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1、Chapter4ApplicationsofDifferentiation4.1MaximumandMinimumvaluesDefinition1Afunctionfhasanabsolutemaximum(orglobalmaximum)atcifforallxinD,whereDisthedomainoff.Thenumberf(c)iscalledthemaximumvalueoffonD.fhasanabsoluteminimum(orglobalminimum)atcifforallxinD,wher
2、eDisthedomainoff.Thenumberf(c)iscalledtheminimumvalueoffonD.Themaximumandminimumvaluesoffarecalledtheextremevaluesoff.1Definition2Letcbeanumberinthedomainofafunctionf.1f(c)isalocal(orrelative)maximumvalueoffifthereisanopenintervalIcontainingcsuchthatf(c)≥f(x)
3、forallxinI.2f(c)isalocal(orrelative)minimumvalueoffifthereisanopenintervalIcontainingcsuchthatf(c)≤f(x)forallxinI.2InFigure1,thefunctiontakesonalocalminimumatc.f(b)isalocalmaximumvalueoff(x);f(d)isamaximumvalueoff(x)on[a,e]andisalsoalocalmaximumvalueoff(x).ya
4、0bcdexFigure1f(a)isaminimumvalueoff(x)on[a,e],butitisnotalocalminimumvalueoff(x)becausethereisnoopenintervalIcontainedin[a,e]suchthatf(a)istheleastvalueoff(x)onI.3Example1Iff(x)=x2,thenf(x)≥f(0)forallx.Therefore,f(0)=0istheabsolute(andlocal)minimumvalueoff.Ho
5、wever,thisfunctionhasnomaximumvalue.Example2ThegraphofthefunctionisshowninFigure2.Wecanseethatf(1)=5isalocalmaximum,whereastheabsolutemaximumisf(-1)=37.Also,f(0)=0isalocalminimumandf(3)=-27isbothalocalandanabsoluteminimum.Figure24Theorem3(Fermat’sTheorem)Iff(
6、x)hasalocalextremevalue(thatis,maximumorminimum)atcandiff’(c)exists,thenf’(c)=0.NoteWecan’texpecttolocateextremevaluessimplybysettingf’(x)=0andsolvingforx.dcFigure35ProofForthesakeofdefiniteness,supposethatfhasalocalmaximumatc.Then,6Example3Iff(x)=x3,thenf’(x
7、)=3x2,sof’(0)=0.Butfhasnomaximumorminimumat0.Example4Thefunctionhasits(localandabsolute)minimumvalueat0,butthatvaluecan’tbefoundbysettingf’(x)=0becausef’(0)doesnotexist.Definition4Acriticalnumber(orpoint)ofafunctionfisanumbercinthedomainoffsuchthateitherf’(c)
8、=0orf’(c)doesnotexist.7Example5FindthecriticalnumbersofSolutionWehave8TheClosedIntervalMethodTofindtheabsolutemaximumandminimumvaluesofacontinuousfunctionfonaclosedinterval[a,b]:1.Findthe
