Monotonicity and Extremum of functions - Definition, Examples

What is a monotonic function? Functions are known as monotonic if they are increasing or decreasing in their entire domain. Examples : f(x) = 2x + 3, ... CheckoutJEEMAINS2022QuestionPaperAnalysis: CheckoutJEEMAINS2022QuestionPaperAnalysis: × DownloadNow JEEIITJEEStudyMaterialMonotonicityandExtremumofFunctions Previous Next MonotonicityandExtremumofFunctions Monotonicityisoneoftheimportantconceptsofapplicationofderivatives.Themonotonicityofafunctiongivesanideaaboutthebehaviourofthefunction.Afunctionissaidtobemonotonicallyincreasingifitsgraphisonlyincreasingwithincreasingvaluesofequation.Similarly,functionismonotonicallydecreasingifitsvaluesareonlydecreasing. Monotonicity:Themostusefulelementtakenintoconsiderationamongstthetotalactivitiesofthefunctionistheirmonotonicbehaviour.Ittellsabouttheincreasingordecreasingbehaviourofthefunction. Extremum:Anextremumofafunctionisthepointwherewegetthemaximumorminimumvalueofthefunctioninsomeinterval. MonotonicityofaFunction Whatisamonotonicfunction?Functionsareknownasmonotoniciftheyareincreasingordecreasingintheirentiredomain. Examples:f(x)=2x+3,f(x)=log(x),f(x)=exaretheexamplesofincreasingfunctionandf(x)=-x5andf(x)=e-xaretheexamplesofdecreasingfunction. Increasingfunction:   Ifx1F(x2)thenfunctionisknownasdecreasingfunctionorstrictlydecreasingfunction. NonmonotonicFunction Thefunctionswhichareincreasingaswellasdecreasingintheirdomainareknownasnonmonotonicfunction. Example:f(x)=sinx,f(x)=|x|areexamplesofnonmonotonicfunction.Butf(x)=sinxisincreasingin[0,Π/2]orwecansayitismonotonicin[0,Π/2]   Monotonicityofafunctionatapoint Afunctionissaidtobemonotonicallyincreasingatx=aifitsatisfies f(a+h)>f(a) f(a–h)f(a) Wherehisverysmallvalue   Note:Wecantalkaboutthemonotonicityofafunctionatx=aonlyifx=aisinthedomainofthefunctionandwedon’tneedtotakecontinuityanddifferentiabilityinconsideration.   MonotonicityinanInterval (a).Foranincreasingfunctioninsomeinterval, Ifdy/dx>0forallthevaluesofxbelongstothatinterval,thenthefunctionisknownasmonotonicallyincreasingorstrictlyincreasingfunction.   (b).Foradecreasingfunctioninsomeinterval, Ifdy/dx<0forallthevaluesofxbelongstothatinterval,thenthefunctionisknownasmonotonicallydecreasingorstrictlydecreasingfunction.   Note:hencetofindtheintervalofmonotonicityforafunctiony=f(x)weneedtofindoutthevalueofdy/dxandhavetosolvetheinequalitydy/dx>00rdy/dx<0.Thesolutionofthisinequalitygivestheintervalofmonotonicity.   Note:Ifdy/dx=0forafunctiony=f(x),stillthefunctioncanbeincreasingatx=a.Considerafunctionf(x)=x3whichisincreasingatx=0althoughdy/dx=0.Thisisbecausef(0+h)>f(0)andf(0–h)dy/dx=1–cos(x) dy/dx≥0ascos(x)havingvalueininterval[-1,1]anddy/dx=0forthediscretevaluesofxanddonotformaninterval,hencewecanincludethisfunctioninmonotonicallyincreasingfunction. Example2:Findtheintervalofmonotonicityforf(x)=x/(log(x)). Solution:f(x)=x/(log(x)) =>dy/dx=(log(x)–1)/(log(x))2 dy/dx>0 logx–1>0 =>x>e f(x)isincreasingforx>e. dy/dx<0 => logx–1<0 =>xf’(x)=x2 +25x+6 Forstrictlyincreasing: f’(x)>0 =>x2+5x+6>0 =>(x+3)(x+2)>0 =>xϵ(-3,∞)U(-2,∞) Forstrictlydecreasing: f’(x)<0 =>x2+5x+6<0 =>xϵ(-3,-2). Example4:Findtheextremumoffunctionf(x)=3x3–9xintheinterval[-1,4] Solution:f(x)=3x3 –9x =>f’(x)=9x2–9=9(x2–1) =>f’(x)=0=>9(x2 –1)=0orx=±1 =>f(-1)=6 =>f(1)=–6 =>f(4)=156 =>Greatestvalue=156andleastvalue=–6. Example5: Inwhatintervalisthefunction \(\begin{array}{l}\sinx-\cosx\end{array}\) increasing? Solution: Wehave, \(\begin{array}{l}f'(x)=\cosx+\sinx\end{array}\) Now, f(x)isincreasingfunctionofx,if \(\begin{array}{l}f'(x)=\cosx+\sinx>0\end{array}\)or \(\begin{array}{l}\sqrt{2}\cos\left(x-\frac{\pi}{4}\right)>0\end{array}\) f'(x)>o if0