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Trazado el gráfico de la función/ (-1+x+3*log(x))*exp(-x)

Gráfico de la función y = (-1+x+3*log(x))*exp(-x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
                            -x
f(x) = (-1 + x + 3*log(x))*e
f(x)=((x−1)+3log⁡(x))e−xf{\left(x \right)} = \left(\left(x - 1\right) + 3 \log{\left(x \right)}\right) e^{- x} 
f = (x - 1 + 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−1)+3log⁡(x))e−x=0\left(\left(x - 1\right) + 3 \log{\left(x \right)}\right) e^{- x} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=1x_{1} = 1
Solución numérica
x1=63.5437579557744x_{1} = 63.5437579557744
x2=43.765250986269x_{2} = 43.765250986269
x3=65.5308475064368x_{3} = 65.5308475064368
x4=71.4972806718954x_{4} = 71.4972806718954
x5=85.4399693179621x_{5} = 85.4399693179621
x6=117.365001953683x_{6} = 117.365001953683
x7=105.387274910551x_{7} = 105.387274910551
x8=69.507700393925x_{8} = 69.507700393925
x9=113.371859425047x_{9} = 113.371859425047
x10=89.4273185602433x_{10} = 89.4273185602433
x11=61.5577076626651x_{11} = 61.5577076626651
x12=35.972042538105x_{12} = 35.972042538105
x13=101.395982098149x_{13} = 101.395982098149
x14=83.4468113664324x_{14} = 83.4468113664324
x15=99.4006248639022x_{15} = 99.4006248639022
x16=75.4783808502049x_{16} = 75.4783808502049
x17=47.7001183488908x_{17} = 47.7001183488908
x18=109.379260910294x_{18} = 109.379260910294
x19=51.6488448469515x_{19} = 51.6488448469515
x20=119.361758585721x_{20} = 119.361758585721
x21=34.0536419608443x_{21} = 34.0536419608443
x22=97.4054781810104x_{22} = 97.4054781810104
x23=45.7306027039507x_{23} = 45.7306027039507
x24=49.6730557371836x_{24} = 49.6730557371836
x25=91.4214581845505x_{25} = 91.4214581845505
x26=1x_{26} = 1
x27=55.607284552587x_{27} = 55.607284552587
x28=32.1582550832069x_{28} = 32.1582550832069
x29=93.4158776258835x_{29} = 93.4158776258835
x30=81.4540373188058x_{30} = 81.4540373188058
x31=59.5728324192605x_{31} = 59.5728324192605
x32=95.4105569720673x_{32} = 95.4105569720673
x33=115.368366558044x_{33} = 115.368366558044
x34=111.375488141096x_{34} = 111.375488141096
x35=41.8050515578526x_{35} = 41.8050515578526
x36=77.4697813285159x_{36} = 77.4697813285159
x37=79.461681215674x_{37} = 79.461681215674
x38=53.6270394661104x_{38} = 53.6270394661104
x39=37.9060797267244x_{39} = 37.9060797267244
x40=67.5188611250771x_{40} = 67.5188611250771
x41=39.8513567828081x_{41} = 39.8513567828081
x42=73.4875287275765x_{42} = 73.4875287275765
x43=121.358629936895x_{43} = 121.358629936895
x44=103.391536265365x_{44} = 103.391536265365
x45=107.38318661929x_{45} = 107.38318661929
x46=57.5892935780553x_{46} = 57.5892935780553
x47=87.4334807874926x_{47} = 87.4334807874926
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 (-1 + x + 3*log(x))*exp(-x).
(3log⁡(0)−1)e−0\left(3 \log{\left(0 \right)} - 1\right) e^{- 0}
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−1)+3log⁡(x))e−x=0\left(1 + \frac{3}{x}\right) e^{- x} - \left(\left(x - 1\right) + 3 \log{\left(x \right)}\right) e^{- x} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=121.360179716374x_{1} = 121.360179716374
x2=85.4433414294184x_{2} = 85.4433414294184
x3=41.8271503681158x_{3} = 41.8271503681158
x4=57.5980637435828x_{4} = 57.5980637435828
x5=107.385208785981x_{5} = 107.385208785981
x6=53.6376352228099x_{6} = 53.6376352228099
x7=43.784331566616x_{7} = 43.784331566616
x8=1.82898090512814x_{8} = 1.82898090512814
x9=105.389382153228x_{9} = 105.389382153228
x10=55.6169001983336x_{10} = 55.6169001983336
x11=81.45780110418x_{11} = 81.45780110418
x12=71.5023959198487x_{12} = 71.5023959198487
x13=47.7148317458726x_{13} = 47.7148317458726
x14=113.373655502183x_{14} = 113.373655502183
x15=119.363364895068x_{15} = 119.363364895068
x16=61.565099277752x_{16} = 61.565099277752
x17=95.4131842359605x_{17} = 95.4131842359605
x18=79.4656675821878x_{18} = 79.4656675821878
x19=69.5131753131619x_{19} = 69.5131753131619
x20=51.6605868008712x_{20} = 51.6605868008712
x21=115.370095755896x_{21} = 115.370095755896
x22=87.4366800256624x_{22} = 87.4366800256624
x23=103.39373415809x_{23} = 103.39373415809
x24=111.377355112312x_{24} = 111.377355112312
x25=89.4303581261606x_{25} = 89.4303581261606
x26=39.8773248877601x_{26} = 39.8773248877601
x27=75.4828777043785x_{27} = 75.4828777043785
x28=67.5247362749294x_{28} = 67.5247362749294
x29=45.7472746099704x_{29} = 45.7472746099704
x30=73.4923194602862x_{30} = 73.4923194602862
x31=32.2218340566274x_{31} = 32.2218340566274
x32=83.4503710757866x_{32} = 83.4503710757866
x33=65.5371699536673x_{33} = 65.5371699536673
x34=77.4740111353968x_{34} = 77.4740111353968
x35=37.9371515926127x_{35} = 37.9371515926127
x36=93.4186322984504x_{36} = 93.4186322984504
x37=99.403022856178x_{37} = 99.403022856178
x38=91.4243499500006x_{38} = 91.4243499500006
x39=49.6861515579525x_{39} = 49.6861515579525
x40=109.381203122272x_{40} = 109.381203122272
x41=101.398276719529x_{41} = 101.398276719529
x42=97.4079868177515x_{42} = 97.4079868177515
x43=36.0100984565334x_{43} = 36.0100984565334
x44=117.366667986509x_{44} = 117.366667986509
x45=59.58086753817x_{45} = 59.58086753817
x46=63.5505826668671x_{46} = 63.5505826668671
x47=34.1017383735187x_{47} = 34.1017383735187
Signos de extremos en los puntos:
(121.36017971637446, 2.65150985983682e-51)

(85.44334142941835, 7.63324125238892e-36)

(41.8271503681158, 3.55577897660977e-17)

(57.598063743582806, 6.64975025652919e-24)

(107.38520878598126, 2.77891387281531e-45)

(53.63763522280991, 3.27792150795723e-22)

(43.78433156661604, 5.2249426126083e-18)

(1.828980905128142, 0.423964996854719)

(105.38938215322818, 2.00995587917191e-44)

(55.61690019833358, 4.67555296922145e-23)

(81.45780110418005, 3.93433751173145e-34)

(71.50239591984865, 7.37240908992889e-30)

(47.7148317458726, 1.10524630013819e-19)

(113.37365550218252, 7.32424942024923e-48)

(119.36336489506783, 1.92332442667664e-50)

(61.565099277752026, 1.33504478496959e-25)

(95.4131842359605, 3.94782053435071e-40)

(79.46566758218779, 2.82068712995554e-33)

(69.51317531316185, 5.25495850037655e-29)

(51.66058680087117, 2.2905259437567e-21)

(115.37009575589634, 1.01086713238945e-48)

(87.43668002566241, 1.06188605032614e-36)

(103.39373415809048, 1.45306982726628e-43)

(111.37735511231215, 5.30462722890151e-47)

(89.43035812616064, 1.47607661833041e-37)

(39.8773248877601, 2.39828457856865e-16)

(75.48287770437854, 1.44539107570609e-31)

(67.52473627492938, 3.74022388410885e-28)

(45.74727460997045, 7.62212735792768e-19)

(73.49231946028624, 1.03292072523445e-30)

(32.2218340566274, 4.2241911527071e-13)

(83.45037107578658, 5.48253241387224e-35)

(65.53716995366733, 2.65790058148143e-27)

(77.47401113539682, 2.02024094708306e-32)

(37.93715159261268, 1.59933873990772e-15)

(93.41863229845042, 2.84598008713052e-39)

(99.40302285617801, 7.58247092953313e-42)

(91.42434995000062, 2.05031716684357e-38)

(49.686151557952456, 1.59455474906753e-20)

(109.38120312227221, 3.84026859952048e-46)

(101.39827671952878, 1.04994089653684e-42)

(97.40798681775154, 5.4728365928505e-41)

(36.01009845653341, 1.05078331397421e-14)

(117.36666798650893, 1.39461699369151e-49)

(59.58086753816999, 9.43309383936488e-25)

(63.55058266686705, 1.88551138664645e-26)

(34.10173837351874, 6.76368063849609e-14)


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=1.82898090512814x_{47} = 1.82898090512814
Decrece en los intervalos
(−∞,1.82898090512814]\left(-\infty, 1.82898090512814\right]
Crece en los intervalos
[1.82898090512814,∞)\left[1.82898090512814, \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)−3−6x−3x2)e−x=0\left(x + 3 \log{\left(x \right)} - 3 - \frac{6}{x} - \frac{3}{x^{2}}\right) e^{- x} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=95.4158739682377x_{1} = 95.4158739682377
x2=87.4399638196228x_{2} = 87.4399638196228
x3=67.5308280530971x_{3} = 67.5308280530971
x4=83.4540304585909x_{4} = 83.4540304585909
x5=2.68583712896287x_{5} = 2.68583712896287
x6=47.7304682325957x_{6} = 47.7304682325957
x7=89.4334758422992x_{7} = 89.4334758422992
x8=71.5076858202293x_{8} = 71.5076858202293
x9=69.5188443342373x_{9} = 69.5188443342373
x10=36.0525831074928x_{10} = 36.0525831074928
x11=41.8510349732532x_{11} = 41.8510349732532
x12=91.4273140998576x_{12} = 91.4273140998576
x13=105.391533946079x_{13} = 105.391533946079
x14=97.4105536475856x_{14} = 97.4105536475856
x15=59.5892561213236x_{15} = 59.5892561213236
x16=111.379259103451x_{16} = 111.379259103451
x17=49.7000133748533x_{17} = 49.7000133748533
x18=57.6072395362685x_{18} = 57.6072395362685
x19=37.9713662213646x_{19} = 37.9713662213646
x20=115.371857882233x_{20} = 115.371857882233
x21=75.4875175819314x_{21} = 75.4875175819314
x22=93.4214541505838x_{22} = 93.4214541505838
x23=77.4783710364664x_{23} = 77.4783710364664
x24=117.368365129067x_{24} = 117.368365129067
x25=103.395979568237x_{25} = 103.395979568237
x26=107.387272780303x_{26} = 107.387272780303
x27=101.400622098728x_{27} = 101.400622098728
x28=43.8048170715467x_{28} = 43.8048170715467
x29=119.365000628187x_{29} = 119.365000628187
x30=39.9056230046219x_{30} = 39.9056230046219
x31=79.4697726517234x_{31} = 79.4697726517234
x32=63.557681057861x_{32} = 63.557681057861
x33=121.361757354465x_{33} = 121.361757354465
x34=51.6729724387995x_{34} = 51.6729724387995
x35=109.383184659107x_{35} = 109.383184659107
x36=85.4468052346057x_{36} = 85.4468052346057
x37=55.6269848309959x_{37} = 55.6269848309959
x38=65.5437352809509x_{38} = 65.5437352809509
x39=34.1564653516795x_{39} = 34.1564653516795
x40=32.2962722088623x_{40} = 32.2962722088623
x41=81.4616735144062x_{41} = 81.4616735144062
x42=113.375486472815x_{42} = 113.375486472815
x43=61.5728009768058x_{43} = 61.5728009768058
x44=73.4972679573231x_{44} = 73.4972679573231
x45=45.7650754098845x_{45} = 45.7650754098845
x46=53.6487778040035x_{46} = 53.6487778040035
x47=99.4054751523751x_{47} = 99.4054751523751

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
[2.68583712896287,∞)\left[2.68583712896287, \infty\right)
Convexa en los intervalos
(−∞,2.68583712896287]\left(-\infty, 2.68583712896287\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(((x−1)+3log⁡(x))e−x)=−∞\lim_{x \to -\infty}\left(\left(\left(x - 1\right) + 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−1)+3log⁡(x))e−x)=0\lim_{x \to \infty}\left(\left(\left(x - 1\right) + 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 (-1 + x + 3*log(x))*exp(-x), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(((x−1)+3log⁡(x))e−xx)=∞\lim_{x \to -\infty}\left(\frac{\left(\left(x - 1\right) + 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−1)+3log⁡(x))e−xx)=0\lim_{x \to \infty}\left(\frac{\left(\left(x - 1\right) + 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−1)+3log⁡(x))e−x=(−x+3log⁡(−x)−1)ex\left(\left(x - 1\right) + 3 \log{\left(x \right)}\right) e^{- x} = \left(- x + 3 \log{\left(- x \right)} - 1\right) e^{x}
- No
((x−1)+3log⁡(x))e−x=−(−x+3log⁡(−x)−1)ex\left(\left(x - 1\right) + 3 \log{\left(x \right)}\right) e^{- x} = - \left(- x + 3 \log{\left(- x \right)} - 1\right) e^{x}
- No
es decir, función
no es
par ni impar
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