Critical point (mathematics) - Wikipedia

When dealing with complex variables, a critical point is, similarly, a point in the function's domain where it is either not holomorphic or the derivative is ... Criticalpoint(mathematics) FromWikipedia,thefreeencyclopedia Jumptonavigation Jumptosearch Pointwherethederivativeofafunctioniszero Forotheruses,seeCriticalpoint. Thisarticleincludesalistofgeneralreferences,butitlackssufficientcorrespondinginlinecitations.Pleasehelptoimprovethisarticlebyintroducingmoreprecisecitations.(January2015)(Learnhowandwhentoremovethistemplatemessage) Theabscissae(x-coordinates)oftheredcirclesarestationarypoints;thebluesquaresareinflectionpoints. Criticalpointisawidetermusedinmanybranchesofmathematics. Whendealingwithfunctionsofarealvariable,acriticalpointisapointinthedomainofthefunctionwherethefunctioniseithernotdifferentiableorthederivativeisequaltozero.[1]Whendealingwithcomplexvariables,acriticalpointis,similarly,apointinthefunction'sdomainwhereitiseithernotholomorphicorthederivativeisequaltozero.[2][3]Likewise,forafunctionofseveralrealvariables,acriticalpointisavalueinitsdomainwherethegradientisundefinedorisequaltozero.[4] Thevalueofthefunctionatacriticalpointisacriticalvalue. ThissortofdefinitionextendstodifferentiablemapsbetweenRmandRn,acriticalpointbeing,inthiscase,apointwheretherankoftheJacobianmatrixisnotmaximal.Itextendsfurthertodifferentiablemapsbetweendifferentiablemanifolds,asthepointswheretherankoftheJacobianmatrixdecreases.Inthiscase,criticalpointsarealsocalledbifurcationpoints. Inparticular,ifCisaplanecurve,definedbyanimplicitequationf(x,y)=0,thecriticalpointsoftheprojectionontothex-axis,paralleltothey-axisarethepointswherethetangenttoCareparalleltothey-axis,thatisthepointswhere ∂ f ∂ y ( x , y ) = 0 {\displaystyle{\frac{\partialf}{\partialy}}(x,y)=0} Inotherwords,thecriticalpointsarethosewheretheimplicitfunctiontheoremdoesnotapply. ThenotionofacriticalpointallowsthemathematicaldescriptionofanastronomicalphenomenonthatwasunexplainedbeforethetimeofCopernicus.Astationarypointintheorbitofaplanetisapointofthetrajectoryoftheplanetonthecelestialsphere,wherethemotionoftheplanetseemstostopbeforerestartingintheotherdirection.Thisoccursbecauseofacriticalpointoftheprojectionoftheorbitintotheeclipticcircle. Contents 1Criticalpointofasinglevariablefunction 1.1Examples 1.2Locationofcriticalpoints 2Criticalpointsofanimplicitcurve 2.1Useofthediscriminant 3Severalvariables 3.1Applicationtooptimization 4Criticalpointofadifferentiablemap 5Applicationtotopology 6Seealso 7References Criticalpointofasinglevariablefunction[edit] Acriticalpointofafunctionofasinglerealvariable,f(x),isavaluex0inthedomainoffwhereitisnotdifferentiableoritsderivativeis0(f′(x0)=0).[1]Acriticalvalueistheimageunderfofacriticalpoint.Theseconceptsmaybevisualizedthroughthegraphoff:atacriticalpoint,thegraphhasahorizontaltangentifyoucanassignoneatall. Noticehow,foradifferentiablefunction,criticalpointisthesameasstationarypoint. Althoughitiseasilyvisualizedonthegraph(whichisacurve),thenotionofcriticalpointofafunctionmustnotbeconfusedwiththenotionofcriticalpoint,insomedirection,ofacurve(seebelowforadetaileddefinition).Ifg(x,y)isadifferentiablefunctionoftwovariables,theng(x,y)=0istheimplicitequationofacurve.Acriticalpointofsuchacurve,fortheprojectionparalleltothey-axis(themap(x,y)→x),isapointofthecurvewhere ∂ g ∂ y ( x , y ) = 0 {\displaystyle{\frac{\partialg}{\partialy}}(x,y)=0} .Thismeansthatthetangentofthecurveisparalleltothey-axis,andthat,atthispoint,gdoesnotdefineanimplicitfunctionfromxtoy(seeimplicitfunctiontheorem).If(x0,y0)issuchacriticalpoint,thenx0isthecorrespondingcriticalvalue.Suchacriticalpointisalsocalledabifurcationpoint,as,generally,whenxvaries,therearetwobranchesofthecurveonasideofx0andzeroontheotherside. Itfollowsfromthesedefinitionsthatadifferentiablefunctionf(x)hasacriticalpointx0withcriticalvaluey0,ifandonlyif(x0,y0)isacriticalpointofitsgraphfortheprojectionparalleltothex-axis,withthesamecriticalvaluey0.Iffisnotdifferentiableatx0duetothetangentbecomingparalleltothey-axis,thenx0isagainacriticalpointoff,butnow(x0,y0)isacriticalpointofitsgraphfortheprojectionparalleltoy-axis. Forexample,thecriticalpointsoftheunitcircleofequationx2+y2-1=0are(0,1)and(0,-1)fortheprojectionparalleltothex-axis,and(1,0)and(-1,0)forthedirectionparalleltothey-axis.Ifoneconsiderstheupperhalfcircleasthegraphofthefunction f ( x ) = 1 − x 2 {\displaystylef(x)={\sqrt{1-x^{2}}}} ,thenx=0isacriticalpointwithcriticalvalue1duetothederivativebeingequalto0,andx=-1andx=1arecriticalpointswithcriticalvalue0duetothederivativebeingundefined. Examples[edit] Thefunctionf(x)=x2+2x+3isdifferentiableeverywhere,withthederivativef′(x)=2x+2.Thisfunctionhasauniquecriticalpoint−1,becauseitistheuniquenumberx0forwhich2x0+2=0.Thispointisaglobalminimumoff.Thecorrespondingcriticalvalueisf(−1)=2.Thegraphoffisaconcaveupparabola,thecriticalpointistheabscissaofthevertex,wherethetangentlineishorizontal,andthecriticalvalueistheordinateofthevertexandmayberepresentedbytheintersectionofthistangentlineandthey-axis. Thefunctionf(x)=x2/3isdefinedforallxanddifferentiableforx≠0,withthederivativef′(x)=2x−1/3/3.Sincefisnotdifferentiableatx=0andf'(x)≠0otherwise,itistheuniquecriticalpoint.Thegraphofthefunctionfhasacuspatthispointwithverticaltangent.Thecorrespondingcriticalvalueisf(0)=0. Theabsolutevaluefunctionf(x)=|x|isdifferentiableeverywhereexceptatcriticalpointx=0,whereithasaglobalminimumpoint,withcriticalvalue0. Thefunctionf(x)=1/xhasnocriticalpoints.Thepointx=0isnotacriticalpointbecauseitisnotincludedinthefunction'sdomain. Locationofcriticalpoints[edit] BytheGauss–Lucastheorem,allofapolynomialfunction'scriticalpointsinthecomplexplanearewithintheconvexhulloftherootsofthefunction.Thusforapolynomialfunctionwithonlyrealroots,allcriticalpointsarerealandarebetweenthegreatestandsmallestroots. Sendov'sconjectureassertsthat,ifallofafunction'srootslieintheunitdiskinthecomplexplane,thenthereisatleastonecriticalpointwithinunitdistanceofanygivenroot. Criticalpointsofanimplicitcurve[edit] Seealso:Algebraiccurve Criticalpointsplayanimportantroleinthestudyofplanecurvesdefinedbyimplicitequations,inparticularforsketchingthemanddeterminingtheirtopology.Thenotionofcriticalpointthatisusedinthissection,mayseemdifferentfromthatofprevioussection.Infactitisthespecializationtoasimplecaseofthegeneralnotionofcriticalpointgivenbelow. Thus,weconsideracurveCdefinedbyanimplicitequation f ( x , y ) = 0 {\displaystylef(x,y)=0} ,wherefisadifferentiablefunctionoftwovariables,commonlyabivariatepolynomial.ThepointsofthecurvearethepointsoftheEuclideanplanewhoseCartesiancoordinatessatisfytheequation.Therearetwostandardprojections π y {\displaystyle\pi_{y}} and π x {\displaystyle\pi_{x}} ,definedby π y ( ( x , y ) ) = x {\displaystyle\pi_{y}((x,y))=x} and π x ( ( x , y ) ) = y , {\displaystyle\pi_{x}((x,y))=y,} thatmapthecurveontothecoordinateaxes.Theyarecalledtheprojectionparalleltothey-axisandtheprojectionparalleltothex-axis,respectively. ApointofCiscriticalfor π y {\displaystyle\pi_{y}} ,ifthetangenttoCexistsandisparalleltothey-axis.Inthatcase,theimagesby π y {\displaystyle\pi_{y}} ofthecriticalpointandofthetangentarethesamepointofthex-axis,calledthecriticalvalue.Thusapointiscriticalfor π y {\displaystyle\pi_{y}} ifitscoordinatesaresolutionofthesystemofequations: f ( x , y ) = ∂ f ∂ y ( x , y ) = 0 {\displaystylef(x,y)={\frac{\partialf}{\partialy}}(x,y)=0} Thisimpliesthatthisdefinitionisaspecialcaseofthegeneraldefinitionofacriticalpoint,whichisgivenbelow. Thedefinitionofacriticalpointfor π x {\displaystyle\pi_{x}} issimilar.IfCisthegraphofafunction y = g ( x ) {\displaystyley=g(x)} ,then(x,y)iscriticalfor π x {\displaystyle\pi_{x}} ifandonlyifxisacriticalpointofg,andthatthecriticalvaluesarethesame. SomeauthorsdefinethecriticalpointsofCasthepointsthatarecriticalforeither π x {\displaystyle\pi_{x}} or π y {\displaystyle\pi_{y}} ,althoughtheydependnotonlyonC,butalsoonthechoiceofthecoordinateaxes.Itdependsalsoontheauthorsifthesingularpointsareconsideredascriticalpoints.Infactthesingularpointsarethepointsthatsatisfy f ( x , y ) = ∂ f ∂ x ( x , y ) = ∂ f ∂ y ( x , y ) = 0 {\displaystylef(x,y)={\frac{\partialf}{\partialx}}(x,y)={\frac{\partialf}{\partialy}}(x,y)=0} , andarethussolutionsofeithersystemofequationscharacterizingthecriticalpoints.Withthismoregeneraldefinition,thecriticalpointsfor π y {\displaystyle\pi_{y}} areexactlythepointswheretheimplicitfunctiontheoremdoesnotapply. Useofthediscriminant[edit] WhenthecurveCisalgebraic,thatiswhenitisdefinedbyabivariatepolynomialf,thenthediscriminantisausefultooltocomputethecriticalpoints. Hereweconsideronlytheprojection π y {\displaystyle\pi_{y}} ;Similarresultsapplyto π x {\displaystyle\pi_{x}} byexchangingxandy. Let Disc y ⁡ ( f ) {\displaystyle\operatorname{Disc}_{y}(f)} bethediscriminantoffviewedasapolynomialinywithcoefficientsthatarepolynomialsinx.Thisdiscriminantisthusapolynomialinxwhichhasthecriticalvaluesof π y {\displaystyle\pi_{y}} amongitsroots. Moreprecisely,asimplerootof Disc y ⁡ ( f ) {\displaystyle\operatorname{Disc}_{y}(f)} iseitheracriticalvalueof π y {\displaystyle\pi_{y}} suchthecorrespondingcriticalpointisapointwhichisnotsingularnoraninflectionpoint,orthex-coordinateofanasymptotewhichisparalleltothey-axisandistangent"atinfinity"toaninflectionpoint(inflexionasymptote). Amultiplerootofthediscriminantcorrespondeithertoseveralcriticalpointsorinflectionasymptotessharingthesamecriticalvalue,ortoacriticalpointwhichisalsoaninflectionpoint,ortoasingularpoint. Severalvariables[edit] Forafunctionofseveralrealvariables,apointP(thatisasetofvaluesfortheinputvariables,whichisviewedasapointinRn)iscriticalifitisapointwherethegradientisundefinedorthegradientiszero.[4]Thecriticalvaluesarethevaluesofthefunctionatthecriticalpoints. Acriticalpoint(wherethefunctionisdifferentiable)maybeeitheralocalmaximum,alocalminimumorasaddlepoint.IfthefunctionisatleasttwicecontinuouslydifferentiablethedifferentcasesmaybedistinguishedbyconsideringtheeigenvaluesoftheHessianmatrixofsecondderivatives. AcriticalpointatwhichtheHessianmatrixisnonsingularissaidtobenondegenerate,andthesignsoftheeigenvaluesoftheHessiandeterminethelocalbehaviorofthefunction.Inthecaseofafunctionofasinglevariable,theHessianissimplythesecondderivative,viewedasa1×1-matrix,whichisnonsingularifandonlyifitisnotzero.Inthiscase,anon-degeneratecriticalpointisalocalmaximumoralocalminimum,dependingonthesignofthesecondderivative,whichispositiveforalocalminimumandnegativeforalocalmaximum.Ifthesecondderivativeisnull,thecriticalpointisgenerallyaninflectionpoint,butmayalsobeanundulationpoint,whichmaybealocalminimumoralocalmaximum. Forafunctionofnvariables,thenumberofnegativeeigenvaluesoftheHessianmatrixatacriticalpointiscalledtheindexofthecriticalpoint.Anon-degeneratecriticalpointisalocalmaximumifandonlyiftheindexisn,or,equivalently,iftheHessianmatrixisnegativedefinite;itisalocalminimumiftheindexiszero,or,equivalently,iftheHessianmatrixispositivedefinite.Fortheothervaluesoftheindex,anon-degeneratecriticalpointisasaddlepoint,thatisapointwhichisamaximuminsomedirectionsandaminimuminothers. Applicationtooptimization[edit] Mainarticle:Mathematicaloptimization ByFermat'stheorem,alllocalmaximaandminimaofacontinuousfunctionoccuratcriticalpoints.Therefore,tofindthelocalmaximaandminimaofadifferentiablefunction,itsuffices,theoretically,tocomputethezerosofthegradientandtheeigenvaluesoftheHessianmatrixatthesezeros.Thisdoesnotworkwellinpracticebecauseitrequiresthesolutionofanonlinearsystemofsimultaneousequations,whichisadifficulttask.Theusualnumericalalgorithmsaremuchmoreefficientforfindinglocalextrema,butcannotcertifythatallextremahavebeenfound. Inparticular,inglobaloptimization,thesemethodscannotcertifythattheoutputisreallytheglobaloptimum. Whenthefunctiontominimizeisamultivariatepolynomial,thecriticalpointsandthecriticalvaluesaresolutionsofasystemofpolynomialequations,andmodernalgorithmsforsolvingsuchsystemsprovidecompetitivecertifiedmethodsforfindingtheglobalminimum. Criticalpointofadifferentiablemap[edit] GivenadifferentiablemapffromRmintoRn,thecriticalpointsoffarethepointsofRm,wheretherankoftheJacobianmatrixoffisnotmaximal.[5]Theimageofacriticalpointunderfisacalledacriticalvalue.Apointinthecomplementofthesetofcriticalvaluesiscalledaregularvalue.Sard'stheoremstatesthatthesetofcriticalvaluesofasmoothmaphasmeasurezero. Someauthors[6]giveaslightlydifferentdefinition:acriticalpointoffisapointofRmwheretherankoftheJacobianmatrixoffislessthann.Withthisconvention,allpointsarecriticalwhenm