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Trazado el gráfico de la función/ log(x)/(x^2-1)^2

Gráfico de la función y = log(x)/(x^2-1)^2

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
         log(x) 
f(x) = ---------
               2
       / 2    \ 
       \x  - 1/
f(x)=log⁡(x)(x2−1)2f{\left(x \right)} = \frac{\log{\left(x \right)}}{\left(x^{2} - 1\right)^{2}} 
f = log(x)/(x^2 - 1)^2
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=−1x_{1} = -1
x2=1x_{2} = 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)(x2−1)2=0\frac{\log{\left(x \right)}}{\left(x^{2} - 1\right)^{2}} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=11868.2197476035x_{1} = 11868.2197476035
x2=11366.8970814154x_{2} = 11366.8970814154
x3=1698.49868265918x_{3} = 1698.49868265918
x4=6579.23377437361x_{4} = 6579.23377437361
x5=11617.6084931168x_{5} = 11617.6084931168
x6=2482.86292765679x_{6} = 2482.86292765679
x7=12118.7335900743x_{7} = 12118.7335900743
x8=5563.27364864577x_{8} = 5563.27364864577
x9=2222.47571815619x_{9} = 2222.47571815619
x10=4030.87563426386x_{10} = 4030.87563426386
x11=12619.4793617895x_{11} = 12619.4793617895
x12=3774.19905515517x_{12} = 3774.19905515517
x13=7338.92619061318x_{13} = 7338.92619061318
x14=10614.1322304007x_{14} = 10614.1322304007
x15=4798.54937071711x_{15} = 4798.54937071711
x16=3517.06763091401x_{16} = 3517.06763091401
x17=3001.23864309133x_{17} = 3001.23864309133
x18=10111.7308804807x_{18} = 10111.7308804807
x19=3259.43347213341x_{19} = 3259.43347213341
x20=4287.13756197522x_{20} = 4287.13756197522
x21=5053.75414404079x_{21} = 5053.75414404079
x22=6325.59228242362x_{22} = 6325.59228242362
x23=5817.6255846709x_{23} = 5817.6255846709
x24=10362.9896936882x_{24} = 10362.9896936882
x25=2742.41176515911x_{25} = 2742.41176515911
x26=11116.0826237799x_{26} = 11116.0826237799
x27=8349.31241693296x_{27} = 8349.31241693296
x28=9608.8489996537x_{28} = 9608.8489996537
x29=12869.7161501915x_{29} = 12869.7161501915
x30=8601.50607064916x_{30} = 8601.50607064916
x31=7844.45519970175x_{31} = 7844.45519970175
x32=6071.72711336117x_{32} = 6071.72711336117
x33=7085.89070502773x_{33} = 7085.89070502773
x34=9105.45330101748x_{34} = 9105.45330101748
x35=7591.77832867796x_{35} = 7591.77832867796
x36=8096.9642616196x_{36} = 8096.9642616196
x37=1961.09403573067x_{37} = 1961.09403573067
x38=6832.66309074712x_{38} = 6832.66309074712
x39=8853.55118055139x_{39} = 8853.55118055139
x40=12369.152632297x_{40} = 12369.152632297
x41=9357.21762138131x_{41} = 9357.21762138131
x42=9860.35199227789x_{42} = 9860.35199227789
x43=5308.65566855231x_{43} = 5308.65566855231
x44=10865.1620768583x_{44} = 10865.1620768583
x45=4543.01901311462x_{45} = 4543.01901311462
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)/(x^2 - 1)^2.
log⁡(0)(−1+02)2\frac{\log{\left(0 \right)}}{\left(-1 + 0^{2}\right)^{2}}
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
−4xlog⁡(x)(x2−1)3+1x(x2−1)2=0- \frac{4 x \log{\left(x \right)}}{\left(x^{2} - 1\right)^{3}} + \frac{1}{x \left(x^{2} - 1\right)^{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=994.79802209141x_{1} = 994.79802209141
x2=1725.61862682028x_{2} = 1725.61862682028
x3=2243.57786782149x_{3} = 2243.57786782149
x4=2862.35500026547x_{4} = 2862.35500026547
x5=3684.05058075643x_{5} = 3684.05058075643
x6=5013.62757253792x_{6} = 5013.62757253792
x7=1933.10706389741x_{7} = 1933.10706389741
x8=4707.30855089308x_{8} = 4707.30855089308
x9=4809.44508841288x_{9} = 4809.44508841288
x10=3991.42048253229x_{10} = 3991.42048253229
x11=4093.7974387415x_{11} = 4093.7974387415
x12=1829.4188438223x_{12} = 1829.4188438223
x13=4502.94054540882x_{13} = 4502.94054540882
x14=5115.67512917555x_{14} = 5115.67512917555
x15=5217.69457234705x_{15} = 5217.69457234705
x16=783.941938903636x_{16} = 783.941938903636
x17=2965.25272187248x_{17} = 2965.25272187248
x18=3376.29065127965x_{18} = 3376.29065127965
x19=2759.39669194708x_{19} = 2759.39669194708
x20=3786.54845541179x_{20} = 3786.54845541179
x21=4400.70720221718x_{21} = 4400.70720221718
x22=4196.13675317173x_{22} = 4196.13675317173
x23=1621.69679041648x_{23} = 1621.69679041648
x24=677.938778750613x_{24} = 677.938778750613
x25=4911.55115403373x_{25} = 4911.55115403373
x26=3478.92338633603x_{26} = 3478.92338633603
x27=464.138325450939x_{27} = 464.138325450939
x28=1517.64209554598x_{28} = 1517.64209554598
x29=3889.0046073518x_{29} = 3889.0046073518
x30=2346.89163159892x_{30} = 2346.89163159892
x31=4605.14067421585x_{31} = 4605.14067421585
x32=3273.60933950581x_{32} = 3273.60933950581
x33=571.405661546538x_{33} = 571.405661546538
x34=2140.1797481115x_{34} = 2140.1797481115
x35=2656.37473156072x_{35} = 2656.37473156072
x36=2450.12608735292x_{36} = 2450.12608735292
x37=1204.5340087759x_{37} = 1204.5340087759
x38=2036.69161044985x_{38} = 2036.69161044985
x39=889.53554472855x_{39} = 889.53554472855
x40=4298.43962916141x_{40} = 4298.43962916141
x41=3581.50944073106x_{41} = 3581.50944073106
x42=1413.44125848094x_{42} = 1413.44125848094
x43=3170.87741092418x_{43} = 3170.87741092418
x44=1309.07836015989x_{44} = 1309.07836015989
x45=3068.09266501104x_{45} = 3068.09266501104
x46=1099.78411393857x_{46} = 1099.78411393857
x47=2553.28576105542x_{47} = 2553.28576105542
Signos de extremos en los puntos:
(994.7980220914103, 7.04806888272018e-12)

(1725.6186268202773, 8.40566342810104e-13)

(2243.5778678214924, 3.04521578902434e-13)

(2862.3550002654683, 1.18573176299208e-13)

(3684.0505807564264, 4.45794190684374e-14)

(5013.627572537918, 1.34842568377356e-14)

(1933.1070638974124, 5.41868206131259e-13)

(4707.3085508930835, 1.7223408865452e-14)

(4809.445088412885, 1.58464078641473e-14)

(3991.4204825322877, 3.26696372705339e-14)

(4093.797438741497, 2.9612377284183e-14)

(1829.4188438222966, 6.70639303189756e-13)

(4502.94054540882, 2.04615977097549e-14)

(5115.675129175551, 1.24695093382705e-14)

(5217.694572347052, 1.15491385373806e-14)

(783.9419389036356, 1.76450395338436e-11)

(2965.252721872483, 1.03408531410574e-13)

(3376.290651279655, 6.25229088814603e-14)

(2759.396691947083, 1.36653405697535e-13)

(3786.5484554117857, 4.00785100226289e-14)

(4400.707202217178, 2.23690371092192e-14)

(4196.1367531717315, 2.69071384227081e-14)

(1621.696790416481, 1.068656957172e-12)

(677.9387787506125, 3.08620492968767e-11)

(4911.55115403373, 1.46053111073136e-14)

(3478.923386336033, 5.56694315186094e-14)

(464.13832545093925, 1.32310891612619e-10)

(1517.6420955459814, 1.38078290041649e-12)

(3889.0046073518006, 3.61357290601579e-14)

(2346.8916315989236, 2.5582155535322e-13)

(4605.140674215851, 1.87546879034831e-14)

(3273.609339505806, 7.04753128457025e-14)

(571.4056615465382, 5.95481429338424e-11)

(2140.179748111498, 3.65525625001973e-13)

(2656.37473156072, 1.58353951135968e-13)

(2450.1260873529204, 2.16549812197637e-13)

(1204.534008775897, 3.3698162114302e-12)

(2036.6916104498491, 4.42794033098202e-13)

(889.5355447285503, 1.08458197297492e-11)

(4298.43962916141, 2.45061429860596e-14)

(3581.509440731058, 4.97366873274973e-14)

(1413.4412584809445, 1.81741418844272e-12)

(3170.8774109241836, 7.97466296802564e-14)

(1309.0783601598894, 2.44391240502862e-12)

(3068.092665011041, 9.06102768098431e-14)

(1099.784113938567, 4.78681852368243e-12)

(2553.2857610554215, 1.84587775340639e-13)


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
4(6x2x2−1−1)log⁡(x)x2−1−8x2−1−1x2(x2−1)2=0\frac{\frac{4 \left(\frac{6 x^{2}}{x^{2} - 1} - 1\right) \log{\left(x \right)}}{x^{2} - 1} - \frac{8}{x^{2} - 1} - \frac{1}{x^{2}}}{\left(x^{2} - 1\right)^{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=1812.89997839524x_{1} = 1812.89997839524
x2=1535.81994441797x_{2} = 1535.81994441797
x3=360.652439045554x_{3} = 360.652439045554
x4=2420.73934366595x_{4} = 2420.73934366595
x5=867.680167056638x_{5} = 867.680167056638
x6=417.514556579136x_{6} = 417.514556579136
x7=643.373182415073x_{7} = 643.373182415073
x8=2365.56307962394x_{8} = 2365.56307962394
x9=1313.68604282886x_{9} = 1313.68604282886
x10=2310.37203616797x_{10} = 2310.37203616797
x11=2696.41368929005x_{11} = 2696.41368929005
x12=2199.94371057308x_{12} = 2199.94371057308
x13=1591.28398479513x_{13} = 1591.28398479513
x14=2255.16574506589x_{14} = 2255.16574506589
x15=1868.24899404412x_{15} = 1868.24899404412
x16=1646.72291419698x_{16} = 1646.72291419698
x17=1258.07527660852x_{17} = 1258.07527660852
x18=811.70858357249x_{18} = 811.70858357249
x19=2641.30512227233x_{19} = 2641.30512227233
x20=699.563276361071x_{20} = 699.563276361071
x21=246.05787320989x_{21} = 246.05787320989
x22=2034.17772415273x_{22} = 2034.17772415273
x23=1146.74751227394x_{23} = 1146.74751227394
x24=2806.59382640276x_{24} = 2806.59382640276
x25=1923.5777823238x_{25} = 1923.5777823238
x26=303.536205111338x_{26} = 303.536205111338
x27=1369.26416496896x_{27} = 1369.26416496896
x28=923.592911867885x_{28} = 923.592911867885
x29=1978.88711809124x_{29} = 1978.88711809124
x30=1424.81145569123x_{30} = 1424.81145569123
x31=979.451943240246x_{31} = 979.451943240246
x32=755.672069557883x_{32} = 755.672069557883
x33=587.090476162373x_{33} = 587.090476162373
x34=2751.50981548484x_{34} = 2751.50981548484
x35=1035.26162795503x_{35} = 1035.26162795503
x36=1480.32955316153x_{36} = 1480.32955316153
x37=1702.1378708284x_{37} = 1702.1378708284
x38=2586.18377272454x_{38} = 2586.18377272454
x39=2089.4502761792x_{39} = 2089.4502761792
x40=2531.04928159199x_{40} = 2531.04928159199
x41=2144.70540703033x_{41} = 2144.70540703033
x42=1202.42985079201x_{42} = 1202.42985079201
x43=1091.02572736451x_{43} = 1091.02572736451
x44=2475.90127116929x_{44} = 2475.90127116929
x45=1757.52990285957x_{45} = 1757.52990285957
x46=474.184224602407x_{46} = 474.184224602407
x47=530.700603570167x_{47} = 530.700603570167
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=−1x_{1} = -1
x2=1x_{2} = 1

lim⁡x→−1−(4(6x2x2−1−1)log⁡(x)x2−1−8x2−1−1x2(x2−1)2)=∞i\lim_{x \to -1^-}\left(\frac{\frac{4 \left(\frac{6 x^{2}}{x^{2} - 1} - 1\right) \log{\left(x \right)}}{x^{2} - 1} - \frac{8}{x^{2} - 1} - \frac{1}{x^{2}}}{\left(x^{2} - 1\right)^{2}}\right) = \infty i
lim⁡x→−1+(4(6x2x2−1−1)log⁡(x)x2−1−8x2−1−1x2(x2−1)2)=∞i\lim_{x \to -1^+}\left(\frac{\frac{4 \left(\frac{6 x^{2}}{x^{2} - 1} - 1\right) \log{\left(x \right)}}{x^{2} - 1} - \frac{8}{x^{2} - 1} - \frac{1}{x^{2}}}{\left(x^{2} - 1\right)^{2}}\right) = \infty i
- los límites son iguales, es decir omitimos el punto correspondiente
lim⁡x→1−(4(6x2x2−1−1)log⁡(x)x2−1−8x2−1−1x2(x2−1)2)=−∞\lim_{x \to 1^-}\left(\frac{\frac{4 \left(\frac{6 x^{2}}{x^{2} - 1} - 1\right) \log{\left(x \right)}}{x^{2} - 1} - \frac{8}{x^{2} - 1} - \frac{1}{x^{2}}}{\left(x^{2} - 1\right)^{2}}\right) = -\infty
lim⁡x→1+(4(6x2x2−1−1)log⁡(x)x2−1−8x2−1−1x2(x2−1)2)=∞\lim_{x \to 1^+}\left(\frac{\frac{4 \left(\frac{6 x^{2}}{x^{2} - 1} - 1\right) \log{\left(x \right)}}{x^{2} - 1} - \frac{8}{x^{2} - 1} - \frac{1}{x^{2}}}{\left(x^{2} - 1\right)^{2}}\right) = \infty
- los límites no son iguales, signo
x2=1x_{2} = 1
- es el punto de flexión

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=−1x_{1} = -1
x2=1x_{2} = 1
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(log⁡(x)(x2−1)2)=0\lim_{x \to -\infty}\left(\frac{\log{\left(x \right)}}{\left(x^{2} - 1\right)^{2}}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞(log⁡(x)(x2−1)2)=0\lim_{x \to \infty}\left(\frac{\log{\left(x \right)}}{\left(x^{2} - 1\right)^{2}}\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)/(x^2 - 1)^2, dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(log⁡(x)x(x2−1)2)=0\lim_{x \to -\infty}\left(\frac{\log{\left(x \right)}}{x \left(x^{2} - 1\right)^{2}}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(log⁡(x)x(x2−1)2)=0\lim_{x \to \infty}\left(\frac{\log{\left(x \right)}}{x \left(x^{2} - 1\right)^{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:
log⁡(x)(x2−1)2=log⁡(−x)(x2−1)2\frac{\log{\left(x \right)}}{\left(x^{2} - 1\right)^{2}} = \frac{\log{\left(- x \right)}}{\left(x^{2} - 1\right)^{2}}
- No
log⁡(x)(x2−1)2=−log⁡(−x)(x2−1)2\frac{\log{\left(x \right)}}{\left(x^{2} - 1\right)^{2}} = - \frac{\log{\left(- x \right)}}{\left(x^{2} - 1\right)^{2}}
- No
es decir, función
no es
par ni impar
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