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# MAT 137Y 2007-08Winter Session, Self Generated Solutions to Problem Set 8.pdf

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Department
Mathematics
Course
MAT137Y1
Professor
All Professors
Semester
Winter

Description
MAT137Y200708WinterSessionSolutionstoProblemSet823021iForfxx1thederivativeisfx6xx1sothecriticalpointsarex01and0theendpointsarex12Forthelocalextremaweseethatfx0whenx21001andfx0whenx2101Hencelocalminimaoccuratthepoints10and10and01ForabsoluteextremawecomparethevaluesofthefunctionattheendpointsandcriticalpointsWehavef10f01f10f227sotheabsolutemaximumis227andtheabsoluteminimaare10and100iiGivenfxx2cosxwehavefx12sinx0sincesinxispositiveon0pTherep2p33pforetherearenolocalextremaLookingattheendpointsf02andf036psotheabsoluteminimumisatx0andtheabsolutemaximumisatx32SHE432038SupposefxAxBxCThenfx2AxB0xB2A2soB4AFurthermoref13impliesABC3andf31implies9A3BC111SolvingforABandCwegetAB2C223SHE44r0r1036Supposefx1x1rxforx1Thenfxr1x1sofx000r200whenx0Furthermorefxrr11xsof0rr10sofhasalocalminimumatx0ByTheorem443f00istheabsoluteminimumoff4SHE452514Bydoingquestion13weseethatAm102mwheremistheslopeofthelinewhich2mintersects25Sincem20wendthatatoneoftheendpointslimAmm0sonoabsolutemaximumexists1222244WemaximizethevolumeVprhwhererhaso3122Vhpahh0ha31a022pThusVhpa3h0hBycheckingtheendpointsweseethemaximum33pp23pavolumeisVa33275Wemaximizethecrosssectionalarea1dh10hdh1010sinq10cosq10sinqAq10h22p100sinqsinqcosq0q20222Aq100cosqcosqsinq100cosq2cosq11002cosq1cosq11p0whencosqq23
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