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Trazado el gráfico de la función/ lambertw(x*sqrt(exp(-x^2)))/x

Gráfico de la función y = lambertw(x*sqrt(exp(-x^2)))/x

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
        /      ______\
        |     /    2 |
        |    /   -x  |
       W\x*\/   e    /
f(x) = ---------------
              x
f(x)=W(xe−x2)xf{\left(x \right)} = \frac{W\left(x \sqrt{e^{- x^{2}}}\right)}{x} 
f = LambertW(x*sqrt(exp(-x^2)))/x
Gráfico de la función
02468-8-6-4-2-1010-1010
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=0x_{1} = 0
Puntos de cruce con el eje de coordenadas X
El gráfico de la función cruce el eje X con f = 0
o sea hay que resolver la ecuación:
W(xe−x2)x=0\frac{W\left(x \sqrt{e^{- x^{2}}}\right)}{x} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=84.4909633958019x_{1} = 84.4909633958019
x2=52.637700071067x_{2} = 52.637700071067
x3=−38.513465780955x_{3} = -38.513465780955
x4=−100.196173603935x_{4} = -100.196173603935
x5=−70.279977507963x_{5} = -70.279977507963
x6=72.5308480554467x_{6} = 72.5308480554467
x7=−42.4650824372664x_{7} = -42.4650824372664
x8=−8.77374395986999x_{8} = -8.77374395986999
x9=−7.59424836185763x_{9} = -7.59424836185763
x10=98.4567018595091x_{10} = 98.4567018595091
x11=68.5472420039917x_{11} = 68.5472420039917
x12=−78.2513543828221x_{12} = -78.2513543828221
x13=−88.2228642387224x_{13} = -88.2228642387224
x14=54.6235015720464x_{14} = 54.6235015720464
x15=−15.353108656251x_{15} = -15.353108656251
x16=25.075319411531x_{16} = 25.075319411531
x17=58.5980034259874x_{17} = 58.5980034259874
x18=17.4609191452313x_{18} = 17.4609191452313
x19=100.452586119961x_{19} = 100.452586119961
x20=−54.3624686511977x_{20} = -54.3624686511977
x21=86.4853904712026x_{21} = 86.4853904712026
x22=−94.2086697842429x_{22} = -94.2086697842429
x23=50.6530165467517x_{23} = 50.6530165467517
x24=−80.2450889728205x_{24} = -80.2450889728205
x25=82.4968062154825x_{25} = 82.4968062154825
x26=19.3349347528308x_{26} = 19.3349347528308
x27=−92.213196234426x_{27} = -92.213196234426
x28=90.4749828917125x_{28} = 90.4749828917125
x29=−17.1926649157673x_{29} = -17.1926649157673
x30=92.4701163765988x_{30} = 92.4701163765988
x31=−20.9625982675926x_{31} = -20.9625982675926
x32=−32.6082664994078x_{32} = -32.6082664994078
x33=−68.2881796267158x_{33} = -68.2881796267158
x34=62.5757561905499x_{34} = 62.5757561905499
x35=38.7780171390331x_{35} = 38.7780171390331
x36=13.821602667061x_{36} = 13.821602667061
x37=76.5161651178913x_{37} = 76.5161651178913
x38=88.4800691857377x_{38} = 88.4800691857377
x39=32.8746479943924x_{39} = 32.8746479943924
x40=−60.3264178164522x_{40} = -60.3264178164522
x41=10.4573998233968x_{41} = 10.4573998233968
x42=−76.2579480048004x_{42} = -76.2579480048004
x43=−56.3496007818346x_{43} = -56.3496007818346
x44=−24.8063277911585x_{44} = -24.8063277911585
x45=−40.4880833510938x_{45} = -40.4880833510938
x46=7.75276413624176x_{46} = 7.75276413624176
x47=94.465455711809x_{47} = 94.465455711809
x48=46.6875743496688x_{48} = 46.6875743496688
x49=−66.2968757729049x_{49} = -66.2968757729049
x50=−84.2334495346616x_{50} = -84.2334495346616
x51=23.1470746269724x_{51} = 23.1470746269724
x52=−26.7456937966605x_{52} = -26.7456937966605
x53=−30.6481072194899x_{53} = -30.6481072194899
x54=70.5388130254423x_{54} = 70.5388130254423
x55=−28.6935014520304x_{55} = -28.6935014520304
x56=−50.3912611159759x_{56} = -50.3912611159759
x57=−86.2280342650635x_{57} = -86.2280342650635
x58=60.5865134772897x_{58} = 60.5865134772897
x59=48.6695879232527x_{59} = 48.6695879232527
x60=15.6187887420463x_{60} = 15.6187887420463
x61=42.7285812456194x_{61} = 42.7285812456194
x62=−64.3061118645014x_{62} = -64.3061118645014
x63=80.502938993158x_{63} = 80.502938993158
x64=−98.2001694780952x_{64} = -98.2001694780952
x65=−10.2223744328282x_{65} = -10.2223744328282
x66=−22.8776155997798x_{66} = -22.8776155997798
x67=−72.272228553208x_{67} = -72.272228553208
x68=30.915155296142x_{68} = 30.915155296142
x69=−13.5609611596815x_{69} = -13.5609611596815
x70=40.7520918320371x_{70} = 40.7520918320371
x71=56.6103032962327x_{71} = 56.6103032962327
x72=28.9612268639749x_{72} = 28.9612268639749
x73=−74.2648962776587x_{73} = -74.2648962776587
x74=−36.5416186428031x_{74} = -36.5416186428031
x75=21.2322589953881x_{75} = 21.2322589953881
x76=−52.3763167396819x_{76} = -52.3763167396819
x77=−62.3159396788963x_{77} = -62.3159396788963
x78=−90.2179231835785x_{78} = -90.2179231835785
x79=36.8067470674045x_{79} = 36.8067470674045
x80=−48.407436882187x_{80} = -48.407436882187
x81=44.7071643259838x_{81} = 44.7071643259838
x82=−11.8385348559139x_{82} = -11.8385348559139
x83=66.5561766635487x_{83} = 66.5561766635487
x84=−19.0655550235483x_{84} = -19.0655550235483
x85=74.5233098703306x_{85} = 74.5233098703306
x86=8.98020794628996x_{86} = 8.98020794628996
x87=−82.2391278966615x_{87} = -82.2391278966615
x88=27.0140819592206x_{88} = 27.0140819592206
x89=−44.4441436501962x_{89} = -44.4441436501962
x90=−34.5730204138554x_{90} = -34.5730204138554
x91=12.0899461362664x_{91} = 12.0899461362664
x92=−46.4250022403361x_{92} = -46.4250022403361
x93=64.5656638089125x_{93} = 64.5656638089125
x94=34.8387597741051x_{94} = 34.8387597741051
x95=−58.3376128604543x_{95} = -58.3376128604543
x96=−96.2043313539657x_{96} = -96.2043313539657
x97=78.5093838275879x_{97} = 78.5093838275879
x98=96.4609881201779x_{98} = 96.4609881201779
Puntos de cruce con el eje de coordenadas Y
El gráfico cruce el eje Y cuando x es igual a 0:
sustituimos x = 0 en LambertW(x*sqrt(exp(-x^2)))/x.
W(0e−02)0\frac{W\left(0 \sqrt{e^{- 0^{2}}}\right)}{0}
Resultado:
f(0)=NaNf{\left(0 \right)} = \text{NaN}
- no hay soluciones de la ecuación
Extremos de la función
Para hallar los extremos hay que resolver la ecuación
ddxf(x)=0\frac{d}{d x} f{\left(x \right)} = 0
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
ddxf(x)=\frac{d}{d x} f{\left(x \right)} =
primera derivada
(−x2e−x22+e−x2)ex22W(xe−x2)x2(W(xe−x2)+1)−W(xe−x2)x2=0\frac{\left(- x^{2} e^{- \frac{x^{2}}{2}} + \sqrt{e^{- x^{2}}}\right) e^{\frac{x^{2}}{2}} W\left(x \sqrt{e^{- x^{2}}}\right)}{x^{2} \left(W\left(x \sqrt{e^{- x^{2}}}\right) + 1\right)} - \frac{W\left(x \sqrt{e^{- x^{2}}}\right)}{x^{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=68.5473054354995x_{1} = 68.5473054354995
x2=−74.2649444149412x_{2} = -74.2649444149412
x3=98.4567232107033x_{3} = 98.4567232107033
x4=72.5309015703522x_{4} = 72.5309015703522
x5=−76.2579924560991x_{5} = -76.2579924560991
x6=−28.6943596043799x_{6} = -28.6943596043799
x7=15.6247818795334x_{7} = 15.6247818795334
x8=−96.2043534607085x_{8} = -96.2043534607085
x9=−24.8076741592432x_{9} = -24.8076741592432
x10=58.5981051512238x_{10} = 58.5981051512238
x11=−34.5735053916839x_{11} = -34.5735053916839
x12=−92.2132213430942x_{12} = -92.2132213430942
x13=−84.2334824940775x_{13} = -84.2334824940775
x14=34.8392502079753x_{14} = 34.8392502079753
x15=52.6378406446428x_{15} = 52.6378406446428
x16=−15.3593372555059x_{16} = -15.3593372555059
x17=25.0766626196802x_{17} = 25.0766626196802
x18=44.7073945322353x_{18} = 44.7073945322353
x19=27.0151495573047x_{19} = 27.0151495573047
x20=10.4810551180426x_{20} = 10.4810551180426
x21=−20.9648808652053x_{21} = -20.9648808652053
x22=90.4750104191475x_{22} = 90.4750104191475
x23=−17.1969684851483x_{23} = -17.1969684851483
x24=−86.228064985273x_{24} = -86.228064985273
x25=−36.5420283825472x_{25} = -36.5420283825472
x26=23.1487959086869x_{26} = 23.1487959086869
x27=−70.2800343339193x_{27} = -70.2800343339193
x28=−22.8793491121891x_{28} = -22.8793491121891
x29=−50.3914159606663x_{29} = -50.3914159606663
x30=−13.5703851876112x_{30} = -13.5703851876112
x31=−98.2001902620909x_{31} = -98.2001902620909
x32=−7.68768138260503x_{32} = -7.68768138260503
x33=21.2345123835376x_{33} = 21.2345123835376
x34=−90.2179499984874x_{34} = -90.2179499984874
x35=−46.4252005910588x_{35} = -46.4252005910588
x36=50.6531744084895x_{36} = 50.6531744084895
x37=−10.247960314217x_{37} = -10.247960314217
x38=−32.6088462722306x_{38} = -32.6088462722306
x39=−68.2882415884721x_{39} = -68.2882415884721
x40=88.4800986212492x_{40} = 88.4800986212492
x41=−88.2228929179074x_{41} = -88.2228929179074
x42=−40.4883834667848x_{42} = -40.4883834667848
x43=94.4654798903143x_{43} = 94.4654798903143
x44=19.3379596244699x_{44} = 19.3379596244699
x45=74.5233591950223x_{45} = 74.5233591950223
x46=84.4909972102885x_{46} = 84.4909972102885
x47=−44.4443699549213x_{47} = -44.4443699549213
x48=42.7288452402976x_{48} = 42.7288452402976
x49=13.8305646213614x_{49} = 13.8305646213614
x50=−26.7467597908737x_{50} = -26.7467597908737
x51=−100.196193168581x_{51} = -100.196193168581
x52=36.8071620516744x_{52} = 36.8071620516744
x53=−8.82065711418988x_{53} = -8.82065711418988
x54=100.45260622169x_{54} = 100.45260622169
x55=−62.3160212977879x_{55} = -62.3160212977879
x56=−82.2391633180928x_{56} = -82.2391633180928
x57=12.1040591364533x_{57} = 12.1040591364533
x58=60.5866054706585x_{58} = 60.5866054706585
x59=−80.2451271074651x_{59} = -80.2451271074651
x60=62.5758396530863x_{60} = 62.5758396530863
x61=96.4610108264976x_{61} = 96.4610108264976
x62=−64.3061861108853x_{62} = -64.3061861108853
x63=92.4701421572582x_{63} = 92.4701421572582
x64=−58.3377124249816x_{64} = -58.3377124249816
x65=70.5388712182941x_{65} = 70.5388712182941
x66=−52.3764545425553x_{66} = -52.3764545425553
x67=−94.2086933283192x_{67} = -94.2086933283192
x68=82.4968425477036x_{68} = 82.4968425477036
x69=48.6697660202548x_{69} = 48.6697660202548
x70=−56.3497113142373x_{70} = -56.3497113142373
x71=−30.6488081209205x_{71} = -30.6488081209205
x72=−60.3265078154327x_{72} = -60.3265078154327
x73=−48.4076116917235x_{73} = -48.4076116917235
x74=28.9620890166341x_{74} = 28.9620890166341
x75=56.6104161729572x_{75} = 56.6104161729572
x76=−54.362591820007x_{76} = -54.362591820007
x77=54.623627287895x_{77} = 54.623627287895
x78=7.83686490645478x_{78} = 7.83686490645478
x79=−11.8535751456397x_{79} = -11.8535751456397
x80=78.5094259967043x_{80} = 78.5094259967043
x81=9.02289695499675x_{81} = 9.02289695499675
x82=76.5162106782208x_{82} = 76.5162106782208
x83=40.7523965392307x_{83} = 40.7523965392307
x84=17.465102009593x_{84} = 17.465102009593
x85=66.5562459808845x_{85} = 66.5562459808845
x86=−19.0686408223086x_{86} = -19.0686408223086
x87=−72.2722807950237x_{87} = -72.2722807950237
x88=30.9158612891769x_{88} = 30.9158612891769
x89=64.5657397621909x_{89} = 64.5657397621909
x90=32.8752332341382x_{90} = 32.8752332341382
x91=80.5029780991381x_{91} = 80.5029780991381
x92=86.4854219951314x_{92} = 86.4854219951314
x93=−78.2513955147154x_{93} = -78.2513955147154
x94=46.6877762829739x_{94} = 46.6877762829739
x95=−38.5138150497611x_{95} = -38.5138150497611
x96=−42.4653421912683x_{96} = -42.4653421912683
x97=−66.2969435075443x_{97} = -66.2969435075443
x98=38.7783713446935x_{98} = 38.7783713446935
Signos de extremos en los puntos:
(68.54730543549952, 4.82031227394403e-1021)

(-74.26494441494121, 2.35362555752321e-1198)

(98.45672321070329, 1.08160717280565e-2105)

(72.5309015703522, 4.43230599003598e-1143)

(-76.25799245609907, 1.68921805793219e-1263)

(-28.69435960437989, 1.61542017513745e-179)

(15.624781879533398, 9.70547957044729e-54)

(-96.2043534607085, 1.74582028324948e-2010)

(-24.807674159243177, 2.30724429290355e-134)

(58.598105151223834, 2.36201653131843e-746)

(-34.57350539168385, 2.74144908586576e-260)

(-92.21322134309419, 3.44034420083174e-1847)

(-84.23348249407753, 1.9037807240392e-1541)

(34.83925020797529, 2.70685386071025e-264)

(52.637840644642765, 2.19260890801306e-602)

(-15.359337255505897, 5.92869648316435e-52)

(25.07666261968023, 2.81413144742492e-137)

(44.70739453223533, 9.47780284440881e-435)

(27.01514955730468, 3.3263545821833e-159)

(10.481055118042562, 1.39903685357388e-24)

(-20.964880865205323, 3.61486344845391e-96)

(90.47501041914748, 3.10352999273789e-1778)

(-17.196968485148272, 6.05066694628083e-65)

(-86.22806498527301, 2.81603434092229e-1615)

(-36.542028382547215, 1.09390932081865e-290)

(23.1487959086869, 4.34523511677173e-117)

(-70.28003433391933, 2.80721192749264e-1073)

(-22.879349112189093, 2.1433986053126e-114)

(-50.39141596066627, 3.97319307435429e-552)

(-13.57038518761119, 1.02606043620113e-40)

(-98.20019026209087, 9.74823328548446e-2095)

(-7.687681382605025, 1.46724088582559e-13)

(21.23451238353765, 1.2227238452565e-98)

(-90.21794999848743, 3.78555856260199e-1768)

(-46.425200591058825, 9.60980233587364e-469)

(50.65317440848955, 7.17385424830634e-558)

(-10.247960314216956, 1.566919888556e-23)

(-32.60884627223058, 1.25700119293546e-231)

(-68.28824158847213, 2.40302793130379e-1013)

(88.48009862124916, 1.0312091820997e-1700)

(-88.22289291790742, 7.62913789956981e-1691)

(-40.48838346678479, 1.06790120243364e-356)

(94.46547989031433, 1.72713638589383e-1938)

(19.337959624469928, 6.2565968724807e-82)

(74.52335919502231, 1.05346980859634e-1206)

(84.49099721028851, 6.99485192550893e-1551)

(-44.44436995492126, 1.1711598112743e-429)

(42.72884524029763, 3.48746159180186e-397)

(13.830564621361363, 2.90465619215689e-42)

(-26.74675979087374, 4.52082659317584e-156)

(-100.19619316858123, 9.96944666258599e-2181)

(36.80716205167437, 6.54704160419386e-295)

(-8.82065711418988, 1.27373129166641e-17)

(100.45260622168976, 6.70917154753397e-2192)

(-62.31602129778792, 5.68916389122711e-844)

(-82.23916331809285, 2.35728369221517e-1469)

(12.10405913645333, 1.5351056199082e-32)

(60.586605470658476, 8.12156590184954e-798)

(-80.24512710746508, 5.34591571744502e-1399)

(62.575839653086256, 5.11439950432004e-851)

(96.46101082649756, 3.19366163385579e-2021)

(-64.30618611088535, 1.08180121625321e-898)

(92.47014215725818, 1.71073379469617e-1857)

(-58.337712424981596, 9.66608459956877e-740)

(70.53887121829413, 3.41544964166352e-1081)

(-52.37645454255525, 2.00214719120909e-596)

(-94.20869332831917, 5.72653123253108e-1928)

(82.49684254770361, 1.42796774324628e-1478)

(48.66976602025484, 4.29847650499965e-515)

(-56.349711314237275, 3.12280478315665e-690)

(-30.648808120920528, 1.05404943814838e-204)

(-60.326507815432706, 5.47962109630034e-791)

(-48.407611691723474, 1.44394288393636e-509)

(28.962089016634078, 7.1833935752502e-183)

(56.610416172957216, 1.25811565057559e-696)

(-54.362591820007005, 1.84768777834024e-642)

(54.623627287895, 1.22729335061317e-648)

(7.836864906454783, 4.60879078917943e-14)

(-11.853575145639718, 3.08489004883961e-31)

(78.50942599670425, 3.65630967428387e-1339)

(9.022896954996751, 2.09634833633054e-18)

(76.51621067822079, 4.58591665963617e-1272)

(40.75239653923073, 2.34974479761189e-361)

(17.46510200959298, 5.80260600504341e-67)

(66.55624598088446, 1.24597695657418e-962)

(-19.06864082230857, 1.10262195767039e-79)

(-72.2722807950237, 6.00621574723043e-1135)

(30.915861289176938, 2.83618465308492e-208)

(64.5657397621909, 5.89862181605406e-906)

(32.8752332341382, 2.04831346410163e-235)

(80.50297809913809, 5.33916631748338e-1408)

(86.48542199513139, 6.27558857446106e-1625)

(-78.25139551471544, 2.22048065090639e-1330)

(46.68777628297386, 4.71668845417238e-474)

(-38.51381504976113, 7.98855740547038e-323)

(-42.46534219126834, 2.6135586050952e-392)

(-66.2969435075443, 3.76747338849005e-955)

(38.77837134469346, 2.89876080074486e-327)


Intervalos de crecimiento y decrecimiento de la función:
Hallemos los intervalos donde la función crece y decrece y también los puntos mínimos y máximos de la función, para lo cual miramos cómo se comporta la función en los extremos con desviación mínima del extremo:
La función no tiene puntos mínimos
La función no tiene puntos máximos
Decrece en todo el eje numérico
Puntos de flexiones
Hallemos los puntos de flexiones, para eso hay que resolver la ecuación
d2dx2f(x)=0\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0
(la segunda derivada es igual a cero),
las raíces de la ecuación obtenida serán los puntos de flexión para el gráfico de la función indicado:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
segunda derivada
(−2−(x2−1)2x2(W(xe−x22)+1)+(x2−1)2W(xe−x22)x2(W(xe−x22)+1)2−x2−1x2W(xe−x22)+1+2(x2−1)x2(W(xe−x22)+1)+2x2)W(xe−x22)x=0\frac{\left(- \frac{2 - \frac{\left(x^{2} - 1\right)^{2}}{x^{2} \left(W\left(x e^{- \frac{x^{2}}{2}}\right) + 1\right)} + \frac{\left(x^{2} - 1\right)^{2} W\left(x e^{- \frac{x^{2}}{2}}\right)}{x^{2} \left(W\left(x e^{- \frac{x^{2}}{2}}\right) + 1\right)^{2}} - \frac{x^{2} - 1}{x^{2}}}{W\left(x e^{- \frac{x^{2}}{2}}\right) + 1} + \frac{2 \left(x^{2} - 1\right)}{x^{2} \left(W\left(x e^{- \frac{x^{2}}{2}}\right) + 1\right)} + \frac{2}{x^{2}}\right) W\left(x e^{- \frac{x^{2}}{2}}\right)}{x} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=52.6379814218934x_{1} = 52.6379814218934
x2=−84.2335154720999x_{2} = -84.2335154720999
x3=40.752701984963x_{3} = 40.752701984963
x4=19.3410179745966x_{4} = 19.3410179745966
x5=17.469342088328x_{5} = 17.469342088328
x6=74.5234085553027x_{6} = 74.5234085553027
x7=−20.9671849932147x_{7} = -20.9671849932147
x8=−42.465602524913x_{8} = -42.465602524913
x9=−58.3378121068688x_{9} = -58.3378121068688
x10=13.8397273670271x_{10} = 13.8397273670271
x11=−88.2229216118497x_{11} = -88.2229216118497
x12=−74.2649925871971x_{12} = -74.2649925871971
x13=−90.2179768265901x_{13} = -90.2179768265901
x14=−94.2087168830181x_{14} = -94.2087168830181
x15=−13.5800309789521x_{15} = -13.5800309789521
x16=−8.87064342038439x_{16} = -8.87064342038439
x17=50.6533325173004x_{17} = 50.6533325173004
x18=100.452626331394x_{18} = 100.452626331394
x19=8.30954292526043x_{19} = 8.30954292526043
x20=−17.2013333747715x_{20} = -17.2013333747715
x21=−92.2132464635871x_{21} = -92.2132464635871
x22=−64.3062604292563x_{22} = -64.3062604292563
x23=15.6308785720717x_{23} = 15.6308785720717
x24=60.5866975645439x_{24} = 60.5866975645439
x25=−11.8690912203368x_{25} = -11.8690912203368
x26=68.5473689211199x_{26} = 68.5473689211199
x27=54.6237531728431x_{27} = 54.6237531728431
x28=94.4655040796695x_{28} = 94.4655040796695
x29=25.0780145396787x_{29} = 25.0780145396787
x30=86.4854535359408x_{30} = 86.4854535359408
x31=−54.3627151560868x_{31} = -54.3627151560868
x32=32.8758206624564x_{32} = 32.8758206624564
x33=98.4567445707167x_{33} = 98.4567445707167
x34=76.51625626973x_{34} = 76.51625626973
x35=−40.4886843197642x_{35} = -40.4886843197642
x36=−30.6495120504912x_{36} = -30.6495120504912
x37=−72.2723330769207x_{37} = -72.2723330769207
x38=−60.3265979135992x_{38} = -60.3265979135992
x39=−34.5739920094063x_{39} = -34.5739920094063
x40=38.7787264992114x_{40} = 38.7787264992114
x41=30.9165702724382x_{41} = 30.9165702724382
x42=−96.2043755770155x_{42} = -96.2043755770155
x43=58.5982069953047x_{43} = 58.5982069953047
x44=80.5030172292918x_{44} = 80.5030172292918
x45=96.4610335425887x_{45} = 96.4610335425887
x46=48.6699444192874x_{46} = 48.6699444192874
x47=−68.288303603488x_{47} = -68.288303603488
x48=−56.3498219863142x_{48} = -56.3498219863142
x49=−98.2002110547164x_{49} = -98.2002110547164
x50=34.8397422727672x_{50} = 34.8397422727672
x51=66.5563153609531x_{51} = 66.5563153609531
x52=36.8075782709927x_{52} = 36.8075782709927
x53=28.9629553384854x_{53} = 28.9629553384854
x54=90.4750379600501x_{54} = 90.4750379600501
x55=−10.2746784818947x_{55} = -10.2746784818947
x56=−24.8090294977817x_{56} = -24.8090294977817
x57=23.1505303409965x_{57} = 23.1505303409965
x58=27.0162231035494x_{58} = 27.0162231035494
x59=−44.4445967203753x_{59} = -44.4445967203753
x60=42.7291098166418x_{60} = 42.7291098166418
x61=−70.2800912059859x_{61} = -70.2800912059859
x62=88.4801280718193x_{62} = 88.4801280718193
x63=−28.6952219982488x_{63} = -28.6952219982488
x64=−15.3656784163536x_{64} = -15.3656784163536
x65=9.06819578916296x_{65} = 9.06819578916296
x66=−82.2391987605035x_{66} = -82.2391987605035
x67=−32.6094282528384x_{67} = -32.6094282528384
x68=−80.2451652658341x_{68} = -80.2451652658341
x69=−22.8810962714784x_{69} = -22.8810962714784
x70=−100.196212741031x_{70} = -100.196212741031
x71=72.5309551260244x_{71} = 72.5309551260244
x72=−26.7478318697802x_{72} = -26.7478318697802
x73=12.1185952351011x_{73} = 12.1185952351011
x74=−66.2970113039634x_{74} = -66.2970113039634
x75=−48.4077868009633x_{75} = -48.4077868009633
x76=64.5658157885216x_{76} = 64.5658157885216
x77=−50.3915710502459x_{77} = -50.3915710502459
x78=82.4968789013096x_{78} = 82.4968789013096
x79=−19.0717620613987x_{79} = -19.0717620613987
x80=−46.4253993116754x_{80} = -46.4253993116754
x81=62.5759232010974x_{81} = 62.5759232010974
x82=44.7076252015549x_{82} = 44.7076252015549
x83=70.5389294580204x_{83} = 70.5389294580204
x84=92.4701679499916x_{84} = 92.4701679499916
x85=−86.2280957220308x_{85} = -86.2280957220308
x86=−62.3161030009634x_{86} = -62.3161030009634
x87=10.5056967958724x_{87} = 10.5056967958724
x88=−36.5424393607068x_{88} = -36.5424393607068
x89=−78.2514366735206x_{89} = -78.2514366735206
x90=46.687978588574x_{90} = 46.687978588574
x91=56.6105291910076x_{91} = 56.6105291910076
x92=−52.376592547093x_{92} = -52.376592547093
x93=−38.5141652677611x_{93} = -38.5141652677611
x94=78.5094681932305x_{94} = 78.5094681932305
x95=−76.2580369380243x_{95} = -76.2580369380243
x96=21.2367863293909x_{96} = 21.2367863293909
x97=84.4910310437484x_{97} = 84.4910310437484
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
x1=0x_{1} = 0

lim⁡x→0−((−2−(x2−1)2x2(W(xe−x22)+1)+(x2−1)2W(xe−x22)x2(W(xe−x22)+1)2−x2−1x2W(xe−x22)+1+2(x2−1)x2(W(xe−x22)+1)+2x2)W(xe−x22)x)=2\lim_{x \to 0^-}\left(\frac{\left(- \frac{2 - \frac{\left(x^{2} - 1\right)^{2}}{x^{2} \left(W\left(x e^{- \frac{x^{2}}{2}}\right) + 1\right)} + \frac{\left(x^{2} - 1\right)^{2} W\left(x e^{- \frac{x^{2}}{2}}\right)}{x^{2} \left(W\left(x e^{- \frac{x^{2}}{2}}\right) + 1\right)^{2}} - \frac{x^{2} - 1}{x^{2}}}{W\left(x e^{- \frac{x^{2}}{2}}\right) + 1} + \frac{2 \left(x^{2} - 1\right)}{x^{2} \left(W\left(x e^{- \frac{x^{2}}{2}}\right) + 1\right)} + \frac{2}{x^{2}}\right) W\left(x e^{- \frac{x^{2}}{2}}\right)}{x}\right) = 2
lim⁡x→0+((−2−(x2−1)2x2(W(xe−x22)+1)+(x2−1)2W(xe−x22)x2(W(xe−x22)+1)2−x2−1x2W(xe−x22)+1+2(x2−1)x2(W(xe−x22)+1)+2x2)W(xe−x22)x)=2\lim_{x \to 0^+}\left(\frac{\left(- \frac{2 - \frac{\left(x^{2} - 1\right)^{2}}{x^{2} \left(W\left(x e^{- \frac{x^{2}}{2}}\right) + 1\right)} + \frac{\left(x^{2} - 1\right)^{2} W\left(x e^{- \frac{x^{2}}{2}}\right)}{x^{2} \left(W\left(x e^{- \frac{x^{2}}{2}}\right) + 1\right)^{2}} - \frac{x^{2} - 1}{x^{2}}}{W\left(x e^{- \frac{x^{2}}{2}}\right) + 1} + \frac{2 \left(x^{2} - 1\right)}{x^{2} \left(W\left(x e^{- \frac{x^{2}}{2}}\right) + 1\right)} + \frac{2}{x^{2}}\right) W\left(x e^{- \frac{x^{2}}{2}}\right)}{x}\right) = 2
- los límites son iguales, es decir omitimos el punto correspondiente

Intervalos de convexidad y concavidad:
Hallemos los intervales donde la función es convexa o cóncava, para eso veamos cómo se comporta la función en los puntos de flexiones:
No tiene corvaduras en todo el eje numérico
Asíntotas verticales
Hay:
x1=0x_{1} = 0
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(W(xe−x2)x)=0\lim_{x \to -\infty}\left(\frac{W\left(x \sqrt{e^{- x^{2}}}\right)}{x}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞(W(xe−x2)x)=0\lim_{x \to \infty}\left(\frac{W\left(x \sqrt{e^{- x^{2}}}\right)}{x}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=0y = 0
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función LambertW(x*sqrt(exp(-x^2)))/x, dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(W(xe−x2)x2)=0\lim_{x \to -\infty}\left(\frac{W\left(x \sqrt{e^{- x^{2}}}\right)}{x^{2}}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(W(xe−x2)x2)=0\lim_{x \to \infty}\left(\frac{W\left(x \sqrt{e^{- x^{2}}}\right)}{x^{2}}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
Paridad e imparidad de la función
Comprobemos si la función es par o impar mediante las relaciones f = f(-x) и f = -f(-x).
Pues, comprobamos:
W(xe−x2)x=−W(−xe−x22)x\frac{W\left(x \sqrt{e^{- x^{2}}}\right)}{x} = - \frac{W\left(- x e^{- \frac{x^{2}}{2}}\right)}{x}
- No
W(xe−x2)x=W(−xe−x22)x\frac{W\left(x \sqrt{e^{- x^{2}}}\right)}{x} = \frac{W\left(- x e^{- \frac{x^{2}}{2}}\right)}{x}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор