《高数双语》课件section 3-4

Section3.4-3.6PropertiesofFunctionsOverviewMonotonicityofFunctionsLocalExtremeValuesofFunctionsGlobalMaximaandMinimaConvexityofFunctionsInflectionsGraphingFunctions23MonotonicityofFunctions(1)Thenecessaryandsufficientconditionforthefunctionftobemonotoneincreasing(decreasing)onIis(2)If()inI

,thenf

isstrictlymonotoneincreasing,anddifferentiableinI.

becontinuousonITheorem

LetThen(decreasing)onI.()inI;MonotonicityofFunctions4Proof(1)SufficiencyForanyiscontinuousonandisdifferentiableinBytheLagrangetheoremandtheconditionwehavewhereHence,thefunctionftobemonotoneincreasing(decreasing)onI.theintervalMonotonicityofFunctions5Proof(continued)(1)NecessitySupposethatfismonotoneincreasing(decreasing)on

I.ForanyxbelongingtotheinteriorI,thenSo,takingsuchthat6MonotonicityofFunctionsSincetherootsoftheequation

areIteasytoseethat

if

and;Example

Discussthemonotonicityofthefunction.Solution,

and.

if.7MonotonicityofFunctionsThen,wehavethefunction

isstrictlymonotoneincreasingintheintervalFinish.Solution(continued)

and;andisstrictlymonotonedecreasingintheinterval.Example

Discussthemonotonicityofthefunction.8MonotonicityofFunctionsExample

ProvethatProofInordertoobtainthegiveninequatity,weneedonlyprovethatLetthenButthesignofisstillnotclearon[0,1).LetusrepeattheproceduretoconsiderthederivativeoffunctionItiseasytoobtainMonotonicityofFunctions9Proof(continued)ByTheorem2.7.1,isstrictlymonotonedecreasingon[0,1).ThusAgainfromTheorem2.7.1,weknowthatorThisisthedesiredconclusion.Finish.Example

Provethat10ExtremeValuesofFunctionsDefinition

SupposethatIfthereisa

suchthat

(),thenwesaythatthefunctionfhasamaximum(minimum)

atMaximumandminimumvaluesaregivenajointname

extremevalue[极值],

iscalledamaximal(minimal)andthepointpoint

or

extremepoint[极值点].11ExtremeValuesofFunctions(1)Iffor

and

for,thenf

hasamaximum

(2)If

for

and

for,thenf

hasaminimum

thenthepoint

isnotanextremepoint.TheoremLetthefunctionfbedifferentiableinaneighborhoodofapoint

,and

atthepoint;(3)If

hasthesamesignonboththeleftandrightsidesofthepoint,

atthepoint;12ExtremeValuesofFunctions13ExtremeValuesofFunctionsExample

FindtheextremevaluesofthefunctionWeneedthreestepstodiscussthisproblem.,itisobviousthatthestationarypointis

and.SinceSolution(1)Findthestationarypointsandpointswherethefunctionisnotdifferentiable.non-differentiabilityareandthepointsof,(2)Inordertodeterminethesignof

intheneighborhoodsofthesepoints,wepartitiontheintervalofdefinitionofthegivenfunctionpointsandmakeatableasfollowing

bythose0460MinimalPointMaximalPointNon-extremePoint14Solution(continued)0460MinimalPointMaximalPointNon-extremePointFinish.Fromthetablewemayseethatthemaximumofthefunctionis(3)Determinetheextremevalues.,andtheminimumis.ExtremeValuesofFunctionsExample

Findtheextremevaluesofthefunction15ExtremeValuesofFunctionsExample

Findtheextremevaluesofthefunction16ExtremeValuesofFunctionsTheorem

hasasecondderivativeatapointand,Thenthefunction

hasamaximum(minimum)

if().Supposethatthefunction,.

atthepoint

Sincefhasasecondderivativeatpoint,intheneighborhood

theTaylorformulaofthefunctionfwiththesecondorderandPeanoremainderwhereSince,wehave.Proofisthefollowing17ExtremeValuesofFunctionsWecanseefromthisthatthesignof,thefirsttermofthelastexpression.,or,thenTheminimumpropertymaybeprovedsimilarly.Proof(continued)

isdeterminedbyHence,when,

isamaximumofthefunction.Finish.,Theorem

hasasecondderivativeatapointand,Thenthefunction

hasamaximum(minimum)

if().Supposethatthefunction,.

atthepoint

ExtremeValuesofFunctions18weneedonlyconsider

inaperiodicinterval.Bythelasttheorem,wemayfindtheextremeLet(2)FindthesecondderivativeofandconsideritssignattheseExample

FindtheextremevalueofthefunctionSolutionSince

isaperiodicfunction,,(1)Findthestationarypointsofthevaluebythefollowingsteps:..Wehavethestationarypoints.stationarypoints(3)Findthemaximumandminimumof.19ExtremeValuesofFunctionsThenthemaximumofthefunction

intheinterval

is;andtheminimumis.

Solution(continued)Finish.Example

Findtheextremevalueofthefunction20NoteIfwehave

atastationarypointTheorem

Supposethatthefunction

is-timesdifferentiableandThen

mustbeanextremepoint,andisamaximumof

if

andaminimumif(2)when

isodd,

isnotanextremepoint.ExtremeValuesofFunctions,thenwehavetoutilizethehigherderivativesofthefunction.

iseven,(1)when21Ifafunction

iscontinuousonaclosedintervalpropertiesofcontinuousfunctions,

musthaveaglobalmaximumitmustbealocalmaximumvalueoralocalminimumvalue;ifwewanttofindtheglobalmaximumvalueorglobalminimumvalue,

and.,thenbytheandaglobalminimumon.Andifanyofthem,say,isin,possibilityisthat

maybeoneoftheendpointsof.Therefore,weneedtofindallstationarypoints,compareallthefunctionvaluesatthesepointsandvalues(1)(2)(3)Globalmaximaandminimaanothernon-differentiablepointsandthen22GlobalmaximaandminimaAswehadseeninlastlecture,bythedefinitionofmaximumandminimalvalueofafunction,theyareonlylocalvalues.Butformanyproblems,weneedtofindthelargestvalueorsmallestvalueinthefixedinterval,andthesevalueisreferredasglobalmaximumandglobalminimum.Ifafunction

iscontinuousonaclosedintervalpropertiesofcontinuousfunctions,

musthaveaglobalmaximum.Andifanyofthem,sayitmustbealocalmaximumvalueoralocalminimumvalue;anotherifwewanttofindtheglobalmaximumvalueorglobalminimumvalue,

and.,thenbytheandaglobalminimumon,isin,possibilityisthat

maybeoneoftheendpointsof.Therefore,weneedtofindallstationarypoints,non-differentiablepointsandthencompareallthefunctionvaluesatthesepointsandvalues(1)(2)(3)Globalmaximaandminima23Note

Itisworthwhiletoindicatethatforsomespecialcases,findingglobalmaximaorminimamaybesimplified.Forinstance,iff(x)ismonotoneincreasing(decreasing)ontheinterval[a,b],thentheglobalmaximumandminimummustbeattainedattheendpointsb(ora)ora(orb,respectively;if

f(x)iscontinuouson[a,b]andhasonlyoneextremepointthenx0mustbetheglobalmaximum(minimum)pointprovideditisalocalmaximum(minimum)point.Globalmaximaandminima24Note

Tosolveapracticalproblem,wemayshouldestablishtheobjectivefunctionfirstlyandthenfindtheglobalmaximumorminimumbyfindingallthestationarypoints,non-differentiablepointsandcomparingthefunctionvaluesandthefunctionvaluesattheendofthegiveninterval.Globalmaximaandminima25

(1)Establishtheobjectivefunction.SolutionTheleastamountofmaterialmeansthesmallestsurfacearea.SupposethatthesurfaceareaofthecontainerisS,theheightisH,andtheradiusofthebottomisR.ThenBythelastequation,wehave,andthentheobjectivefunctionisExample

AcylindricalcontainerwithvolumeV0andwithoutcoveristobemadeofasheetofiron.Howshouldwedesignitifwewishtousetheleastamountofmaterial?Globalmaximaandminima26Solution(continued)(2)Findtheglobalminimum.Let;weobtainthestationarypointSincethenweknowthatitistheglobalminimumpoint.Wehave..

istheminimum.Sincethispointisuniquein,Example

AcylindricalcontainerwithvolumeV0andwithoutcoveristobemadeofasheetofiron.Howshouldwedesignitifwewishtousetheleastamountofmaterial?27GlobalmaximaandminimaSolution(continued)While,wehaveTherefore,theamountofmaterialisminimizedprovidedtheheightHand.Finish.theradiusofthebottomRareequal.Example

AcylindricalcontainerwithvolumeV0andwithoutcoveristobemadeofasheetofiron.Howshouldwedesignitifwewishtousetheleastamountofmaterial?28Globalmaximaandminima

Itiseasytoseethattheclosestpointfromustotheenemyisthebestpositionforustoshoottheenemy.Example

Supposethattheenemy’scargostraighttothenorthfrompointAwiththevelocityof1km/minandthewidthoftheriveris0.5

kilometres.AtankofourarmygoalongtheriversideanddirecttotheeastfrompointBwiththevelocityof2km/min.(Seetherightfigure)Thequestioniswhereisthebestpositiontoshootenemy?29Globalmaximaandminima(1)Findtherelationbetweenthepositionofourtankandtheenemy’scar.SolutionSupposethattisthetimewhenourtankbegintochasetheenemyfromBandthedistancebetweentheenemyandusThen(2)Findtheglobalminimalvalueof.

.Let,wefindthestationarypointItiseasytoseethatthispointisthepointofminimumvalue.afterwebegantochasetheenemy,isthebesttimeweshoottheenemy.Finish..

isTherefore,1.5minuets,30ConvexityoffunctionsConcaveFunctionConvexFunctionofbeingconvexuporconvexdownforthegraphofafunctioniscalledtheconvexityConvexityisanotherimportantpropertyoffunctions.Ingenerally,afunction,whosethegraphisconvexdown,iscalledaconcavefunction,whileafunctionwhosegraphisconvexup,iscalledconvexfunction.Thepropertyofthefunction.31Convexityoffunctions32ConvexityoffunctionsAndthen,wehavethefollowingdefinition.Definition(convexfunction)If

and

theinequalityholds,thenif,

and,wehavethenIftheinequalityofthesetwo

iscalledconvexfunctionorstrictlyconvexfunction,respectively.Let.

iscalledaconcavefunctionon;,

fiscalledastrictlyconcavefunctionon.inequalityisreversed,thenonItisnotveryeasytojudgetheconvexityofagivenfunctiondirectlyfromthedefinition.33ConvexityoffunctionsThisconclusionareeasilyunderstoodgeometrically.34ConvexityoffunctionsWeproveonlythecaseofstrictconvexity;theotherproofsaresimilar.,Itiseasytoprovethat,,

wehavetheinequalitySupposethat;notethat.ItisenoughtoproveAddingtwotermsandthenusingthemeanvaluetheoremwehaveProofSupposethat..Convexityoffunctions35Proof(continued)?where,whileSubstitutingtheseintotheaboveequalityandusingthemeanvaluetheorem,,Convexityoffunctions36Proof(continued)whereFinish..Theconclusionisproved.37Convexityoffunctions(1)(2)Example

Studytheconvexityofthefollowingfunctions(1)Sincesothepowerfunction(2)SincesothelogarithmfunctionFinish.Solution

isstrictlyconvexintheinterval.,,

isstrictlyconcavein.Convexityoffunctions38Theorem

If

iscontinuousandstrictlyconvexinanintervalI,thenthenitmustbetheglobalminimumpointsinI.hasatmostoneglobalminimumpoint,andifthereexistsauniquelocalminimumpointinI

and,sothatBythestrictconvexityof

wehaveBecause,acontradictionappears.ProofSupposethatthereexisttwominimumpoints..Convexityoffunctions39Proof(continued)Secondly,letitistheglobalminimumpointin.Supposethatthereexists,

suchthat.Bythedefinitionofconvexfunctionwehave

bealocalminimumpoint;,wewillprovethat.Theorem

If

iscontinuousandstrictlyconvexinanintervalI,thenthenitmustbetheglobalminimumpointsinI.hasatmostoneglobalminimumpoint,andifthereexistsauniquelocalminimumpointinI40Convexityoffunctions

isarbitraryon,and

canbetakenarbitrarilyclosetoInthiscase,

contradictsassumptionthat

isalocalNotethat

iscontinuousin

andProof(continued)

bytaking

closeenoughto1.Finish.minimum.So,mustbetheglobalminimumpointsinI.Theorem

If

iscontinuousandstrictlyconvexinanintervalI,thenthenitmustbetheglobalminimumpointsinI.hasatmostoneglobalminimumpoint,andifthereexistsauniquelocalminimumpointinI41ConvexityoffunctionsInflectionpointsWeknowthatthegraphofaconvexfunctionisconvexdown,andthe

graphofaconcavefunctionisconvexup.Concerningthetransitionpointonthecurvebetweenconvexdownandconvexup.42ConvexityoffunctionsAssumethat;ifthereexists

suchthatthecurve

isconvexdown(up)intheinterval