoff(x)=x
3
−x. f
(x)=3x
2
−1,f

(x)=6x.
Since f

(0)=0,thereispotentiallyaninﬂectionpoint at zero. . Since e f

(x) >0 when
x>0andf(x)<0whenx<0theconcavitydoeschangefromdowntoupatzero,and
thecurveisconcavedownforallx<0andconcaveupforallx>0.
Notethatweneedtocomputeandanalyzethesecondderivativetounderstandcon-
cavity,sowemayaswelltrytousethesecondderivativetestformaximaandminima.If
forsomereasonthisfailswecanthentryoneoftheothertests.
Exercises
Describetheconcavityofthefunctionsin1–18.
1. y=x
2
−x
2. y=2+3x−x
3
3. y=x
3
−9x
2
+24x
4. y=x
4
−2x
2
+3
5. y=3x
4
−4x
3
6. y=(x
2
−1)/x
7. y=3x
2
−(1/x
2
)
8. y=sinx+cosx
9. y=4x+
1−x
10. y=(x+1)/
p
5x+35
11. y=x
5
−x
12. y=6x+sin3x
13. y=x+1/x
14. y=x
2
+1/x
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102
Chapter5 Curve e Sketching
15. y=(x+5)
1/4
16. y=tan
2
x
17. y=cos
2
x−sin
2
x
18. y=sin
3
x
19. Identifytheintervals s onwhichthegraphofthefunctionf(x)=x
4
−4x
3
+10isofoneof
thesefourshapes:concaveupandincreasing;concaveupanddecreasing;concavedownand
increasing;concavedownanddecreasing.
20. Describetheconcavityofy=x
3
+bx
2
+cx+d. Youwillneedtoconsider r diﬀerentcases,
dependingonthevaluesofthecoeﬃcients.
21. Let t nbeaninteger greaterthanorequaltotwo,andsuppose f f isapolynomialofdegree
n. How w many inﬂectionpointscan f have? ? Hint: : Use e thesecondderivativetestandthe
fundamentaltheoremofalgebra.
Averticalasymptoteisaplacewherethefunctionbecomesinﬁnite,typicallybecausethe
functionf(x)=1/xhasaverticalasymptoteatx=0,andthefunctiontanxhasavertical
asymptoteatx=π/2(andalsoatx=−π/2,x=3π/2,etc.).Whenevertheformulafora
seeifthedenominatorcaneverbezero,andthencheckingthelimitatsuchpoints. Note
thatthereisnotalwaysaverticalasymptotewherethederivativeiszero: f(x)=(sinx)/x
hasa zero denominator at x =0,butsince lim
x→0
(sinx)/x=1 there isnoasymptote
there.
Ahorizontalasymptoteisahorizontallinetowhichf(x)getscloserandcloserasx
approaches∞ (or as s x x approaches−∞). . For r example, , the e reciprocalfunctionhasthe
x-axisforahorizontalasymptote. Horizontalasymptotescanbeidentiﬁedbycomputing
thelimitslim
x→∞
f(x)andlim
x→−∞
f(x). Sincelim
x→∞
1/x =lim
x→−∞
1/x=0,the
liney=0(thatis,thex-axis)isahorizontalasymptoteinbothdirections.
Some functionshave asymptotesthat t are e neither r horizontalnor r vertical, , but some
other line. . Such h asymptotesare somewhatmore diﬃcult to identifyand wewillignore
them.
If the domain of f the e function does not extend out to inﬁnity, , we e should also o ask
what happensasx approachesthe boundaryof thedomain. . Forexample,thefunction
y=f(x)=1/
r2−xhasdomain−r<x<r,andybecomesinﬁniteasxapproaches
eitherror−r.Inthiscasewemightalsoidentifythisbehaviorbecausewhenx=±rthe
denominatorofthefunctioniszero.
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5.5 AsymptotesandOther r Thingsto o LookFor
103
Ifthereareanypointswherethederivativefailstoexist(acusporcorner),thenwe
shouldtakespecialnoteofwhatthefunctiondoesatsuchapoint.
Finally,itisworthwhiletonoticeanysymmetry. Afunctionf(x)thathasthesame
value for r −x x as for x, , i.e., f(−x) = f(x), , is called an n “even n function.” ” Its s graph h is
symmetric withrespecttothe y-axis. . Someexamplesofevenfunctionsare: : x
n
whenn
isan even number, cosx, and sin
2
x. On n the other hand, a function that satisﬁes s the
propertyf(−x)=−f(x)iscalledan“oddfunction.” Itsgraphissymmetricwithrespect
totheorigin.Someexamplesofoddfunctionsare:x
n
whennisanoddnumber,sinx,and
tanx.Ofcourse,mostfunctionsareneitherevennorodd,anddonothaveanyparticular
symmetry.
Exercises
Sketchthecurves. Identifyclearly y anyinterestingfeatures,includinglocalmaximumandmini-
mumpoints,inﬂectionpoints,asymptotes,andintercepts.
1. y=x
5
−5x
4
+5x
3
2. y=x
3
−3x
2
−9x+5
3. y=(x−1)
2
(x+3)
2/3
4. x
2
+x
2
y
2
=a
2
y
2
,a>0.
5. y=xe
x
6. y=(e
x
+e
−x
)/2
7. y=e
−x
cosx
8. y=e
x
−sinx
9. y=e
x
/x
10. y=4x+
1−x
11. y=(x+1)/
p
5x+35
12. y=x
5
−x
13. y=6x+sin3x
14. y=x+1/x
15. y=x
2
+1/x
16. y=(x+5)
1/4
17. y=tan
2
x
18. y=cos
2
x−sin
2
x
19. y=sin
3
x
20. y=x(x
2
+1)
21. y=x
3
+6x
2
+9x
22. y=x/(x
2
−9)
23. y=x
2
/(x
2
+9
24. y=2
x−x
25. y=3sin(x)−sin
3
(x),forx∈[0,2π]
26. y=(x−1)/(x
2
)
For each of the e following ﬁve functions, , identify y any y vertical and horizontal asymptotes, , and
identifyintervalsonwhichthefunctionisconcaveupandincreasing;concaveupanddecreasing;
concavedownandincreasing;concavedownanddecreasing.
27. f(θ)=sec(θ)
28. f(x)=1/(1+x
2
)
29. f(x)=(x−3)/(2x−2)
30. f(x)=1/(1−x
2
)
31. f(x)=1+1/(x
2
)
32. Let t f(x) = = 1/(x
2
−a
2
), where e a a ≥ 0. . Find d any y vertical l andhorizontal asymptotes s and
theintervalsuponwhichthegivenfunctionisconcave upandincreasing;concave upand
104
Chapter5 Curve e Sketching
decreasing; concave downandincreasing; concave down and decreasing. . Discuss s how the
valueofaaﬀectsthesefeatures.
6
ApplicationsoftheDerivative
Oftenthisinvolvesﬁndingthemaximumorminimumvalueofsomefunction:theminimum
ﬁndingtheappropriatefunctionandthenusingtechniquesofcalculustoﬁndthemaximum
ortheminimumvaluerequired.
Generallysuchaproblemwillhavethefollowingmathematicalform:Findthelargest
(orsmallest)valueoff(x)whena≤x≤b. Sometimesaorbareinﬁnite,butfrequently
therealworldimposessomeconstraintonthevaluesthatxmayhave.
Suchaproblemdiﬀersintwowaysfromthelocalmaximumandminimumproblems
weencounteredwhengraphingfunctions: Weareinterestedonlyinthefunctionbetween
aandb,andwewanttoknowthelargestorsmallestvaluethatf(x)takeson,notmerely
valuesthat are the largestor smallest in asmall l interval. . That t is, , weseeknot t alocal
maximumor minimum but a a global maximumor minimum, , sometimesalso o called an
absolutemaximumorminimum.
Anyglobalmaximumorminimummustofcoursebealocalmaximumorminimum.
Ifweﬁndallpossiblelocalextrema,thentheglobalmaximum,ifitexists,mustbethe
largestofthelocalmaximaandtheglobalminimum,ifitexists,mustbethesmallestof
atwhichf
105
106
Chapter6 Applications s oftheDerivative
maximumorminimumcanoccur iftheendpointsaandbarenot inﬁnite,namely,ata
andb.Wehavenotpreviouslyconsideredsuchpointsbecausewehavenotbeeninterested
inlimitingafunctiontoasmallinterval.Anexampleshouldmakethisclear.
−2
1
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Figure 6.1
Thefunctiong(x)orf(x)=x
2
truncatedto[−2,1]
EXAMPLE 6.1
Findthemaximumandminimumvaluesoff(x)=x
2
ontheinterval
[−2,1],showninﬁgure6.1. Wecomputef
(x)=2x,whichiszeroatx=0andisalways
deﬁned.
Sincef
(1)=2wewouldnotnormallyﬂagx=1asapointofinterest,butitisclear
fromthegraphthatwhenf(x)isrestrictedto [−2,1]thereisalocalmaximumatx=1.
Likewisewewouldnotnormallypayattentiontox=−2,butsincewehavetruncatedf
at−2wehaveintroducedanewlocalmaximumthereaswell. Inatechnicalsensenothing
newisgoingonhere: Whenwetruncatef weactuallycreateanewfunction,let’scallit
g,thatisdeﬁnedonlyontheinterval[−2,1]. Ifwetrytocomputethederivativeofthis