alevel资源-纯数学课件

Chapter7ApplicationofDifferentiationMaximumandminimumpointsyx3Ifx1<3,thenf(x1)>f(3);Ifx2<3,thenf(x2)>f(3).Foreveryvalueofxotherthan3,f(x)isgreaterthanf(3)=-5Thepoint(3,-5)istheminimumpointonthegraphofy=f(x)Aminimumpointdoesnothavetobethelowestpointonthewholegraph,butisthelowestpointinitsimmediateneighborhood.yx10Thepoint(0,0)isaminimumpoint.Thisisshownbythefactf(x)>0foreverynumberx<1exceptx=0althoughf(x)<0whenx>1DefinitionAfunctionf(x)hasaminimumatx=qifthereisanintervalp<x<rcontainingqinwhichf(x)>f(q)foreveryvalueofxexceptqItisamaximumiff(x)<f(q)foreveryvalueofxintheintervalexceptqThepoint(q,f(q))iscalledaminimumpoint,oramaximumpointMinimumandmaximumpointsaresometimesalsocalledturningpoints.pqrminimumy=f(x)yxpqrmaximumy=f(x)yxIf(q,f(q))isaminimumormaximumpointofthegraphofy=f(x),theneitherf’(q)=0orf’(q)=isundefined.AlltheseresultscanbesummedupasaprocedureTofindtheminimumandmaximumpointsonthegraphofy=f(x):Step1Decidethedomaininwhichyouareinterested.Step2Findanexpressionforf’(x).Step3Listthevalueofxinthedomainforwhichf’(x)iseither0orundefined.Step4Takingeachofthesevaluesofxinturn,findthesignoff’(x)inintervalsimmediatelytotheleftandtotherightofthatvalue.Step5Ifthesesignsare–and+respectively,thegraphhasaminimumpoint.Ifthesignsarethesame,ithasneither.Step6Foreachvalueofxwhichgivesaminimumormax