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

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

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       /  -1    \            
       |--------|    x       
       \x*log(2)/   2 *log(2)
f(x) = ---------- + ---------
           x                2
                    / x    \ 
                    \2  - 1/
f(x)=2xlog⁡(2)(2x−1)2+(−1)1xlog⁡(2)xf{\left(x \right)} = \frac{2^{x} \log{\left(2 \right)}}{\left(2^{x} - 1\right)^{2}} + \frac{\left(-1\right) \frac{1}{x \log{\left(2 \right)}}}{x} 
f = (2^x*log(2))/(2^x - 1)^2 + (-1/(x*log(2)))/x
Gráfico de la función
02468-8-6-4-2-1010-0.100.00
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:
2xlog⁡(2)(2x−1)2+(−1)1xlog⁡(2)x=0\frac{2^{x} \log{\left(2 \right)}}{\left(2^{x} - 1\right)^{2}} + \frac{\left(-1\right) \frac{1}{x \log{\left(2 \right)}}}{x} = 0
Resolvermos esta ecuación
Solución no hallada,
puede ser que el gráfico no cruce el eje X
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*log(2)))/x + (2^x*log(2))/(2^x - 1)^2.
(−1)10log⁡(2)0+20log⁡(2)(−1+20)2\frac{\left(-1\right) \frac{1}{0 \log{\left(2 \right)}}}{0} + \frac{2^{0} \log{\left(2 \right)}}{\left(-1 + 2^{0}\right)^{2}}
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
−2⋅22xlog⁡(2)2(2x−1)3+2xlog⁡(2)2(2x−1)2+1x1log⁡(2)x2+1x3log⁡(2)=0- \frac{2 \cdot 2^{2 x} \log{\left(2 \right)}^{2}}{\left(2^{x} - 1\right)^{3}} + \frac{2^{x} \log{\left(2 \right)}^{2}}{\left(2^{x} - 1\right)^{2}} + \frac{\frac{1}{x} \frac{1}{\log{\left(2 \right)}}}{x^{2}} + \frac{1}{x^{3} \log{\left(2 \right)}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=38207.8534501551x_{1} = 38207.8534501551
x2=−38076.6205637343x_{2} = -38076.6205637343
x3=16170.2072339581x_{3} = 16170.2072339581
x4=−42314.6370890542x_{4} = -42314.6370890542
x5=−27057.7844226496x_{5} = -27057.7844226496
x6=−18581.7719305674x_{6} = -18581.7719305674
x7=40750.6632690734x_{7} = 40750.6632690734
x8=−40619.4303623627x_{8} = -40619.4303623627
x9=−21972.1741460331x_{9} = -21972.1741460331
x10=−34686.2081625031x_{10} = -34686.2081625031
x11=41598.2666205299x_{11} = 41598.2666205299
x12=−22819.7753899879x_{12} = -22819.7753899879
x13=−13496.208530477x_{13} = -13496.208530477
x14=30579.4268759524x_{14} = 30579.4268759524
x15=−16038.9752597959x_{15} = -16038.9752597959
x16=29547.9598087961x_{16} = 29547.9598087961
x17=−24514.9784989544x_{17} = -24514.9784989544
x18=−15191.3791645157x_{18} = -15191.3791645157
x19=39903.0599551918x_{19} = 39903.0599551918
x20=25493.8129993218x_{20} = 25493.8129993218
x21=34817.4410146354x_{21} = 34817.4410146354
x22=36512.6471277453x_{22} = 36512.6471277453
x23=19560.6044007384x_{23} = 19560.6044007384
x24=−23667.376848405x_{24} = -23667.376848405
x25=29731.8242890716x_{25} = 29731.8242890716
x26=35665.0440431628x_{26} = 35665.0440431628
x27=−28752.9890385016x_{27} = -28752.9890385016
x28=42445.8700073105x_{28} = 42445.8700073105
x29=22103.4066980435x_{29} = 22103.4066980435
x30=33122.2351429581x_{30} = 33122.2351429581
x31=−29600.5915120748x_{31} = -29600.5915120748
x32=18014.2722496163x_{32} = 18014.2722496163
x33=−30448.1940837816x_{33} = -30448.1940837816
x34=21255.8056536129x_{34} = 21255.8056536129
x35=−14343.7877245408x_{35} = -14343.7877245408
x36=−25362.5803223458x_{36} = -25362.5803223458
x37=20408.2048754459x_{37} = 20408.2048754459
x38=17865.4045854108x_{38} = 17865.4045854108
x39=26341.4150027076x_{39} = 26341.4150027076
x40=−31295.7967456477x_{40} = -31295.7967456477
x41=−36381.4142572736x_{41} = -36381.4142572736
x42=−38924.2237877047x_{42} = -38924.2237877047
x43=31427.029551779x_{43} = 31427.029551779
x44=−17734.1723154465x_{44} = -17734.1723154465
x45=32274.6323095441x_{45} = 32274.6323095441
x46=−20276.9724109851x_{46} = -20276.9724109851
x47=−37229.0173858502x_{47} = -37229.0173858502
x48=24646.2111494812x_{48} = 24646.2111494812
x49=−19429.3719896177x_{49} = -19429.3719896177
x50=18713.0042788702x_{50} = 18713.0042788702
x51=15322.6107477011x_{51} = 15322.6107477011
x52=33969.8380463588x_{52} = 33969.8380463588
x53=−21124.5731426564x_{53} = -21124.5731426564
x54=−26210.1823018011x_{54} = -26210.1823018011
x55=17017.8054621743x_{55} = 17017.8054621743
x56=14475.0183435899x_{56} = 14475.0183435899
x57=28884.2217989658x_{57} = 28884.2217989658
x58=−39771.8270548144x_{58} = -39771.8270548144
x59=−32143.3994905401x_{59} = -32143.3994905401
x60=−27905.3866720052x_{60} = -27905.3866720052
x61=37360.2502645683x_{61} = 37360.2502645683
x62=27189.0171452783x_{62} = 27189.0171452783
x63=−16886.5733019871x_{63} = -16886.5733019871
x64=22951.007978519x_{64} = 22951.007978519
x65=39055.4566813312x_{65} = 39055.4566813312
x66=−32991.0023120587x_{66} = -32991.0023120587
x67=−33838.605204445x_{67} = -33838.605204445
x68=−41467.0337078697x_{68} = -41467.0337078697
x69=23798.6094695966x_{69} = 23798.6094695966
x70=13627.4365761369x_{70} = 13627.4365761369
x71=−35533.8111815333x_{71} = -35533.8111815333
x72=28036.6194144113x_{72} = 28036.6194144113
Signos de extremos en los puntos:
                                                     6.85006545436918e-10 
(38207.85345015511, 1.95003124382029e-11502*log(2) - --------------------)
                                                            log(2)        

                                                      6.89736497791236e-10 
(-38076.62056373434, 6.23845145583425e-11463*log(2) - --------------------)
                                                             log(2)        

                                                     3.824448600147e-9 
(16170.207233958079, 1.91684270294788e-4868*log(2) - -----------------)
                                                           log(2)      

                                                      5.58494317912367e-10 
(-42314.63708905421, 1.05920630939428e-12738*log(2) - --------------------)
                                                             log(2)        

                                                    1.36588940218638e-9 
(-27057.7844226496, 6.24126428177133e-8146*log(2) - -------------------)
                                                           log(2)       

                                                     2.89618132864276e-9 
(-18581.77193056739, 2.13440245719703e-5594*log(2) - -------------------)
                                                            log(2)       

                                                      6.02185974759399e-10 
(40750.663269073375, 6.72996501325709e-12268*log(2) - --------------------)
                                                             log(2)        

                                                      6.06083334933965e-10 
(-40619.43036236271, 2.15305014407515e-12228*log(2) - --------------------)
                                                             log(2)        

                                                     2.07135212900612e-9 
(-21972.17414603311, 5.20609503153121e-6615*log(2) - -------------------)
                                                            log(2)       

                                                      8.31163279124204e-10 
(-34686.20816250315, 2.57577095526037e-10442*log(2) - --------------------)
                                                             log(2)        

                                                      5.77895791397205e-10 
(41598.266620529896, 4.72040859813683e-12523*log(2) - --------------------)
                                                             log(2)        

                                                      1.9203362013414e-9 
(-22819.775389987895, 3.65690180571392e-6870*log(2) - ------------------)
                                                            log(2)       

                                                      5.49005177452242e-9 
(-13496.208530477039, 1.72347342140578e-4063*log(2) - -------------------)
                                                             log(2)       

                                                     1.06940280010589e-9 
(30579.426875952435, 4.73434742513088e-9206*log(2) - -------------------)
                                                            log(2)       

                                                      3.88728842380603e-9 
(-16038.975259795929, 6.12839978871031e-4829*log(2) - -------------------)
                                                             log(2)       

                                                    1.14536788237407e-9 
(29547.95980879607, 1.50586791013925e-8895*log(2) - -------------------)
                                                           log(2)       

                                                      1.66393733575689e-9 
(-24514.978498954428, 1.80355236854597e-7380*log(2) - -------------------)
                                                             log(2)       

                                                      4.33316865564797e-9 
(-15191.379164515733, 8.69352637265602e-4574*log(2) - -------------------)
                                                             log(2)       

                                                     6.28040424639244e-10 
(39903.05995519176, 9.59477310380349e-12013*log(2) - --------------------)
                                                            log(2)        

                                                   1.53861658072523e-9 
(25493.81299932179, 3.9589804378129e-7675*log(2) - -------------------)
                                                          log(2)       

                                                      8.2490949586614e-10 
(34817.441014635384, 8.05160263628955e-10482*log(2) - -------------------)
                                                             log(2)       

                                                     7.50089973137431e-10 
(36512.64712774534, 3.96272061626916e-10992*log(2) - --------------------)
                                                            log(2)        

                                                   2.61357796679761e-9 
(19560.60440073839, 4.6918277639248e-5889*log(2) - -------------------)
                                                          log(2)       

                                                     1.78525290340964e-9 
(-23667.37684840496, 2.56832480865958e-7125*log(2) - -------------------)
                                                            log(2)       

                                                     1.13124555296289e-9 
(29731.824289071596, 6.74626161081472e-8951*log(2) - -------------------)
                                                            log(2)       

                                                    7.8616638570971e-10 
(35665.04404316275, 5.6486716568567e-10737*log(2) - -------------------)
                                                           log(2)       

                                                     1.20957834320997e-9 
(-28752.98903850157, 3.07492419241799e-8656*log(2) - -------------------)
                                                            log(2)       

                                                      5.55046183297051e-10 
(42445.870007310485, 3.31082133711202e-12778*log(2) - --------------------)
                                                             log(2)        

                                                     2.04682904282434e-9 
(22103.406698043458, 1.62771186325996e-6654*log(2) - -------------------)
                                                            log(2)       

                                                    9.11508510683117e-10 
(33122.23514295815, 1.63568071745269e-9971*log(2) - --------------------)
                                                           log(2)        

                                                      1.14129843198264e-9 
(-29600.591512074796, 2.15806970843624e-8911*log(2) - -------------------)
                                                             log(2)       

                                                     3.08153110672662e-9 
(18014.272249616253, 1.45781632158299e-5423*log(2) - -------------------)
                                                            log(2)       

                                                     1.07864099346126e-9 
(-30448.194083781593, 1.5144919984926e-9166*log(2) - -------------------)
                                                            log(2)       

                                                   2.21332331490622e-9 
(21255.8056536129, 2.31694784843999e-6399*log(2) - -------------------)
                                                          log(2)       

                                                      4.86040323386044e-9 
(-14343.787724540842, 1.22925922572975e-4318*log(2) - -------------------)
                                                             log(2)       

                                                     1.55458018974783e-9 
(-25362.58032234585, 1.26635519654776e-7635*log(2) - -------------------)
                                                            log(2)       

                                                    2.40099020924727e-9 
(20408.20487544592, 3.29742430589014e-6144*log(2) - -------------------)
                                                           log(2)       

                                                    3.13310022921006e-9 
(17865.40458541082, 9.49150582299392e-5379*log(2) - -------------------)
                                                           log(2)       

                                                    1.44119185434066e-9 
(26341.41500270757, 2.77943135046213e-7930*log(2) - -------------------)
                                                           log(2)       

                                                      1.02100524927618e-9 
(-31295.796745647705, 1.06277506613512e-9421*log(2) - -------------------)
                                                             log(2)       

                                                      7.5551109264959e-10 
(-36381.41425727359, 1.26772158054758e-10952*log(2) - -------------------)
                                                             log(2)       

                                                      6.60024532891633e-10 
(-38924.22378770465, 4.37604685528308e-11718*log(2) - --------------------)
                                                             log(2)        

                                                     1.01249603731428e-9 
(31427.029551778956, 3.32223459400286e-9461*log(2) - -------------------)
                                                            log(2)       

                                                      3.17964146575754e-9 
(-17734.172315446496, 3.03518227155602e-5339*log(2) - -------------------)
                                                             log(2)       

                                                     9.60013617709051e-10 
(32274.632309544144, 2.33117996167287e-9716*log(2) - --------------------)
                                                            log(2)        

                                                     2.43216917220451e-9 
(-20276.972410985134, 1.0545885274891e-6104*log(2) - -------------------)
                                                            log(2)       

                                                      7.21500859181443e-10 
(-37229.01738585019, 8.89319393713213e-11208*log(2) - --------------------)
                                                             log(2)        

                                                     1.64626471649394e-9 
(24646.211149481227, 5.63851237723975e-7420*log(2) - -------------------)
                                                            log(2)       

                                                     2.64900314259698e-9 
(-19429.37198961769, 1.50049382154929e-5849*log(2) - -------------------)
                                                            log(2)       

                                                     2.85570252938712e-9 
(18713.004278870234, 6.67425924341018e-5634*log(2) - -------------------)
                                                            log(2)       

                                                     4.25926304299294e-9 
(15322.610747701096, 2.71990076178069e-4613*log(2) - -------------------)
                                                            log(2)       

                                                     8.66588751114778e-10 
(33969.83804635884, 1.14762552336153e-10226*log(2) - --------------------)
                                                            log(2)        

                                                     2.24090845839737e-9 
(-21124.57314265637, 7.41034559724779e-6360*log(2) - -------------------)
                                                            log(2)       

                                                      1.45565989443495e-9 
(-26210.182301801062, 8.89068732051473e-7891*log(2) - -------------------)
                                                             log(2)       

                                                     3.45297067878783e-9 
(17017.805462174292, 1.34925958272026e-5123*log(2) - -------------------)
                                                            log(2)       

                                                     4.77267382204263e-9 
(14475.018343589922, 3.84849381629277e-4358*log(2) - -------------------)
                                                            log(2)       

                                                     1.19861209981928e-9 
(28884.221798965846, 9.61251653449618e-8696*log(2) - -------------------)
                                                            log(2)       

                                                     6.32191883108117e-10 
(-39771.82705481437, 3.0695459614702e-11973*log(2) - --------------------)
                                                            log(2)        

                                                    9.67868572523961e-10 
(-32143.39949054005, 7.4574566433761e-9677*log(2) - --------------------)
                                                           log(2)        

                                                      1.28417411189181e-9 
(-27905.386672005163, 4.38097852742267e-8401*log(2) - -------------------)
                                                             log(2)       

                                                      7.16441024846644e-10 
(37360.250264568254, 2.77987233669376e-11247*log(2) - --------------------)
                                                             log(2)        

                                                    1.35273579723854e-9 
(27189.01714527827, 1.95113198761798e-8185*log(2) - -------------------)
                                                           log(2)       

                                                      3.50684798831999e-9 
(-16886.573301987122, 4.31431770562263e-5084*log(2) - -------------------)
                                                             log(2)       

                                                   1.89843823509753e-9 
(22951.00797851898, 1.1433198467791e-6909*log(2) - -------------------)
                                                          log(2)       

                                                     6.55596398492552e-10 
(39055.45668133125, 1.36786917889077e-11757*log(2) - --------------------)
                                                            log(2)        

                                                     9.18774598280675e-10 
(-32991.00231205869, 5.23259414087004e-9932*log(2) - --------------------)
                                                            log(2)        

                                                      8.73323392084263e-10 
(-33838.60520444496, 3.67131822970089e-10187*log(2) - --------------------)
                                                             log(2)        

                                                       5.81559374098447e-10 
(-41467.033707869734, 1.51015916197173e-12483*log(2) - --------------------)
                                                              log(2)        

                                                     1.76561835593508e-9 
(23798.609469596635, 8.02961196970894e-7165*log(2) - -------------------)
                                                            log(2)       

                                                   5.38482583414486e-9 
(13627.4365761369, 5.40538428057278e-4103*log(2) - -------------------)
                                                          log(2)       

                                                       7.91984021111587e-10 
(-35533.811181533325, 1.80706634777702e-10697*log(2) - --------------------)
                                                              log(2)        

                                                    1.27218042081069e-9 
(28036.61941441128, 1.36955427247036e-8440*log(2) - -------------------)
                                                           log(2)


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
Crece 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
6⋅23xlog⁡(2)3(2x−1)4−6⋅22xlog⁡(2)3(2x−1)3+2xlog⁡(2)3(2x−1)2−6x4log⁡(2)=0\frac{6 \cdot 2^{3 x} \log{\left(2 \right)}^{3}}{\left(2^{x} - 1\right)^{4}} - \frac{6 \cdot 2^{2 x} \log{\left(2 \right)}^{3}}{\left(2^{x} - 1\right)^{3}} + \frac{2^{x} \log{\left(2 \right)}^{3}}{\left(2^{x} - 1\right)^{2}} - \frac{6}{x^{4} \log{\left(2 \right)}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=3288.06101797592x_{1} = 3288.06101797592
x2=−8269.80534815666x_{2} = -8269.80534815666
x3=−4126.49361514019x_{3} = -4126.49361514019
x4=−5216.83456632016x_{4} = -5216.83456632016
x5=4160.26232565819x_{5} = 4160.26232565819
x6=−3254.30312210557x_{6} = -3254.30312210557
x7=10920.4122349142x_{7} = 10920.4122349142
x8=−4998.76581682109x_{8} = -4998.76581682109
x9=1791.01946841018x_{9} = 1791.01946841018
x10=4378.3297181762x_{10} = 4378.3297181762
x11=−1961.7209829434x_{11} = -1961.7209829434
x12=−1760.59176267991x_{12} = -1760.59176267991
x13=10702.3423321896x_{13} = 10702.3423321896
x14=9830.06286657826x_{14} = 9830.06286657826
x15=9393.92323510645x_{15} = 9393.92323510645
x16=5032.5348338162x_{16} = 5032.5348338162
x17=−9796.29364921871x_{17} = -9796.29364921871
x18=3.95948798497865x_{18} = 3.95948798497865
x19=−7179.457423917x_{19} = -7179.457423917
x20=7867.43524860009x_{20} = 7867.43524860009
x21=3070.10932107662x_{21} = 3070.10932107662
x22=8521.64422667933x_{22} = 8521.64422667933
x23=4596.39779480085x_{23} = 4596.39779480085
x24=−10232.4333452831x_{24} = -10232.4333452831
x25=6340.94900694958x_{25} = 6340.94900694958
x26=2852.33327312915x_{26} = 2852.33327312915
x27=−3690.36754923305x_{27} = -3690.36754923305
x28=3942.19654378071x_{28} = 3942.19654378071
x29=5250.60360676676x_{29} = 5250.60360676676
x30=8085.50487717376x_{30} = 8085.50487717376
x31=−6743.31854652688x_{31} = -6743.31854652688
x32=−2171.20906409384x_{32} = -2171.20906409384
x33=9175.85344891741x_{33} = 9175.85344891741
x34=−9578.22382729971x_{34} = -9578.22382729971
x35=10484.2724430874x_{35} = 10484.2724430874
x36=1993.76913046767x_{36} = 1993.76913046767
x37=−7833.66607062896x_{37} = -7833.66607062896
x38=−4562.6288503238x_{38} = -4562.6288503238
x39=−3908.42822028792x_{39} = -3908.42822028792
x40=8739.71394300349x_{40} = 8739.71394300349
x41=−10450.5032172484x_{41} = -10450.5032172484
x42=8957.78368434389x_{42} = 8957.78368434389
x43=−5871.04149781434x_{43} = -5871.04149781434
x44=5468.67250142715x_{44} = 5468.67250142715
x45=−10014.3634889257x_{45} = -10014.3634889257
x46=−8705.94474424403x_{46} = -8705.94474424403
x47=9611.99304144018x_{47} = 9611.99304144018
x48=−5434.90344116473x_{48} = -5434.90344116473
x49=7649.36565432205x_{49} = 7649.36565432205
x50=−6525.24918722325x_{50} = -6525.24918722325
x51=−8924.01448131404x_{51} = -8924.01448131404
x52=−6961.38796041714x_{52} = -6961.38796041714
x53=5686.74150029258x_{53} = 5686.74150029258
x54=2634.97329995587x_{54} = 2634.97329995587
x55=−7615.59648269236x_{55} = -7615.59648269236
x56=−9360.15402441224x_{56} = -9360.15402441224
x57=6122.87976238584x_{57} = 6122.87976238584
x58=−9142.08424191854x_{58} = -9142.08424191854
x59=−2385.16850411536x_{59} = -2385.16850411536
x60=2204.14147363214x_{60} = 2204.14147363214
x61=−5652.97242274738x_{61} = -5652.97242274738
x62=−7397.52693263738x_{62} = -7397.52693263738
x63=3506.08395368876x_{63} = 3506.08395368876
x64=6777.08768635722x_{64} = 6777.08768635722
x65=−3.95948798497865x_{65} = -3.95948798497865
x66=7431.29609735788x_{66} = 7431.29609735788
x67=8303.57453734178x_{67} = 8303.57453734178
x68=10048.1327092966x_{68} = 10048.1327092966
x69=4814.4662109971x_{69} = 4814.4662109971
x70=6995.15710932559x_{70} = 6995.15710932559
x71=−10886.6430042501x_{71} = -10886.6430042501
x72=−4344.56084851685x_{72} = -4344.56084851685
x73=7213.22658109075x_{73} = 7213.22658109075
x74=3724.13487183387x_{74} = 3724.13487183387
x75=−8051.73569336837x_{75} = -8051.73569336837
x76=−10668.5731038643x_{76} = -10668.5731038643
x77=−8487.87503252295x_{77} = -8487.87503252295
x78=−6307.17988817526x_{78} = -6307.17988817526
x79=−2818.63443997569x_{79} = -2818.63443997569
x80=2418.5530329695x_{80} = 2418.5530329695
x81=−3472.31925563555x_{81} = -3472.31925563555
x82=−2601.37220473245x_{82} = -2601.37220473245
x83=−3036.36860543918x_{83} = -3036.36860543918
x84=−4780.69722370166x_{84} = -4780.69722370166
x85=10266.2025684753x_{85} = 10266.2025684753
x86=6559.01831705229x_{86} = 6559.01831705229
x87=5904.81059066272x_{87} = 5904.81059066272
x88=−6089.11065587517x_{88} = -6089.11065587517
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−(6⋅23xlog⁡(2)3(2x−1)4−6⋅22xlog⁡(2)3(2x−1)3+2xlog⁡(2)3(2x−1)2−6x4log⁡(2))=log⁡(2)3120\lim_{x \to 0^-}\left(\frac{6 \cdot 2^{3 x} \log{\left(2 \right)}^{3}}{\left(2^{x} - 1\right)^{4}} - \frac{6 \cdot 2^{2 x} \log{\left(2 \right)}^{3}}{\left(2^{x} - 1\right)^{3}} + \frac{2^{x} \log{\left(2 \right)}^{3}}{\left(2^{x} - 1\right)^{2}} - \frac{6}{x^{4} \log{\left(2 \right)}}\right) = \frac{\log{\left(2 \right)}^{3}}{120}
lim⁡x→0+(6⋅23xlog⁡(2)3(2x−1)4−6⋅22xlog⁡(2)3(2x−1)3+2xlog⁡(2)3(2x−1)2−6x4log⁡(2))=log⁡(2)3120\lim_{x \to 0^+}\left(\frac{6 \cdot 2^{3 x} \log{\left(2 \right)}^{3}}{\left(2^{x} - 1\right)^{4}} - \frac{6 \cdot 2^{2 x} \log{\left(2 \right)}^{3}}{\left(2^{x} - 1\right)^{3}} + \frac{2^{x} \log{\left(2 \right)}^{3}}{\left(2^{x} - 1\right)^{2}} - \frac{6}{x^{4} \log{\left(2 \right)}}\right) = \frac{\log{\left(2 \right)}^{3}}{120}
- 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:
Cóncava en los intervalos
[−3.95948798497865,3.95948798497865]\left[-3.95948798497865, 3.95948798497865\right]
Convexa en los intervalos
(−∞,−3.95948798497865]∪[3.95948798497865,∞)\left(-\infty, -3.95948798497865\right] \cup \left[3.95948798497865, \infty\right)
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→−∞(2xlog⁡(2)(2x−1)2+(−1)1xlog⁡(2)x)=0\lim_{x \to -\infty}\left(\frac{2^{x} \log{\left(2 \right)}}{\left(2^{x} - 1\right)^{2}} + \frac{\left(-1\right) \frac{1}{x \log{\left(2 \right)}}}{x}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞(2xlog⁡(2)(2x−1)2+(−1)1xlog⁡(2)x)=0\lim_{x \to \infty}\left(\frac{2^{x} \log{\left(2 \right)}}{\left(2^{x} - 1\right)^{2}} + \frac{\left(-1\right) \frac{1}{x \log{\left(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 (-1/(x*log(2)))/x + (2^x*log(2))/(2^x - 1)^2, dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(2xlog⁡(2)(2x−1)2+(−1)1xlog⁡(2)xx)=0\lim_{x \to -\infty}\left(\frac{\frac{2^{x} \log{\left(2 \right)}}{\left(2^{x} - 1\right)^{2}} + \frac{\left(-1\right) \frac{1}{x \log{\left(2 \right)}}}{x}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(2xlog⁡(2)(2x−1)2+(−1)1xlog⁡(2)xx)=0\lim_{x \to \infty}\left(\frac{\frac{2^{x} \log{\left(2 \right)}}{\left(2^{x} - 1\right)^{2}} + \frac{\left(-1\right) \frac{1}{x \log{\left(2 \right)}}}{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:
2xlog⁡(2)(2x−1)2+(−1)1xlog⁡(2)x=−1x2log⁡(2)+2−xlog⁡(2)(−1+2−x)2\frac{2^{x} \log{\left(2 \right)}}{\left(2^{x} - 1\right)^{2}} + \frac{\left(-1\right) \frac{1}{x \log{\left(2 \right)}}}{x} = - \frac{1}{x^{2} \log{\left(2 \right)}} + \frac{2^{- x} \log{\left(2 \right)}}{\left(-1 + 2^{- x}\right)^{2}}
- No
2xlog⁡(2)(2x−1)2+(−1)1xlog⁡(2)x=1x2log⁡(2)−2−xlog⁡(2)(−1+2−x)2\frac{2^{x} \log{\left(2 \right)}}{\left(2^{x} - 1\right)^{2}} + \frac{\left(-1\right) \frac{1}{x \log{\left(2 \right)}}}{x} = \frac{1}{x^{2} \log{\left(2 \right)}} - \frac{2^{- x} \log{\left(2 \right)}}{\left(-1 + 2^{- x}\right)^{2}}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор