Understanding Local Maxima, Minima, and Concavity in Functions
School
University of Toronto**We aren't endorsed by this school
Course
MATH 1001
Subject
Mathematics
Date
Dec 12, 2024
Pages
5
Uploaded by ProfLightningLeopard20
43DerivativesshapeofagraphFirstderivativetestSpposethatcisacriticalnumberofacontinuousfunctionfIfFchangessignfrompositivetonegativeatxcthenFhasalocalmaxatcIffchangesfromnegativetopositiveatcthenfhasalocalminatcIfthesignoffdoesn'tchangeatxcthencisneitheramaxnoraminnnaathisisanumberlineusedlocalminatxCtorecordthesignofthenolocalmaxorlocalminatxcderivativeofflocalmaxatxCEgFindthelocalmaximaandlocalminimaoffixx62135SolutionApplythe1stderivativetestSowemustcomputefandfondthecriticalnumbersfx3212135
Criticalnumbersarewhenfx0032121350x24450x95icritrialnumbersare9and5Usethe1stderivtesttodetermineiftheyaremaxesorminsf1F10730sbin0_toftofindthesignoffbetweenand9fo135nuBythe1stderivativetestthereisalocalmaxatx5andalocalminat9WhereisfincreasingWhereisfdecreasingincreasingonx5decreasingon5,9increasingon90DefIfthegraphoffliesaboveitstangentlinesonanintervalIthenfisconcaveupwardonIIfthegraphoffliesbelowitstangentlinesonanintervalIthenfisconcavedownwardonI
ConcavityTestaIffx0onIthenthegraphoffisconcaveuponIbIffx0onIthenthegraphoffisconcavedownonItangentlinesbelowthegraphtangentlinesareabovegraphconcaveupcocavedownFxisincreasingFx0fxisdecreasingFx20DefApointPonacurveyflatitaninflectionpointiffiscontinuousatPandthecorechangesfromconcaveuptoconcavedownorfromconcavedowntoconcaveupatPEgFttPisaninflectionpointrcopeSecondDerivativeTestSupposefiscontinuousnearxCIffcOandfa0thenfhasalocalminatxCIffaoandfecothenfhasalocalmaxatxCEgfix2x122WhatarelocalmaxminsWhataretheinflectionpointsWhereisfconcaveupconcavedown
SolutionComputefffx8248x238xxBxtrafx242242411761findcriticalpointsCriticalnumbersarewhenFx0criticalnumbersareXOV353Use2ndderu testtodeterminewhichcriticalnumbersaremaxsminsfo2401o1240ithegraphisconcavedownnear0thereisalocalmaxatx0f5324531510igraphitconcaveupnearXBthereisalocalminatsFs24531531soigraphisconcaveupnearx53thereisalocalminat53Findtheinflectionpointconcavityfpf0724ifx0whenfixeliconcavitychangesatboth1andIwehaveoflectionpointwhen1andwhen1Alsothegraphisconcaveupwhena1concavedownwhenexclandconcaveupwhenol
EgFlxfx43fx122ObserveFo0yetthereisnoinflectionpointatxOffx0fxsowhenwhenCOso84.5SummaryofcurvesketchingGivenafunctionflxwewishtosketchthegraphofyflxbyfindingthefollowinginformationADomainofFloBxinterceptsandyinterceptofyflxCsymmetryegisfanevenfunctionoddfunctionperiodicDHorizontalverticalasymptotesEIntervalsofincreasedecreaseFLocalmaxandminsofyfxGconcavityinflectionpointsofyfxHPutallofthisinfotogetherandsketchthegraphEgExample2Sketchthegraphofy1