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Trazado el gráfico de la función/ ln|x|/((3^x+3)(x+1))

Gráfico de la función y = ln|x|/((3^x+3)(x+1))

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
           log(|x|)    
f(x) = ----------------
       / x    \        
       \3  + 3/*(x + 1)
f(x)=log⁡(∣x∣)(3x+3)(x+1)f{\left(x \right)} = \frac{\log{\left(\left|{x}\right| \right)}}{\left(3^{x} + 3\right) \left(x + 1\right)} 
f = log(|x|)/(((3^x + 3)*(x + 1)))
Gráfico de la función
02468-8-6-4-2-10101.0-1.0
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=−1x_{1} = -1
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:
log⁡(∣x∣)(3x+3)(x+1)=0\frac{\log{\left(\left|{x}\right| \right)}}{\left(3^{x} + 3\right) \left(x + 1\right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=1x_{1} = 1
Solución numérica
x1=103.152197238873x_{1} = 103.152197238873
x2=59.0398928158645x_{2} = 59.0398928158645
x3=40.9150989315417x_{3} = 40.9150989315417
x4=28.7311071856708x_{4} = 28.7311071856708
x5=73.0912736807879x_{5} = 73.0912736807879
x6=87.1251967527365x_{6} = 87.1251967527365
x7=63.0570951445221x_{7} = 63.0570951445221
x8=117.169562779696x_{8} = 117.169562779696
x9=69.078877801029x_{9} = 69.078877801029
x10=111.162674521812x_{10} = 111.162674521812
x11=101.149308770164x_{11} = 101.149308770164
x12=83.1167364508841x_{12} = 83.1167364508841
x13=24.6454779046952x_{13} = 24.6454779046952
x14=26.6860656128057x_{14} = 26.6860656128057
x15=57.0303004899535x_{15} = 57.0303004899535
x16=65.0648381980633x_{16} = 65.0648381980633
x17=38.8930624434456x_{17} = 38.8930624434456
x18=85.1210713172098x_{18} = 85.1210713172098
x19=81.1121756258823x_{19} = 81.1121756258823
x20=91.1328777211616x_{20} = 91.1328777211616
x21=79.1073705173126x_{21} = 79.1073705173126
x22=89.1291277039413x_{22} = 89.1291277039413
x23=61.0488005931824x_{23} = 61.0488005931824
x24=46.9686147548521x_{24} = 46.9686147548521
x25=107.157638593541x_{25} = 107.157638593541
x26=95.139883111464x_{26} = 95.139883111464
x27=48.9832010929708x_{27} = 48.9832010929708
x28=71.0852624486007x_{28} = 71.0852624486007
x29=77.1023007513339x_{29} = 77.1023007513339
x30=97.1431599383692x_{30} = 97.1431599383692
x31=42.9348192626393x_{31} = 42.9348192626393
x32=30.7722159162088x_{32} = 30.7722159162088
x33=50.9965151435447x_{33} = 50.9965151435447
x34=32.808474707418x_{34} = 32.808474707418
x35=109.160204327129x_{35} = 109.160204327129
x36=34.8402940490066x_{36} = 34.8402940490066
x37=53.008716890151x_{37} = 53.008716890151
x38=44.9525644903x_{38} = 44.9525644903
x39=115.16734902733x_{39} = 115.16734902733
x40=113.165054456732x_{40} = 113.165054456732
x41=36.8682945374015x_{41} = 36.8682945374015
x42=119.171699941278x_{42} = 119.171699941278
x43=99.1462989636874x_{43} = 99.1462989636874
x44=67.0720834442505x_{44} = 67.0720834442505
x45=93.13645911837x_{45} = 93.13645911837
x46=105.15497161971x_{46} = 105.15497161971
x47=75.0969436095681x_{47} = 75.0969436095681
x48=55.0199407708778x_{48} = 55.0199407708778
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 log(|x|)/(((3^x + 3)*(x + 1))).
log⁡(∣0∣)30+3\frac{\log{\left(\left|{0}\right| \right)}}{3^{0} + 3}
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
(−3x(x+1)log⁡(3)−3x−3)log⁡(∣x∣)(3x+3)2(x+1)2+1(3x+3)(x+1)sign⁡(x)∣x∣=0\frac{\left(- 3^{x} \left(x + 1\right) \log{\left(3 \right)} - 3^{x} - 3\right) \log{\left(\left|{x}\right| \right)}}{\left(3^{x} + 3\right)^{2} \left(x + 1\right)^{2}} + \frac{\frac{1}{\left(3^{x} + 3\right) \left(x + 1\right)} \operatorname{sign}{\left(x \right)}}{\left|{x}\right|} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=48.9769730430882x_{1} = 48.9769730430882
x2=46.9617831630105x_{2} = 46.9617831630105
x3=81.1100665430803x_{3} = 81.1100665430803
x4=93.1348792110655x_{4} = 93.1348792110655
x5=91.131224650882x_{5} = 91.131224650882
x6=99.1449114000728x_{6} = 99.1449114000728
x7=119.1707503366x_{7} = 119.1707503366
x8=115.166330379211x_{8} = 115.166330379211
x9=109.159066951714x_{9} = 109.159066951714
x10=67.068930091267x_{10} = 67.068930091267
x11=34.8270754262926x_{11} = 34.8270754262926
x12=97.1417123946586x_{12} = 97.1417123946586
x13=65.0614733819287x_{13} = 65.0614733819287
x14=103.150918887023x_{14} = 103.150918887023
x15=57.0258318259641x_{15} = 57.0258318259641
x16=61.0449428974447x_{16} = 61.0449428974447
x17=89.1273962183452x_{17} = 89.1273962183452
x18=95.1383715758753x_{18} = 95.1383715758753
x19=53.0034781633461x_{19} = 53.0034781633461
x20=107.156456956734x_{20} = 107.156456956734
x21=73.0886467503272x_{21} = 73.0886467503272
x22=75.0944628685852x_{22} = 75.0944628685852
x23=24.6327499511412x_{23} = 24.6327499511412
x24=38.8826834561625x_{24} = 38.8826834561625
x25=105.153743063549x_{25} = 105.153743063549
x26=69.075916417789x_{26} = 69.075916417789
x27=63.0534966284006x_{27} = 63.0534966284006
x28=26.6660060546255x_{28} = 26.6660060546255
x29=117.168579568878x_{29} = 117.168579568878
x30=59.0357466066597x_{30} = 59.0357466066597
x31=83.114732684953x_{31} = 83.114732684953
x32=77.0999542229932x_{32} = 77.0999542229932
x33=113.163998403934x_{33} = 113.163998403934
x34=44.945037375884x_{34} = 44.945037375884
x35=30.7551436310896x_{35} = 30.7551436310896
x36=55.0151101695965x_{36} = 55.0151101695965
x37=40.9058243391679x_{37} = 40.9058243391679
x38=71.0824758748379x_{38} = 71.0824758748379
x39=111.161578947793x_{39} = 111.161578947793
x40=85.1191651056892x_{40} = 85.1191651056892
x41=32.7934533504587x_{41} = 32.7934533504587
x42=87.1233810813533x_{42} = 87.1233810813533
x43=79.1051475041507x_{43} = 79.1051475041507
x44=1.69382002578544x_{44} = 1.69382002578544
x45=42.9264856626939x_{45} = 42.9264856626939
x46=28.7119573074522x_{46} = 28.7119573074522
x47=36.8566133751686x_{47} = 36.8566133751686
x48=101.147977504935x_{48} = 101.147977504935
x49=50.9908139364942x_{49} = 50.9908139364942
Signos de extremos en los puntos:
(48.976973043088186, 3.33714857748905e-25)

(46.96178316301053, 3.14793230430263e-24)

(81.11006654308034, 1.06980716070928e-40)

(93.13487921106551, 1.76240286013667e-46)

(91.13122465088196, 1.61937309782801e-45)

(99.14491140007284, 2.27856619218883e-49)

(119.17075033659992, 5.50531067989844e-59)

(115.1663303792109, 4.60233204029224e-57)

(109.15906695171353, 3.52615482987876e-54)

(67.06893009126698, 6.1784487160008e-34)

(34.827075426292566, 2.39512069642525e-18)

(97.1417123946586, 2.09060939161621e-48)

(65.06147338192874, 5.73496805925914e-33)

(103.15091888702344, 2.71020766959669e-51)

(57.02583182596408, 4.31419834608406e-29)

(61.04494289744473, 4.95981883080521e-31)

(89.12739621834523, 1.4887468674289e-44)

(95.13837157587533, 1.91904027938389e-47)

(53.00347816334612, 3.77852134256581e-27)

(107.1564569567342, 3.22879152718831e-53)

(73.08864675032716, 7.77549636488103e-37)

(75.09446286858518, 8.41085882299435e-38)

(24.632749951141168, 2.20850893240523e-13)

(38.88268345616246, 2.5763634846079e-20)

(105.15374306354933, 2.95759057187296e-52)

(69.07591641778902, 6.6637371063996e-35)

(63.053496628400644, 5.32981029271488e-32)

(26.666006054625463, 2.24624787800383e-14)

(117.16857956887849, 5.03285973971297e-58)

(59.03574660665968, 4.62213956100171e-30)

(83.11473268495303, 1.16082423344428e-41)

(77.09995422299319, 9.10588932439467e-39)

(113.16399840393436, 4.20994444827565e-56)

(44.94503737588402, 2.97806164734966e-23)

(30.755143631089577, 2.28585046472187e-16)

(55.01511016959649, 4.03368179000527e-28)

(40.905824339167914, 2.69283793253635e-21)

(71.08247587483794, 7.19467550719518e-36)

(111.16157894779325, 3.85225976937881e-55)

(85.11916510568922, 1.26042348449353e-42)

(32.79345335045869, 2.33113662033544e-17)

(87.12338108135334, 1.36942902378677e-43)

(79.1051475041507, 9.86622662496345e-40)

(1.6938200257854417, 0.0207470107155686)

(42.92648566269395, 2.82659548796775e-22)

(28.711957307452153, 2.25930862379777e-15)

(36.85661337516865, 2.47705121954204e-19)

(101.14797750493487, 2.4845119151544e-50)

(50.990813936494185, 3.54688094430096e-26)


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:
x49=1.69382002578544x_{49} = 1.69382002578544
Decrece en los intervalos
(−∞,1.69382002578544]\left(-\infty, 1.69382002578544\right]
Crece en los intervalos
[1.69382002578544,∞)\left[1.69382002578544, \infty\right)
Asíntotas verticales
Hay:
x1=−1x_{1} = -1
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(log⁡(∣x∣)(3x+3)(x+1))=0\lim_{x \to -\infty}\left(\frac{\log{\left(\left|{x}\right| \right)}}{\left(3^{x} + 3\right) \left(x + 1\right)}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞(log⁡(∣x∣)(3x+3)(x+1))=0\lim_{x \to \infty}\left(\frac{\log{\left(\left|{x}\right| \right)}}{\left(3^{x} + 3\right) \left(x + 1\right)}\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 log(|x|)/(((3^x + 3)*(x + 1))), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(1(3x+3)(x+1)log⁡(∣x∣)x)=0\lim_{x \to -\infty}\left(\frac{\frac{1}{\left(3^{x} + 3\right) \left(x + 1\right)} \log{\left(\left|{x}\right| \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(1(3x+3)(x+1)log⁡(∣x∣)x)=0\lim_{x \to \infty}\left(\frac{\frac{1}{\left(3^{x} + 3\right) \left(x + 1\right)} \log{\left(\left|{x}\right| \right)}}{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:
log⁡(∣x∣)(3x+3)(x+1)=log⁡(∣x∣)(1−x)(3+3−x)\frac{\log{\left(\left|{x}\right| \right)}}{\left(3^{x} + 3\right) \left(x + 1\right)} = \frac{\log{\left(\left|{x}\right| \right)}}{\left(1 - x\right) \left(3 + 3^{- x}\right)}
- No
log⁡(∣x∣)(3x+3)(x+1)=−log⁡(∣x∣)(1−x)(3+3−x)\frac{\log{\left(\left|{x}\right| \right)}}{\left(3^{x} + 3\right) \left(x + 1\right)} = - \frac{\log{\left(\left|{x}\right| \right)}}{\left(1 - x\right) \left(3 + 3^{- x}\right)}
- No
es decir, función
no es
par ni impar
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