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Rolle中值定理应用中辅助函数的构造.docx

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Rolle中值定理应用中辅助函数的构造 Rolle'stheoremisafundamentaltheoremincalculus,whichisoftenemployedtosolveoptimizationproblems.ItisoftennecessarytouseauxiliaryfunctionsintheapplicationofRolle'stheoremtosolveoptimizationproblems.Inthispaper,wewillexploretheconstructionofauxiliaryfunctionsintheapplicationofRolle'stheorem. Rolle'stheoremisaspecialcaseofthemeanvaluetheorem.AccordingtoRolle'stheorem,ifafunctionf(x)iscontinuousontheclosedinterval[a,b]anddifferentiableontheopeninterval(a,b),andf(a)=f(b),thenthereexistsatleastonepointcintheopeninterval(a,b)suchthatf′(c)=0. OneofthemostcommonapplicationsofRolle'stheoremistofindthemaximumorminimumofafunction.Tofindthemaximumorminimumofafunction,itisnecessarytofindthecriticalpointsofthefunction.Recallthatacriticalpointofafunctionf(x)isanypointxwheref′(x)=0orf′(x)doesnotexist.Themaximumorminimumofthefunctioncanbefoundatthecriticalpoints. Auxiliaryfunctionsareusedtosimplifytheproblemandmakeitmoremanageable.TheycanprovideawaytoframetheprobleminawaythatmakesiteasiertoapplyRolle'stheorem.Onecommonwayinwhichauxiliaryfunctionscanbeusedistocreateanewfunctiong(x)thathasthesamecriticalpointsasf(x)butiseasiertoworkwith. Letusconsiderasimpleexampletoillustratetheconstructionofauxiliaryfunctions.Supposewewanttofindthemaximumvalueofthefunctionf(x)=x^2-3x+2ontheinterval[0,3].ToapplyRolle'stheorem,weneedtofindthecriticalpointsofthefunctionf(x).Wecanbeginbytakingthederivativeoff(x),whichgivesusf′(x)=2x-3.Tofindthecriticalpoints,weneedtosolvetheequationf′(x)=0.Solvingthisequation,wegetx=3/2,whichistheonlycriticalpointofthefunctionontheinterval[0,3]. Now,wecanconstructtheauxiliaryfunctiong(x)=f(x)-mx+b,wheremandbareconstants.Wewanttochoosetheconstantsmandbinsuchawaythattheauxiliaryfunctiong(x)hasthesamecriticalpointsasf(x).Onewaytoensurethatg(x)hasthesamecriticalpointasf(x)istosetg′(x)=0atthecriticalpoint.Therefore,wehave: g′(x)=f′(x)-m=0 Substitutingthecriticalpointx=3/2intog′(x),weget: f′(3/2)-m=0 Solvingform,weget: m=2(3/2)-3=0 Now,wecanfindbbysubstitutin