Stationary point - Wikipedia

In mathematics, particularly in calculus, a stationary point of a differentiable function of one variable is a point on the graph of the function where the ... Stationarypoint FromWikipedia,thefreeencyclopedia Jumptonavigation Jumptosearch Thisarticleisaboutstationarypointsofareal-valueddifferentiablefunctionofonerealvariable.Forthebroaderterm,seeCriticalpoint(mathematics). Nottobeconfusedwithafixedpointwherex=f(x). Thisarticleincludesalistofgeneralreferences,butitlackssufficientcorrespondinginlinecitations.Pleasehelptoimprovethisarticlebyintroducingmoreprecisecitations.(March2016)(Learnhowandwhentoremovethistemplatemessage) Thestationarypointsaretheredcircles.Inthisgraph,theyareallrelativemaximaorrelativeminima.Thebluesquaresareinflectionpoints. Inmathematics,particularlyincalculus,astationarypointofadifferentiablefunctionofonevariableisapointonthegraphofthefunctionwherethefunction'sderivativeiszero.[1][2][3]Informally,itisapointwherethefunction"stops"increasingordecreasing(hencethename). Foradifferentiablefunctionofseveralrealvariables,astationarypointisapointonthesurfaceofthegraphwhereallitspartialderivativesarezero(equivalently,thegradientiszero). Stationarypointsareeasytovisualizeonthegraphofafunctionofonevariable:theycorrespondtothepointsonthegraphwherethetangentishorizontal(i.e.,paralleltothex-axis).Forafunctionoftwovariables,theycorrespondtothepointsonthegraphwherethetangentplaneisparalleltothexyplane. Contents 1Turningpoints 2Classification 3Curvesketching 3.1Example 4Seealso 5References 6Externallinks Turningpoints[edit] Aturningpointisapointatwhichthederivativechangessign.[2]Aturningpointmaybeeitherarelativemaximumorarelativeminimum(alsoknownaslocalminimumandmaximum).Ifthefunctionisdifferentiable,thenaturningpointisastationarypoint;howevernotallstationarypointsareturningpoints.Ifthefunctionistwicedifferentiable,thestationarypointsthatarenotturningpointsarehorizontalinflectionpoints.Forexample,thefunction x ↦ x 3 {\displaystylex\mapstox^{3}} hasastationarypointatx=0,whichisalsoaninflectionpoint,butisnotaturningpoint.[3] Classification[edit] Agraphinwhichlocalextremaandglobalextremahavebeenlabeled. Seealso:maximaandminima Isolatedstationarypointsofa C 1 {\displaystyleC^{1}} realvaluedfunction f : R → R {\displaystylef\colon\mathbb{R}\to\mathbb{R}} areclassifiedintofourkinds,bythefirstderivativetest: alocalminimum(minimalturningpointorrelativeminimum)isonewherethederivativeofthefunctionchangesfromnegativetopositive; alocalmaximum(maximalturningpointorrelativemaximum)isonewherethederivativeofthefunctionchangesfrompositivetonegative; Saddlepoints(stationarypointsthatareneitherlocalmaximanorminima:theyareinflectionpoints.Theleftisa"risingpointofinflection"(derivativeispositiveonbothsidesoftheredpoint);therightisa"fallingpointofinflection"(derivativeisnegativeonbothsidesoftheredpoint). arisingpointofinflection(orinflexion)isonewherethederivativeofthefunctionispositiveonbothsidesofthestationarypoint;suchapointmarksachangeinconcavity; afallingpointofinflection(orinflexion)isonewherethederivativeofthefunctionisnegativeonbothsidesofthestationarypoint;suchapointmarksachangeinconcavity. Thefirsttwooptionsarecollectivelyknownas"localextrema".Similarlyapointthatiseitheraglobal(orabsolute)maximumoraglobal(orabsolute)minimumiscalledaglobal(orabsolute)extremum.Thelasttwooptions—stationarypointsthatarenotlocalextremum—areknownassaddlepoints. ByFermat'stheorem,globalextremamustoccur(fora C 1 {\displaystyleC^{1}} function)ontheboundaryoratstationarypoints. Curvesketching[edit] Theroots,stationarypoints,inflectionpointandconcavityofacubicpolynomialx3−3x2−144x+432(blackline)anditsfirstandsecondderivatives(redandblue). Determiningthepositionandnatureofstationarypointsaidsincurvesketchingofdifferentiablefunctions.Solvingtheequationf'(x)=0returnsthex-coordinatesofallstationarypoints;they-coordinatesaretriviallythefunctionvaluesatthosex-coordinates. Thespecificnatureofastationarypointatxcaninsomecasesbedeterminedbyexaminingthesecondderivativef''(x): Iff''(x)<0,thestationarypointatxisconcavedown;amaximalextremum. Iff''(x)>0,thestationarypointatxisconcaveup;aminimalextremum. Iff''(x)=0,thenatureofthestationarypointmustbedeterminedbywayofothermeans,oftenbynotingasignchangearoundthatpoint. Amorestraightforwardwayofdeterminingthenatureofastationarypointisbyexaminingthefunctionvaluesbetweenthestationarypoints(ifthefunctionisdefinedandcontinuousbetweenthem). Asimpleexampleofapointofinflectionisthefunctionf(x)=x3.Thereisaclearchangeofconcavityaboutthepointx=0,andwecanprovethisbymeansofcalculus.Thesecondderivativeoffistheeverywhere-continuous6x,andatx=0,f′′=0,andthesignchangesaboutthispoint.Sox=0isapointofinflection. Moregenerally,thestationarypointsofarealvaluedfunction f : R n → R {\displaystylef\colon\mathbb{R}^{n}\to\mathbb{R}} arethose pointsx0wherethederivativeineverydirectionequalszero,orequivalently,thegradientiszero. Example[edit] Forthefunctionf(x)=x4wehavef'(0)=0andf''(0)=0.Eventhoughf''(0)=0,thispointisnotapointofinflection.Thereasonisthatthesignoff'(x)changesfromnegativetopositive. Forthefunctionf(x)=sin(x)wehavef'(0)≠0andf''(0)=0.Butthisisnotastationarypoint,ratheritisapointofinflection.Thisisbecausetheconcavitychangesfromconcavedownwardstoconcaveupwardsandthesignoff'(x)doesnotchange;itstayspositive. Forthefunctionf(x)=x3wehavef'(0)=0andf''(0)=0.Thisisbothastationarypointandapointofinflection.Thisisbecausetheconcavitychangesfromconcavedownwardstoconcaveupwardsandthesignoff'(x)doesnotchange;itstayspositive. Seealso[edit] Optimization(mathematics) Fermat'stheorem Derivativetest Fixedpoint(mathematics) Saddlepoint References[edit] ^Chiang,AlphaC.(1984).FundamentalMethodsofMathematicalEconomics(3rd ed.).NewYork:McGraw-Hill.p. 236.ISBN 0-07-010813-7. ^abSaddler,David;Shea,Julia;Ward,Derek(2011),"12BStationaryPointsandTurningPoints",Cambridge2UnitMathematicsYear11,CambridgeUniversityPress,p. 318,ISBN 9781107679573 ^ab"Turningpointsandstationarypoints".TCSFREEhighschoolmathematics'How-toLibrary'.Retrieved30October2011. Externallinks[edit] InflectionPointsofFourthDegreePolynomials—asurprisingappearanceofthegoldenratioatcut-the-knot vteCalculusPrecalculus Binomialtheorem Concavefunction Continuousfunction Factorial Finitedifference Freevariablesandboundvariables Graphofafunction Linearfunction Radian Rolle'stheorem Secant Slope Tangent Limits Indeterminateform Limitofafunction One-sidedlimit Limitofasequence Orderofapproximation (ε,δ)-definitionoflimit Differentialcalculus Derivative Secondderivative Partialderivative Differential Differentialoperator Meanvaluetheorem Notation Leibniz'snotation Newton'snotation Rulesofdifferentiation linearity Power Sum Chain L'Hôpital's Product GeneralLeibniz'srule Quotient Othertechniques Implicitdifferentiation Inversefunctionsanddifferentiation Logarithmicderivative Relatedrates Stationarypoints Firstderivativetest Secondderivativetest Extremevaluetheorem Maximaandminima Furtherapplications Newton'smethod Taylor'stheorem Differentialequation Ordinarydifferentialequation Partialdifferentialequation Stochasticdifferentialequation Integralcalculus Antiderivative Arclength Riemannintegral Basicproperties Constantofintegration Fundamentaltheoremofcalculus Differentiatingundertheintegralsign Integrationbyparts Integrationbysubstitution trigonometric Euler Weierstrass Partialfractionsinintegration Quadraticintegral Trapezoidalrule Volumes Washermethod Shellmethod Integralequation Integro-differentialequation Vectorcalculus Derivatives Curl Directionalderivative Divergence Gradient Laplacian Basictheorems Lineintegrals Green's Stokes' Gauss' Multivariablecalculus Divergencetheorem Geometric Hessianmatrix Jacobianmatrixanddeterminant Lagrangemultiplier Lineintegral Matrix Multipleintegral Partialderivative Surfaceintegral Volumeintegral Advancedtopics Differentialforms Exteriorderivative GeneralizedStokes'theorem Tensorcalculus Sequencesandseries Arithmetico–geometricsequence Typesofseries Alternating Binomial Fourier Geometric Harmonic Infinite Power Maclaurin Taylor Telescoping Testsofconvergence Abel's Alternatingseries Cauchycondensation Directcomparison Dirichlet's Integral Limitcomparison Ratio Root Term Specialfunctionsandnumbers Bernoullinumbers e(mathematicalconstant) Exponentialfunction Naturallogarithm Stirling'sapproximation Historyofcalculus Adequality BrookTaylor ColinMaclaurin Generalityofalgebra GottfriedWilhelmLeibniz Infinitesimal Infinitesimalcalculus IsaacNewton Fluxion LawofContinuity LeonhardEuler MethodofFluxions TheMethodofMechanicalTheorems Lists Differentiationrules Listofintegralsofexponentialfunctions Listofintegralsofhyperbolicfunctions Listofintegralsofinversehyperbolicfunctions Listofintegralsofinversetrigonometricfunctions Listofintegralsofirrationalfunctions Listofintegralsoflogarithmicfunctions Listofintegralsofrationalfunctions Listofintegralsoftrigonometricfunctions Secant Secantcubed Listoflimits Listsofintegrals Miscellaneoustopics Complexcalculus Contourintegral Differentialgeometry Manifold Curvature ofcurves ofsurfaces Tensor Euler–Maclaurinformula Gabriel'shorn IntegrationBee Proofthat22/7exceedsπ Regiomontanus'anglemaximizationproblem Steinmetzsolid Retrievedfrom"https://en.wikipedia.org/w/index.php?title=Stationary_point&oldid=1085098959" Categories:DifferentialcalculusHiddencategories:Articleslackingin-textcitationsfromMarch2016Allarticleslackingin-textcitations Navigationmenu Personaltools NotloggedinTalkContributionsCreateaccountLogin Namespaces ArticleTalk English Views ReadEditViewhistory More Search Navigation MainpageContentsCurrenteventsRandomarticleAboutWikipediaContactusDonate Contribute HelpLearntoeditCommunityportalRecentchangesUploadfile Tools WhatlinkshereRelatedchangesUploadfileSpecialpagesPermanentlinkPageinformationCitethispageWikidataitem Print/export DownloadasPDFPrintableversion Languages العربيةČeštinaEspañolEsperantoفارسیFrançaisBahasaIndonesiaNederlandsPolskiRomânăSlovenščinaதமிழ்TürkçeУкраїнськаاردو中文 Editlinks