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

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

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
                 x
       / 1 + x  \ 
f(x) = |--------| 
       \-1 + 2*x/
f(x)=(x+12x−1)xf{\left(x \right)} = \left(\frac{x + 1}{2 x - 1}\right)^{x} 
f = ((x + 1)/(2*x - 1))^x
Gráfico de la función
02468-8-6-4-2-101005000
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=0.5x_{1} = 0.5
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+12x−1)x=0\left(\frac{x + 1}{2 x - 1}\right)^{x} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=124.487880113x_{1} = 124.487880113
x2=108.488410578875x_{2} = 108.488410578875
x3=50.5022671014178x_{3} = 50.5022671014178
x4=130.487736897813x_{4} = 130.487736897813
x5=126.487829737126x_{5} = 126.487829737126
x6=72.491797748857x_{6} = 72.491797748857
x7=52.5003204207963x_{7} = 52.5003204207963
x8=96.4890391883861x_{8} = 96.4890391883861
x9=74.4914233429987x_{9} = 74.4914233429987
x10=66.4932266480552x_{10} = 66.4932266480552
x11=82.490266612904x_{11} = 82.490266612904
x12=106.488497827462x_{12} = 106.488497827462
x13=76.4910879155628x_{13} = 76.4910879155628
x14=60.4953490466866x_{14} = 60.4953490466866
x15=78.4907861504449x_{15} = 78.4907861504449
x16=100.488798839193x_{16} = 100.488798839193
x17=70.4922175658637x_{17} = 70.4922175658637
x18=56.497405352682x_{18} = 56.497405352682
x19=88.4896496614179x_{19} = 88.4896496614179
x20=86.4898371005996x_{20} = 86.4898371005996
x21=122.487933399261x_{21} = 122.487933399261
x22=46.5077627336444x_{22} = 46.5077627336444
x23=62.4945386463219x_{23} = 62.4945386463219
x24=118.488049646891x_{24} = 118.488049646891
x25=104.488591211381x_{25} = 104.488591211381
x26=98.488914508241x_{26} = 98.488914508241
x27=110.488328935558x_{27} = 110.488328935558
x28=92.4893196077744x_{28} = 92.4893196077744
x29=54.4987276863062x_{29} = 54.4987276863062
x30=102.488691325473x_{30} = 102.488691325473
x31=64.4938376056205x_{31} = 64.4938376056205
x32=80.4905136205167x_{32} = 80.4905136205167
x33=114.488180619574x_{33} = 114.488180619574
x34=116.488113141577x_{34} = 116.488113141577
x35=128.487782062312x_{35} = 128.487782062312
x36=90.489477724809x_{36} = 90.489477724809
x37=112.488252423888x_{37} = 112.488252423888
x38=44.511770238091x_{38} = 44.511770238091
x39=120.487989826054x_{39} = 120.487989826054
x40=94.4891738519814x_{40} = 94.4891738519814
x41=48.5046880614612x_{41} = 48.5046880614612
x42=58.4962937146654x_{42} = 58.4962937146654
x43=68.49269064341x_{43} = 68.49269064341
x44=42.5171682924135x_{44} = 42.5171682924135
x45=84.4900419934582x_{45} = 84.4900419934582
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)/(-1 + 2*x))^x.
(1−1+0⋅2)0\left(\frac{1}{-1 + 0 \cdot 2}\right)^{0}
Resultado:
f(0)=1f{\left(0 \right)} = 1
Punto:
(0, 1)
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
(x+12x−1)x(x(2x−1)(−2(x+1)(2x−1)2+12x−1)x+1+log⁡(x+12x−1))=0\left(\frac{x + 1}{2 x - 1}\right)^{x} \left(\frac{x \left(2 x - 1\right) \left(- \frac{2 \left(x + 1\right)}{\left(2 x - 1\right)^{2}} + \frac{1}{2 x - 1}\right)}{x + 1} + \log{\left(\frac{x + 1}{2 x - 1} \right)}\right) = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=78.4909928314846x_{1} = 78.4909928314846
x2=128.487815526633x_{2} = 128.487815526633
x3=112.48830587262x_{3} = 112.48830587262
x4=114.48823081463x_{4} = 114.48823081463
x5=120.488031723741x_{5} = 120.488031723741
x6=106.488562914107x_{6} = 106.488562914107
x7=74.4916787095585x_{7} = 74.4916787095585
x8=58.4970311808328x_{8} = 58.4970311808328
x9=124.487917476664x_{9} = 124.487917476664
x10=118.48809409066x_{10} = 118.48809409066
x11=52.5015859524306x_{11} = 52.5015859524306
x12=86.4899783694111x_{12} = 86.4899783694111
x13=116.488160343558x_{13} = 116.488160343558
x14=54.4997712773588x_{14} = 54.4997712773588
x15=42.5217350910816x_{15} = 42.5217350910816
x16=56.4982775703649x_{16} = 56.4982775703649
x17=66.4936388270172x_{17} = 66.4936388270172
x18=104.488660930511x_{18} = 104.488660930511
x19=92.4894290548019x_{19} = 92.4894290548019
x20=90.489596591564x_{20} = 90.489596591564
x21=126.487865079088x_{21} = 126.487865079088
x22=46.5102782564629x_{22} = 46.5102782564629
x23=62.4950813502932x_{23} = 62.4950813502932
x24=44.5150972769133x_{24} = 44.5150972769133
x25=70.4925386396084x_{25} = 70.4925386396084
x26=76.491317193376x_{26} = 76.491317193376
x27=98.4890010939741x_{27} = 98.4890010939741
x28=72.4920834126443x_{28} = 72.4920834126443
x29=48.5066457313568x_{29} = 48.5066457313568
x30=122.487972942868x_{30} = 122.487972942868
x31=100.488879245506x_{31} = 100.488879245506
x32=102.488766131894x_{32} = 102.488766131894
x33=82.4904363941597x_{33} = 82.4904363941597
x34=108.488471437747x_{34} = 108.488471437747
x35=50.5038262641685x_{35} = 50.5038262641685
x36=68.4930533930123x_{36} = 68.4930533930123
x37=84.4901966327898x_{37} = 84.4901966327898
x38=60.4959788878618x_{38} = 60.4959788878618
x39=110.4883859278x_{39} = 110.4883859278
x40=94.4892748594052x_{40} = 94.4892748594052
x41=64.4943089109692x_{41} = 64.4943089109692
x42=80.4907006235634x_{42} = 80.4907006235634
x43=−1.5769193563577x_{43} = -1.5769193563577
x44=130.487768615883x_{44} = 130.487768615883
x45=88.4897790765283x_{45} = 88.4897790765283
x46=96.4891326108787x_{46} = 96.4891326108787
Signos de extremos en los puntos:
(78.49099283148459, 1.05014634346105e-23)

(128.48781552663283, 9.36477770172146e-39)

(112.4883058726204, 6.13271093478345e-34)

(114.48823081462974, 1.53334523071786e-34)

(120.48803172374079, 2.39656682358129e-36)

(106.48856291410704, 3.92351212097174e-32)

(74.49167870955849, 1.67901594213546e-22)

(58.49703118083283, 1.09482244528553e-17)

(124.48791747666395, 1.4981202637301e-37)

(118.48809409066045, 9.58535770084247e-36)

(52.50158595243062, 6.97984717665841e-16)

(86.48997836941109, 4.10679029620736e-26)

(116.48816034355843, 3.83376220925777e-35)

(54.49977127735879, 1.74759883463734e-16)

(42.52173509108155, 7.03692167546217e-13)

(56.49827757036489, 4.37454977332933e-17)

(66.49363882701724, 4.28991951827825e-20)

(104.48866093051102, 1.56919444569848e-31)

(92.48942905480189, 6.42106922434544e-28)

(90.48959659156404, 2.56790431343069e-27)

(126.48786507908787, 3.74561226241293e-38)

(46.510278256462925, 4.4363474658769e-14)

(62.49508135029324, 6.85461347843343e-19)

(44.51509727691328, 1.76800990576483e-13)

(70.49253863960843, 2.68407895727006e-21)

(76.49131719337599, 4.19912993910605e-23)

(98.48900109397407, 1.00383248216796e-29)

(72.49208341264426, 6.71327284364859e-22)

(48.50664573135677, 1.11223670414643e-14)

(122.48797294286774, 5.99196068969012e-37)

(100.48887924550584, 2.50997957083157e-30)

(102.48876613189415, 6.27588899085674e-31)

(82.49043639415972, 6.56743029975234e-25)

(108.48847143774695, 9.81002732764053e-33)

(50.503826264168545, 2.78684927704995e-15)

(68.49305339301229, 1.07308665493174e-20)

(84.49019663278976, 1.64230296853545e-25)

(60.49597888786176, 2.73961427704645e-18)

(110.48838592779981, 2.45280287781477e-33)

(94.4892748594052, 1.60557374123156e-28)

(64.49430891096922, 1.71488017503667e-19)

(80.49070062356336, 2.62620160864513e-24)

(-1.5769193563577026, 22.4878221663397)

(130.48776861588257, 2.34137367921008e-39)

(88.48977907652834, 1.02693775457712e-26)

(96.48913261087871, 4.01465351063248e-29)


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:
Puntos mínimos de la función:
x1=−1.5769193563577x_{1} = -1.5769193563577
La función no tiene puntos máximos
Decrece en los intervalos
[−1.5769193563577,∞)\left[-1.5769193563577, \infty\right)
Crece en los intervalos
(−∞,−1.5769193563577]\left(-\infty, -1.5769193563577\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+12x−1)x((x(2(x+1)2x−1−1)x+1−log⁡(x+12x−1))2+(2(x+1)2x−1−1)(2x2x−1+xx+1−2)x+1)=0\left(\frac{x + 1}{2 x - 1}\right)^{x} \left(\left(\frac{x \left(\frac{2 \left(x + 1\right)}{2 x - 1} - 1\right)}{x + 1} - \log{\left(\frac{x + 1}{2 x - 1} \right)}\right)^{2} + \frac{\left(\frac{2 \left(x + 1\right)}{2 x - 1} - 1\right) \left(\frac{2 x}{2 x - 1} + \frac{x}{x + 1} - 2\right)}{x + 1}\right) = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=112.488361769439x_{1} = 112.488361769439
x2=94.4893816335819x_{2} = 94.4893816335819
x3=58.4978551931985x_{3} = 58.4978551931985
x4=80.4909008890193x_{4} = 80.4909008890193
x5=86.4901287101181x_{5} = 86.4901287101181
x6=88.4899165499547x_{6} = 88.4899165499547
x7=108.488535211517x_{7} = 108.488535211517
x8=44.5191380920686x_{8} = 44.5191380920686
x9=116.488209617178x_{9} = 116.488209617178
x10=114.488283259663x_{10} = 114.488283259663
x11=76.4915639596199x_{11} = 76.4915639596199
x12=106.488631191204x_{12} = 106.488631191204
x13=78.4912147027176x_{13} = 78.4912147027176
x14=54.5009548505291x_{14} = 54.5009548505291
x15=64.4948269632x_{15} = 64.4948269632
x16=118.488140445162x_{16} = 118.488140445162
x17=46.5132739997232x_{17} = 46.5132739997232
x18=90.4897226422455x_{18} = 90.4897226422455
x19=56.4992589376319x_{19} = 56.4992589376319
x20=52.5030345548304x_{20} = 52.5030345548304
x21=128.487850296657x_{21} = 128.487850296657
x22=130.487801549111x_{22} = 130.487801549111
x23=124.487956354075x_{23} = 124.487956354075
x24=126.487901825944x_{24} = 126.487901825944
x25=42.5274230818693x_{25} = 42.5274230818693
x26=98.4890923649325x_{26} = 98.4890923649325
x27=84.4903615246397x_{27} = 84.4903615246397
x28=82.4906178102729x_{28} = 82.4906178102729
x29=70.4928873710458x_{29} = 70.4928873710458
x30=50.5056304580541x_{30} = 50.5056304580541
x31=66.4940899117473x_{31} = 66.4940899117473
x32=102.488844785967x_{32} = 102.488844785967
x33=68.4934488136196x_{33} = 68.4934488136196
x34=104.48873414931x_{34} = 104.48873414931
x35=72.4923926620456x_{35} = 72.4923926620456
x36=122.488014119897x_{36} = 122.488014119897
x37=60.496678356244x_{37} = 60.496678356244
x38=92.4895449280203x_{38} = 92.4895449280203
x39=74.4919543237358x_{39} = 74.4919543237358
x40=48.5089406247563x_{40} = 48.5089406247563
x41=96.4892312239376x_{41} = 96.4892312239376
x42=100.488963892263x_{42} = 100.488963892263
x43=120.488075386671x_{43} = 120.488075386671
x44=110.488445588716x_{44} = 110.488445588716
x45=62.4956807863732x_{45} = 62.4956807863732
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=0.5x_{1} = 0.5

lim⁡x→0.5−((x+12x−1)x((x(2(x+1)2x−1−1)x+1−log⁡(x+12x−1))2+(2(x+1)2x−1−1)(2x2x−1+xx+1−2)x+1))=∞i\lim_{x \to 0.5^-}\left(\left(\frac{x + 1}{2 x - 1}\right)^{x} \left(\left(\frac{x \left(\frac{2 \left(x + 1\right)}{2 x - 1} - 1\right)}{x + 1} - \log{\left(\frac{x + 1}{2 x - 1} \right)}\right)^{2} + \frac{\left(\frac{2 \left(x + 1\right)}{2 x - 1} - 1\right) \left(\frac{2 x}{2 x - 1} + \frac{x}{x + 1} - 2\right)}{x + 1}\right)\right) = \infty i
lim⁡x→0.5+((x+12x−1)x((x(2(x+1)2x−1−1)x+1−log⁡(x+12x−1))2+(2(x+1)2x−1−1)(2x2x−1+xx+1−2)x+1))=∞\lim_{x \to 0.5^+}\left(\left(\frac{x + 1}{2 x - 1}\right)^{x} \left(\left(\frac{x \left(\frac{2 \left(x + 1\right)}{2 x - 1} - 1\right)}{x + 1} - \log{\left(\frac{x + 1}{2 x - 1} \right)}\right)^{2} + \frac{\left(\frac{2 \left(x + 1\right)}{2 x - 1} - 1\right) \left(\frac{2 x}{2 x - 1} + \frac{x}{x + 1} - 2\right)}{x + 1}\right)\right) = \infty
- los límites no son iguales, signo
x1=0.5x_{1} = 0.5
- 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=0.5x_{1} = 0.5
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(x+12x−1)x=∞\lim_{x \to -\infty} \left(\frac{x + 1}{2 x - 1}\right)^{x} = \infty
Tomamos como el límite
es decir,
no hay asíntota horizontal a la izquierda
lim⁡x→∞(x+12x−1)x=0\lim_{x \to \infty} \left(\frac{x + 1}{2 x - 1}\right)^{x} = 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)/(-1 + 2*x))^x, dividida por x con x->+oo y x ->-oo
lim⁡x→−∞((x+12x−1)xx)=−∞\lim_{x \to -\infty}\left(\frac{\left(\frac{x + 1}{2 x - 1}\right)^{x}}{x}\right) = -\infty
Tomamos como el límite
es decir,
no hay asíntota inclinada a la izquierda
lim⁡x→∞((x+12x−1)xx)=0\lim_{x \to \infty}\left(\frac{\left(\frac{x + 1}{2 x - 1}\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:
(x+12x−1)x=(1−x−2x−1)−x\left(\frac{x + 1}{2 x - 1}\right)^{x} = \left(\frac{1 - x}{- 2 x - 1}\right)^{- x}
- No
(x+12x−1)x=−(1−x−2x−1)−x\left(\frac{x + 1}{2 x - 1}\right)^{x} = - \left(\frac{1 - x}{- 2 x - 1}\right)^{- x}
- No
es decir, función
no es
par ni impar
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