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Trazado el gráfico de la función/ (x+3*log(x))*exp(-x)

Gráfico de la función y = (x+3*log(x))*exp(-x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
                       -x
f(x) = (x + 3*log(x))*e
f(x)=(x+3log⁡(x))e−xf{\left(x \right)} = \left(x + 3 \log{\left(x \right)}\right) e^{- x} 
f = (x + 3*log(x))*exp(-x)
Gráfico de la función
02468-8-6-4-2-1010-1010
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:
(x+3log⁡(x))e−x=0\left(x + 3 \log{\left(x \right)}\right) e^{- x} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=3W(13)x_{1} = 3 W\left(\frac{1}{3}\right)
Solución numérica
x1=35.9456429112861x_{1} = 35.9456429112861
x2=95.4081677497676x_{2} = 95.4081677497676
x3=47.6884044661701x_{3} = 47.6884044661701
x4=67.5137701123205x_{4} = 67.5137701123205
x5=91.4188409638624x_{5} = 91.4188409638624
x6=71.4928046690434x_{6} = 71.4928046690434
x7=101.393882064671x_{7} = 101.393882064671
x8=117.363456626696x_{8} = 117.363456626696
x9=85.4369421508359x_{9} = 85.4369421508359
x10=87.4306005687442x_{10} = 87.4306005687442
x11=111.37376424255x_{11} = 111.37376424255
x12=51.6392649464172x_{12} = 51.6392649464172
x13=61.5514152075878x_{13} = 61.5514152075878
x14=93.4133783695243x_{14} = 93.4133783695243
x15=89.4245746511602x_{15} = 89.4245746511602
x16=0.77288295914921x_{16} = 0.77288295914921
x17=109.377470458736x_{17} = 109.377470458736
x18=37.8837629238526x_{18} = 37.8837629238526
x19=83.443625454375x_{19} = 83.443625454375
x20=41.7883407748457x_{20} = 41.7883407748457
x21=57.5819356999676x_{21} = 57.5819356999676
x22=79.4581369399788x_{22} = 79.4581369399788
x23=53.6183053390345x_{23} = 53.6183053390345
x24=113.370198388513x_{24} = 113.370198388513
x25=107.381325625307x_{25} = 107.381325625307
x26=55.5992846391501x_{26} = 55.5992846391501
x27=45.7175240594065x_{27} = 45.7175240594065
x28=39.8321733818371x_{28} = 39.8321733818371
x29=121.357188500642x_{29} = 121.357188500642
x30=119.360266576691x_{30} = 119.360266576691
x31=65.5253986737185x_{31} = 65.5253986737185
x32=49.6624935079526x_{32} = 49.6624935079526
x33=75.4744124088742x_{33} = 75.4744124088742
x34=105.385339055191x_{34} = 105.385339055191
x35=32.1184864193087x_{35} = 32.1184864193087
x36=115.366764963302x_{36} = 115.366764963302
x37=97.4031917670114x_{37} = 97.4031917670114
x38=81.4506795427504x_{38} = 81.4506795427504
x39=103.389520864996x_{39} = 103.389520864996
x40=73.4833182521218x_{40} = 73.4833182521218
x41=99.3984346595711x_{41} = 99.3984346595711
x42=69.5029321480043x_{42} = 69.5029321480043
x43=34.0217235797127x_{43} = 34.0217235797127
x44=63.5379108248778x_{44} = 63.5379108248778
x45=43.7505346636486x_{45} = 43.7505346636486
x46=59.5660396447811x_{46} = 59.5660396447811
x47=77.4660341711108x_{47} = 77.4660341711108
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 (x + 3*log(x))*exp(-x).
e−0⋅3log⁡(0)e^{- 0} \cdot 3 \log{\left(0 \right)}
Resultado:
f(0)=∞~f{\left(0 \right)} = \tilde{\infty}
signof no cruza Y
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
(1+3x)e−x−(x+3log⁡(x))e−x=0\left(1 + \frac{3}{x}\right) e^{- x} - \left(x + 3 \log{\left(x \right)}\right) e^{- x} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=35.9811834917543x_{1} = 35.9811834917543
x2=77.4701558559905x_{2} = 77.4701558559905
x3=65.5315276358455x_{3} = 65.5315276358455
x4=75.4787910959706x_{4} = 75.4787910959706
x5=1.56699936884542x_{5} = 1.56699936884542
x6=69.5082499196375x_{6} = 69.5082499196375
x7=49.6750385184589x_{7} = 49.6750385184589
x8=71.4977774891721x_{8} = 71.4977774891721
x9=34.0663098460343x_{9} = 34.0663098460343
x10=79.4620241027007x_{10} = 79.4620241027007
x11=117.365094915664x_{11} = 117.365094915664
x12=97.4056498837243x_{12} = 97.4056498837243
x13=61.5585644340847x_{13} = 61.5585644340847
x14=113.371963487868x_{14} = 113.371963487868
x15=89.4275473608043x_{15} = 89.4275473608043
x16=99.4007853548267x_{16} = 99.4007853548267
x17=73.4879794581974x_{17} = 73.4879794581974
x18=63.5445194849918x_{18} = 63.5445194849918
x19=109.37937792943x_{19} = 109.37937792943
x20=67.519471235957x_{20} = 67.519471235957
x21=53.6284943925703x_{21} = 53.6284943925703
x22=55.608546448343x_{22} = 55.608546448343
x23=105.387407149144x_{23} = 105.387407149144
x24=81.4543520716443x_{24} = 81.4543520716443
x25=51.6505357161093x_{25} = 51.6505357161093
x26=57.5903958475771x_{26} = 57.5903958475771
x27=45.7334184240217x_{27} = 45.7334184240217
x28=115.368464860384x_{28} = 115.368464860384
x29=103.391677119764x_{29} = 103.391677119764
x30=43.768672819252x_{30} = 43.768672819252
x31=87.4337277715938x_{31} = 87.4337277715938
x32=107.383310935749x_{32} = 107.383310935749
x33=59.5738014177174x_{33} = 59.5738014177174
x34=37.9129492508678x_{34} = 37.9129492508678
x35=41.8092780509524x_{35} = 41.8092780509524
x36=95.4107409664157x_{36} = 95.4107409664157
x37=39.856680385664x_{37} = 39.856680385664
x38=121.358713331272x_{38} = 121.358713331272
x39=85.4402364888013x_{39} = 85.4402364888013
x40=119.361846589643x_{40} = 119.361846589643
x41=91.4216705602647x_{41} = 91.4216705602647
x42=32.1768362765376x_{42} = 32.1768362765376
x43=93.4160751261564x_{43} = 93.4160751261564
x44=101.396132340273x_{44} = 101.396132340273
x45=111.375598426946x_{45} = 111.375598426946
x46=83.4471010141094x_{46} = 83.4471010141094
x47=47.7024673187512x_{47} = 47.7024673187512
Signos de extremos en los puntos:
(35.98118349175432, 1.10450563362969e-14)

(77.47015585599048, 2.05060753705144e-32)

(65.53152763584546, 2.70741029878888e-27)

(75.47879109597062, 1.46783440634749e-31)

(1.5669993688454187, 0.608167140025688)

(69.50824991963749, 5.34557666485822e-29)

(49.67503851845891, 1.63875283032658e-20)

(71.49777748917214, 7.4950106234455e-30)

(34.0663098460343, 7.16181512569531e-14)

(79.46202410270074, 2.86177487151084e-33)

(117.36509491566378, 1.40748553170105e-49)

(97.40564988372432, 5.5353251792155e-41)

(61.55856443408466, 1.36209839819635e-25)

(113.37196348786802, 7.39451910626098e-48)

(89.42754736080431, 1.4947139472568e-37)

(99.40078535482674, 7.66703058114809e-42)

(73.48797945819744, 1.04950845491397e-30)

(63.54451948499176, 1.92210872012007e-26)

(109.37937792943009, 3.87864051227573e-46)

(67.51947123595703, 3.80720406063906e-28)

(53.628494392570325, 3.35874755583896e-22)

(55.608546448343034, 4.78486794334859e-23)

(105.38740714914354, 2.03090985918538e-44)

(81.4543520716443, 3.98993245773348e-34)

(51.65053571610925, 2.35029236682257e-21)

(57.590395847577085, 6.79760569939251e-24)

(45.73341842402173, 7.86392281146536e-19)

(115.36846486038449, 1.02037641684105e-48)

(103.39167711976444, 1.46855429561531e-43)

(43.768672819252, 5.40382520576156e-18)

(87.43372777159384, 1.07565938701008e-36)

(107.38331093574938, 2.80726948064315e-45)

(59.573801417717405, 9.63308937575565e-25)

(37.912949250867825, 1.67187071859432e-15)

(41.80927805095239, 3.68814386856124e-17)

(95.4107409664157, 3.99399918441605e-40)

(39.85668038566399, 2.49625239853701e-16)

(121.3587133312717, 2.67507641971962e-51)

(85.44023648880128, 7.73502988839825e-36)

(119.3618465896432, 1.94073896738367e-50)

(91.4216705602647, 2.0755364312549e-38)

(32.1768362765376, 4.51950718513269e-13)

(93.41607512615644, 2.88010606010239e-39)

(101.3961323402728, 1.06138359503294e-42)

(111.37559842694574, 5.35655382358895e-47)

(83.44710101410942, 5.55775781462981e-35)

(47.7024673187512, 1.13793496971854e-19)


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
Puntos máximos de la función:
x47=1.56699936884542x_{47} = 1.56699936884542
Decrece en los intervalos
(−∞,1.56699936884542]\left(-\infty, 1.56699936884542\right]
Crece en los intervalos
[1.56699936884542,∞)\left[1.56699936884542, \infty\right)
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
(x+3log⁡(x)−2−6x−3x2)e−x=0\left(x + 3 \log{\left(x \right)} - 2 - \frac{6}{x} - \frac{3}{x^{2}}\right) e^{- x} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=103.3938795677x_{1} = 103.3938795677
x2=111.377468673631x_{2} = 111.377468673631
x3=39.8833266179636x_{3} = 39.8833266179636
x4=36.020723382258x_{4} = 36.020723382258
x5=81.4581293688572x_{5} = 81.4581293688572
x6=53.6391997859506x_{6} = 53.6391997859506
x7=119.363455316025x_{7} = 119.363455316025
x8=49.6883027713006x_{8} = 49.6883027713006
x9=99.4031887795118x_{9} = 99.4031887795118
x10=65.5378886433624x_{10} = 65.5378886433624
x11=95.4133747638047x_{11} = 95.4133747638047
x12=45.7503652932959x_{12} = 45.7503652932959
x13=32.2447032885795x_{13} = 32.2447032885795
x14=55.6182521655207x_{14} = 55.6182521655207
x15=93.4188369884936x_{15} = 93.4188369884936
x16=2.39850478388758x_{16} = 2.39850478388758
x17=63.5513892036866x_{17} = 63.5513892036866
x18=73.4927921960631x_{18} = 73.4927921960631
x19=89.4305956988942x_{19} = 89.4305956988942
x20=101.398431931161x_{20} = 101.398431931161
x21=57.5992407733395x_{21} = 57.5992407733395
x22=113.373762593967x_{22} = 113.373762593967
x23=91.4245702571095x_{23} = 91.4245702571095
x24=51.6624126721263x_{24} = 51.6624126721263
x25=97.4081644714375x_{25} = 97.4081644714375
x26=117.366763550593x_{26} = 117.366763550593
x27=61.5660089408849x_{27} = 61.5660089408849
x28=109.381323689162x_{28} = 109.381323689162
x29=75.4833073118349x_{29} = 75.4833073118349
x30=37.944999973952x_{30} = 37.944999973952
x31=105.389518575299x_{31} = 105.389518575299
x32=85.4436194207933x_{32} = 85.4436194207933
x33=83.4506727953749x_{33} = 83.4506727953749
x34=77.4744027706525x_{34} = 77.4744027706525
x35=107.385336951583x_{35} = 107.385336951583
x36=87.4369367383724x_{36} = 87.4369367383724
x37=71.5029178602067x_{37} = 71.5029178602067
x38=59.5818991600012x_{38} = 59.5818991600012
x39=41.8318647601129x_{39} = 41.8318647601129
x40=43.7881151801198x_{40} = 43.7881151801198
x41=69.5137536619685x_{41} = 69.5137536619685
x42=79.4660256450971x_{42} = 79.4660256450971
x43=34.1168106345989x_{43} = 34.1168106345989
x44=47.7173940452843x_{44} = 47.7173940452843
x45=121.360265358969x_{45} = 121.360265358969
x46=115.370196863584x_{46} = 115.370196863584
x47=67.5253796286827x_{47} = 67.5253796286827

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:
Cóncava en los intervalos
[2.39850478388758,∞)\left[2.39850478388758, \infty\right)
Convexa en los intervalos
(−∞,2.39850478388758]\left(-\infty, 2.39850478388758\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞((x+3log⁡(x))e−x)=−∞\lim_{x \to -\infty}\left(\left(x + 3 \log{\left(x \right)}\right) e^{- x}\right) = -\infty
Tomamos como el límite
es decir,
no hay asíntota horizontal a la izquierda
lim⁡x→∞((x+3log⁡(x))e−x)=0\lim_{x \to \infty}\left(\left(x + 3 \log{\left(x \right)}\right) e^{- 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 (x + 3*log(x))*exp(-x), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞((x+3log⁡(x))e−xx)=∞\lim_{x \to -\infty}\left(\frac{\left(x + 3 \log{\left(x \right)}\right) e^{- x}}{x}\right) = \infty
Tomamos como el límite
es decir,
no hay asíntota inclinada a la izquierda
lim⁡x→∞((x+3log⁡(x))e−xx)=0\lim_{x \to \infty}\left(\frac{\left(x + 3 \log{\left(x \right)}\right) e^{- x}}{x}\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:
(x+3log⁡(x))e−x=(−x+3log⁡(−x))ex\left(x + 3 \log{\left(x \right)}\right) e^{- x} = \left(- x + 3 \log{\left(- x \right)}\right) e^{x}
- No
(x+3log⁡(x))e−x=−(−x+3log⁡(−x))ex\left(x + 3 \log{\left(x \right)}\right) e^{- x} = - \left(- x + 3 \log{\left(- x \right)}\right) e^{x}
- No
es decir, función
no es
par ni impar
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