Monotonically Increasing Function - Video & Lesson Transcript

When a function is increasing on its entire domain or decreasing on its entire domain, we say that the function is strictly monotonic, and we ... forTeachers forSchools forWorkingScholars® forCollegeCredit LogIn SignUp Menu forWorkingScholars® forCollegeCredit Plans Plans Courses Courses FindCoursesbySubject Science Math Business Psychology History English SocialScience Humanities Spanish ProfessionalDevelopment ByEducationLevel College HighSchool MiddleSchool TestPrep PopularTests… PopularTests SAT ACT TOEFL GMAT GRE CLEP GED ASVAB RealEstate TeacherCertification… TeacherCertification AEPA CEOE CSET/CBEST FTCE GACE ILTS MTEL MTLE MTTC MoGEA/MEGA NES NYSTCE OAE Praxis RICA TExES VCLA WEST Nursing… NursingTests HESI TEAS NCLEX PAXRN CNE OtherTests… OtherTests AP ASCP ASWB Accuplacer CHSPE CTEL DSST Finance FSA HSPT ISEE MBLEx NCE PERT PHR SHSAT TASC TECEP UExcel Exploreover 4,600 videocourses BrowseAllCourses SAT ACT TOEFL GMAT GRE CLEP GED ASVAB RealEstate AEPA CEOE CSET/CBEST FTCE GACE ILTS MTEL MTLE MTTC MoGEA/MEGA NES NYSTCE OAE Praxis RICA TExES VCLA WEST HESI TEAS NCLEX PAXRN CNE AP ASCP ASWB Accuplacer CHSPE CTEL DSST Finance FSA HSPT ISEE MBLEx NCE PERT PHR SHSAT TASC TECEP UExcel Credit Credit CreditOptions CreditOptions OnlineCollegeCredit HighSchool&GED CertificatesofCompletion HowitWorks TransferCredit TransferCredit AllTransferCredit FindYourSchool CollegeCoursesforCredit TransferringCredit HowItWorks EarnTransferableCredit&GetyourDegreefast Learnmore ContactSupport Lesson Transcript RileyKench,LauraPennington Author RileyKench Rileyhastutoredcollegiatemathematicsforsevenyears.TheyhaveaMasterofArtsdegreeinMathematicsfromCentralMichiganUniversityandaBachelorofSciencedegreeinMathematicsfromCentralMichiganUniversity. Viewbio Instructor LauraPennington LaurareceivedherMaster'sdegreeinPureMathematicsfromMichiganStateUniversity,andherBachelor'sdegreeinMathematicsfromGrandValleyStateUniversity.Shehas20yearsofexperienceteachingcollegiatemathematicsatvariousinstitutions. Viewbio Inthislesson,learnaboutmonotonicfunctionsandhowtoidentifythem.Moreover,understandthenatureofincreasinganddecreasingmonotonicfunctionswithexamplesandequations. Updated:01/07/2022 TableofContentsWhatisaMonotonicFunction?Non-MonotonicFunctionsLessonSummaryShow Createanaccount WhatisaMonotonicFunction?Amonotonicfunctionisafunctionthatiseitheralwaysincreasingoralwaysdecreasing.HereisanimageinFigure1showingamonotonicallyincreasingfunction: Fig.1:MonotonicallyIncreasingFunction NoticethatintheimageinFigure1,thegraphisalwaysincreasinginthepositivedirectiononitsdomain.Thisisamonotonicallyincreasingfunction.Amonotonicallydecreasingfunctionwouldbeafunctionthatisdecreasinginthepositivedirectiononitsdomain.Thisisthemonotonicdecreasingfunctiondefinition.Thenatureofafunctiondetermineswhetheritwillbemonotonicallyincreasing,monotonicallydecreasing,orneither. Ifthefunctionisalwaysincreasingonitsdomain,thefunctionissaidtobemonotonicallyincreasing.Thismeansthatforany{eq}a,b{/eq}inthedomainofthefunctionsuchthat{eq}af(b){/eq}.Anysubsequentoutputvalueisstrictlylessthananypreviousoutputvalue. Noticethatinthecasesforstrictlymonotonic,itisnotpossibleforthegraphofthefunctiontobehorizontalanywhereinitsdomainsincetheoutputvaluesarealwaysstrictlygreaterthanorstrictlylessthanthepreviousoutputvalues.ConditionforaMonotonicFunctionThereisaconditionforknowingifafunctionismonotonic.Thisconditioninvolvesthesignofthederivativeofthefunction. Amonotonicfunctionwillhaveaderivativethatiseitheralwayspositive(monotonicallyincreasing)oralwaysnegative(monotonicallydecreasing).Thatis,ifthederivativeofafunctionisalwayspositiveornegative,thenthefunctionismonotonic.Moreover,ifthederivativeofafunctionisalwayspositiveornegativeandnotzeroanywhere,thenthefunctionisstrictlymonotonic. HereisanimageofafunctioninFigure2thatismonotonicallyincreasing: Fig.2:MonotonicallyIncreasingFunction2 InFigure2,theequationofthefunctionisgivenas{eq}f(x)=0.5x+2{/eq}.Thederivativeofthisfunctionis{eq}f'(x)=0.5>0{/eq}.So,thisfunctionismonotonicallyincreasingsinceitsderivativeisalwayspositivebutitisalsostrictlymonotonicsincethederivativeisneverzero.HereisanimageofafunctioninFigure3thatismonotonicallydecreasing: Fig.3:MonotonicallyDecreasingFunction InFigure3,theequationofthefunctionis{eq}f(x)=1-x{/eq}.Thederivativeofthisfunctionis{eq}f'(x)=-1<0{/eq}.Thisfunctionismonotonicallydecreasingsinceitsderivativeisalwaysnegative.Moreover,sincethederivativeisneverzero,thefunctionisstrictlymonotonic.Ultimately,thederivativeofafunctiondeterminesifthefunctionismonotonic.MonotonicFunctionExamplesHerearesomemonotonicfunctionexamples:Example1:Isthefunction{eq}f(x)=x^3{/eq}monotonic?Strictlymonotonic?Solution:Well,thederivativeofthisfunctionis{eq}f'(x)=3x^2{/eq}.Noticethat{eq}x^2\geq0{/eq}forallxwhichmeansthat{eq}3x^2\geq0{/eq}forallx.Therefore,theconditionforamonotonicfunctionimpliesthatthisfunctionismonotonicallyincreasingsinceitsderivativeisalwaysgreaterthanorequaltozero.However,thisfunctionisnotstrictlymonotonicsinceat{eq}x=0{/eq},thederivativeiszero.Example2:Isthefunction{eq}f(x)=-x^5-2x^3+1{/eq}monotonic?Strictlymonotonic?Solution:Thederivativeofthisfunctionis{eq}f'(x)=-5x^4-6x^2{/eq}.Now,{eq}x^4{/eq}and{eq}x^2{/eq}arebothalwaysgreaterthanorequaltozeroontheirdomains.But,sincetheyarebeingmultipliedbynegativenumbers,theterms{eq}-5x^4{/eq}and{eq}-6x^2{/eq}arebothnegativemakingthederivativealwayslessthanorequaltozero.Therefore,theconditionformonotonicfunctionsimpliesthatthisfunctionismonotonicallydecreasing.However,sincethederivativecanbezero,thisfunctionisnotstrictlymonotonic.Example3: MonotonicityThemonotonicityofafunctiontellsusifthefunctionisincreasingordecreasing.Afunctionisincreasingwhenitsgraphrisesfromlefttoright.Intechnicalterms,afunctionisincreasingonanintervalIifforanyx1andx2inI,x1islessthanx2impliesthatf(x1)islessthanf(x2).Basically,thissaysthatwhenx1islessthanx2,thenthefunctionevaluatedatx1islessthanthefunctionevaluatedatx2.Afunctionisdecreasingwhenitsgraphfallsfromlefttoright.Again,thetechnicaldefinitionsaysthatafunctionisdecreasingonanintervalIifforanyx1andx2inI,x1islessthanx2impliesthatf(x1)isgreaterthanf(x2).Inotherwords,afunctionisdecreasingonanintervalIwhenitisthecasethatwheneverx1islessthanx2,f(x1)isgreaterthanf(x2).Themonotonicityofafunctionisdirectlyrelatedtothefunction'sderivative.Afunctionisincreasingwhenitsderivativeispositive,andafunctionisdecreasingwhenitsderivativeisnegative.Forexample,considerthefunctionf(x)=x2.Thederivativeisf'(x)equals2x.Nowconsiderwhere2xispositiveandnegative.2xisgreaterthan0.Dividebothsidesby2.xisgreaterthan0.Therefore,fispositivewhenxisgreaterthan0,sofisincreasingontheintervalxisgreaterthan0.Similarly,considerwhere2xisnegative:2xislessthan0Dividebothsidesby2xislessthan0Thus,f'isnegativewhenxislessthan0,andfisdecreasingontheintervalxislessthan0. Anerroroccurredtryingtoloadthisvideo. Tryrefreshingthepage,orcontactcustomersupport. YoumustcCreateanaccounttocontinue watching Registertoviewthislesson Areyouastudentorateacher? Iamastudent Iamateacher CreateYourAccountToContinueWatching Asamember,you'llalsogetunlimitedaccesstoover84,000 lessonsinmath,English,science,history,andmore.Plus,getpracticetests,quizzes,andpersonalizedcoachingtohelpyou succeed. Getunlimitedaccesstoover84,000lessons. Tryitnow Itonlytakesafewminutestosetupandyoucancancelanytime. Alreadyregistered?Loginherefor access  Back WhatteachersaresayingaboutStudy.com Tryitnow Alreadyregistered? Loginhereforaccess Comingupnext: WhatIsanExponentialFunction? You'reonaroll.Keepupthegoodwork! TakeQuiz Watch NextLesson  Replay Justcheckingin.Areyoustillwatching? Yes!Keepplaying. Yournextlessonwillplayin 10seconds 0:01Monotonicity 2:11MonotonicFunction 2:57Non-MonotonicFunction 3:44Examples 4:41LessonSummary Save Save Save Wanttowatchthisagainlater? LoginorsignuptoaddthislessontoaCustomCourse. LoginorSignup Timeline Autoplay Autoplay Speed Speed 84Kviews Video Quiz Course 84K views Isthefunction{eq}f(x)=e^x{/eq}monotonic?Strictlymonotonic?Solution:Thederivativeofthisfunctionis{eq}f'(x)=e^x{/eq}.Thisisanexponentialfunctionthatisneverzeroonitsdomain.So,{eq}e^x>0{/eq}forallxinitsdomain.Therefore,thisfunctionismonotonicallyincreasing.Moreover,sincethederivativeisneverzero,thisfunctionisstrictlymonotonicaswell.Non-MonotonicFunctionsAnon-monotonicfunctionisanyfunctionthatisnotmonotonic.Inessence,anon-monotonicfunctionisnotentirelyincreasingordecreasingonitsdomain;anon-monotonicfunctionwillbothincreaseanddecreaseonitsdomain.Aconditiontoshowifafunctionisnon-monotonicistoshowthatthederivativeisbothpositiveandnegativeonthedomainofthefunctionratherthanitbeingfixed.Herearesomeexamplesofnon-monotonicfunctions: {eq}f(x)=x^2{/eq}isnon-monotonicsinceitsderivative{eq}f'(x)=2x{/eq}isbothpositiveandnegative.Infact,allevenfunctionsarenon-monotonicsincetheyarealwaysincreasinganddecreasingontheirdomains.HereisanimageinFigure4: Fig.4:Graphofx^2 {eq}f(x)=sin(x){/eq}isnon-monotonicsinceitsderivative{eq}f'(x)=cos(x){/eq}isbothpositiveandnegativeonitsdomain.Infact,sincethederivativesofsineandcosinearerelated,bothofthesetrigonometricfunctionsarenon-monotonic.HereisanimageinFigure5: Fig.5:Graphofsin(x) {eq}f(x)=\vertx\vert{/eq}isnon-monotonicsinceitsderivative{eq}f'(x)=\frac{x}{\vertx\vert}{/eq}isbothpositiveandnegative.Thedenominatorofthederivativeisalwayspositivebutthenumeratorcanbebothpositiveandnegative.Therefore,thefunctionmustbenon-monotonic.HereisanimageinFigure6: Fig.6:GraphoftheAbsoluteValueFunction LessonSummaryAmonotonicfunctionisafunctionthatiseitheralwaysincreasingoralwaysdecreasingonitsdomain.Thenatureofafunctionwilldetermineifitismonotonic: Amonotonicallyincreasingfunctionfhastheconditionthatforany{eq}a,b{/eq}inthedomainsuchthat{eq}a1?'s':''}} Ch1. Functions:HomeworkHelp Functions:Identification,Notation&PracticeProblems 9:24 Transformations:HowtoShiftGraphsonaPlane 7:12 WhatIsDomainandRangeinaFunction? 8:32 HowtoAdd,Subtract,MultiplyandDivideFunctions 6:43 HowtoComposeFunctions 6:52 InverseFunctions 6:05 ApplyingFunctionOperationsPracticeProblems 5:17 WhatisFunctionNotation:Definition&Examples 9:26 DiscontinuousFunctions:Properties&Examples 3:52 MonotonicFunction:Definition&Examples 5:17 Goto Functions:HomeworkHelp Ch2. EvaluatingExponentialandLogarithmic... Goto EvaluatingExponentialandLogarithmicFunctions:HomeworkHelp Ch3. Inequalities:HomeworkHelp Goto Inequalities:HomeworkHelp Ch4. FoundationsandLinearEquations:... Goto FoundationsandLinearEquations:HomeworkHelp Ch5. GraphingandRationalEquations:... Goto GraphingandRationalEquations:HomeworkHelp Ch6. FactoringandGraphingQuadratic... Goto FactoringandGraphingQuadraticEquations:HomeworkHelp Ch7. ExponentsandPolynomials:Homework... Goto ExponentsandPolynomials:HomeworkHelp Ch8. PiecewiseandCompositeFunctions:... Goto PiecewiseandCompositeFunctions:HomeworkHelp Ch9. GeometryandTrigonometry:Homework... Goto GeometryandTrigonometry:HomeworkHelp Ch10. UsingScientificCalculators:Homework... Goto UsingScientificCalculators:HomeworkHelp Ch11. Trigonometry:HomeworkHelp Goto Trigonometry:HomeworkHelp MonotonicallyIncreasingFunction|MonotonicFunctionExampleRelatedStudyMaterials RecommendedLessonsandCoursesforYou RelatedLessons RelatedCourses RelatedLessons UsingMonotonicity&ConcavitytoAnalyzeCurves GraphTheoryConceptsandTerminology IncreasingFunctionGraph&Examples|WhenisaFunctionIncreasing? LinearApproximationsUsingDifferentials:Definition&Examples HowtoFindtheArcLengthofaFunction L'Hospital'sRuleOverview&Examples|WhentoUseL'Hospital'sRule ContinuousFunctionsTheorems InfiniteSeries&PartialSums:Explanation,Examples&Types Rolle'sTheorem|ASpecialCaseoftheMeanValueTheorem PowerSeriesinX&theIntervalofConvergence GraphingaSystemofQuadraticInequalities:Examples&Process ThePhillipsCurveModel&Graph|WhatisthePhillipsCurve? L'Hopital'sRuleFormula&Examples|HowDoesL'Hopital'sRuleWork? HowtoFindVolumesofRevolutionWithIntegration HowtoUseRiemannSumstoCalculateIntegrals ConvergentSequenceFormula&Examples|WhatisaConvergentSequence? MarginalRateofSubstitution:Definition,Formula&Examples WhatisaPartialDerivative?|PartialDerivativeExamples,Rules,Formula&Calculation OpenSetvs.ClosedSetExamples&Overview|WhatareOpen&ClosedSets? RatioTestforConvergence&Divergence|Rules,Formula&Examples RelatedCourses Math103:Precalculus APCalculusAB&BC:TutoringSolution BusinessCalculus:Help&Review THEATest:Practice&StudyGuide HSCMathematics:ExamPrep&Syllabus Math105:PrecalculusAlgebra TExESMathematics7-12(235):Practice&StudyGuide Precalculus:HighSchool Trigonometry:HighSchool QuantitativeAnalysis HoltGeometry:OnlineTextbookHelp Precalculus:TutoringSolution Calculus:HelpandReview IntroductiontoStatistics:HelpandReview IntroductiontoStatistics:TutoringSolution HighSchoolAlgebraII:HomeworkHelpResource HighSchoolAlgebraII:HelpandReview HighSchoolAlgebraI:HelpandReview HighSchoolAlgebraI:HomeworkHelpResource HighSchoolAlgebraI:TutoringSolution Related RecentlyUpdated Popular ExploreSubjects HomeworkHelp BrowsebyCourses MathCourses PrecalculusCourses TransferableCreditCourses CertificatesCourses StudyCourses NYRegentsExam-Geometry:HelpandReview NYRegentsExam-Geometry:TutoringSolution NYRegentsExam-IntegratedAlgebra:HelpandReview NYRegentsExam-IntegratedAlgebra:TutoringSolution UExcelStatistics:StudyGuide&TestPrep CollegeMathematics:CertificateProgram Calculus:CertificateProgram PrecalculusAlgebra:CertificateProgram IntroductiontoStatistics:CertificateProgram Precalculus:CertificateProgram CollegeAlgebra:CertificateProgram CLEPCalculus:StudyGuide&TestPrep Precalculus:HelpandReview BrowsebyLessons ConsistencyinWritingStyle&Tone SATPrepEssayTopics&Rubric SATWriting&LanguageTest:CommandofEvidence SATWriting&LanguageTest:AnalysisQuestions-History&Science SATWriting&LanguageTest:ExpressionofIdeas Quiz&Worksheet-PracticewithMass&Volume Quiz&Worksheet-CreatingCoordinatePlanes Quiz&Worksheet-AnglesofElevation&Depression Quiz&Worksheet-SolvingWordProblemswithDiagrams Quiz&Worksheet-Multi-StepEquationswithFractions&Decimals ACTWriting:EssaySkills ACTReading:WordMeaningsandChoice ACTMath:Overview ACTMath:Number&Quantity ACTMath:Pre-Algebra LatestCourses ComputerScience:HelpandReview GeneralHistoryLessons GeneralScienceLessons TExESScienceofTeachingReading(293):Practice&StudyGuide NextGenNCLEX-PNStudyGuide&Practice ProgramminginPython:HelpandReview Chance&FundamentalCountingTechniques Transformations,GeometricModels&Nets BasicPropertiesofShapesinMath Patterns&MathematicalInvestigations WhatistheCPACETest? WhatisthePennsylvaniaCoreAssessmentExam? ShouldIHomeschoolMyChild? HowtoRegisterforthePennsylvaniaCoreAssessmentExam HowtoPickYourHomeschoolCurriculum RoleofStudentSupportinOpen&DistanceLearning TExESPrincipalExamRedesign(068vs.268) LatestLessons TheDifferenceBetweenAffectvs.Effect SafieinFrankenstein ADHD&SpeechDelay ZodiacLessonforKids:Signs&Constellations TheOldManandtheSeaEnding:Meaning,Explanation&Analysis GroverCleveland'sFailures CyclicVomitingSyndrome:SignsandTreatment Quiz&Worksheet-ClinicalSupervision Quiz&Worksheet-PregabalinImpact&Withdrawal Quiz&Worksheet-LifeUndertheShangDynasty Quiz&Worksheet-DayoftheDeadFactsforKids Flashcards-RealEstateMarketingBasics Flashcards-PromotionalMarketinginRealEstate MathWorksheets|PrintableMathWorksheetsforTeachers MusicLessonPlans PopularCourses GACEMiddleGradesMathematics(013):Practice&StudyGuide McDougalLittellPre-Algebra:OnlineTextbookHelp PraxisMarketingEducation(5561):Practice&StudyGuide PraxisSpanish(5195):Practice&StudyGuide OhioAssessmentsforEducators-MiddleGradesMathematics(030):Practice&StudyGuide InformalTechnicalReports:HelpandReview WorkandEnergyfortheMCAT:TutoringSolution Quiz&Worksheet-PrayingMantisasaSymbol Quiz&Worksheet-CharacteristicsofKingdomPlantae Quiz&Worksheet-Chitin Quiz&Worksheet-CharacteristicsofSignalingMolecules Quiz&Worksheet-History&ComponentsoftheSymphony PopularLessons AminoGroup:Definition,Structure&Formula WhatIstheTango?-Definition,Origin&History SoundsExperimentsLessonforKids Greenwich,CTAdultEducation DescriptiveWritingPrompts SeniorProjectIdeas CreativeWritingCompetitionsforHighSchoolStudents CreativeWritingPromptsforKids ScienceFictionWritingPrompts SolarSystemProjectIdeas ScienceWritingPrompts ExpelledfromSchool:NextSteps Math Math-Videos Math-Quizzes Math-Questions&Answers SocialSciences SocialSciences-Videos SocialSciences-Quizzes SocialSciences-Questions&Answers Science Science-Videos Science-Quizzes Science-Questions&Answers Business Business-Videos Business-Quizzes Business-Questions&Answers Humanities Humanities-Videos Humanities-Quizzes Humanities-Questions&Answers Education Education-Videos Education-Quizzes Education-Questions&Answers History History-Videos History-Quizzes History-Questions&Answers ArtandDesign ArtandDesign-Videos ArtandDesign-Quizzes ArtandDesign-Questions&Answers TechandEngineering TechandEngineering-Videos TechandEngineering-Quizzes TechandEngineering-Questions&Answers HealthandMedicine HealthandMedicine-Videos HealthandMedicine-Quizzes HealthandMedicine-Questions&Answers Showthattheequationhasexactlyonerealroot.2x-1-\sinx=0 Supposethatf'(x)=\frac{2}{1+x^4}-\frac{3}{4+x^4}.Whichislarger:f(0)orf(1)? ThefunctionA(x)=0.004x^3-0.05x^2+0.18x+0.05approximatesthebloodalcoholconcentrationinaperson'sbloodstreamxhoursafterdrinking8ouncesofahardliquor.Thefunctiononlyapplies Determinewhetherthefollowingsequenceismonotonic.A_n=\frac{n+1}{n} Considerthefunctionf(x)=(x^2+1)e^x,forallxbelongstoR.Usemonotonicitytoshowthatfisone-to-one.Thenevaluate(f^{-1})'(2e). f(x)iscontinuousfor?0.5?x??0.2andalsohasthefollowingvalues:x -0.5 -0.4 -0.3 -0.2f(x) 1 1.1 1.3 1.6Whichofthefollowingmaybetrueaboutf(x)?I.f(x)maybemonotonicf Usea_{n+1}/a_ntoshowthatthegivensequencea_nisstrictlyincreasingorstrictlydecreasing.(a).{2^n/(1+2^n)}^{+infinity}_{n=1}(b).{10/(2n)!}^{+infinity}_{n=1} AboundedsequenceisonewherethereissomeintegerMsuchthat-M\leqa_{n}\leqMForalln.Amonotonicsequenceisonewhereeithera_{n+1}>a_{n}foralln(increasing)ora_{n+1}