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

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

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
               x        
f(x) = -----------------
        /       2\      
        \(x - 1) /      
       x                
       -----------*2 - 1
          x + 2
f(x)=x2x(x−1)2x+2−1f{\left(x \right)} = \frac{x}{2 \frac{x^{\left(x - 1\right)^{2}}}{x + 2} - 1} 
f = x/(2*(x^((x - 1)^2)/(x + 2)) - 1)
Gráfico de la función
02468-8-6-4-2-1010-5050
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:
x2x(x−1)2x+2−1=0\frac{x}{2 \frac{x^{\left(x - 1\right)^{2}}}{x + 2} - 1} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=0x_{1} = 0
Solución numérica
x1=−80x_{1} = -80
x2=−70x_{2} = -70
x3=−98x_{3} = -98
x4=14.4933422109288x_{4} = 14.4933422109288
x5=92.25x_{5} = 92.25
x6=24.3688825015203x_{6} = 24.3688825015203
x7=22.3831848241789x_{7} = 22.3831848241789
x8=50.25x_{8} = 50.25
x9=−62x_{9} = -62
x10=80.25x_{10} = 80.25
x11=62.25x_{11} = 62.25
x12=−72x_{12} = -72
x13=−48x_{13} = -48
x14=96.25x_{14} = 96.25
x15=−74x_{15} = -74
x16=16.4531867266646x_{16} = 16.4531867266646
x17=78.25x_{17} = 78.25
x18=52.25x_{18} = 52.25
x19=−56x_{19} = -56
x20=68.25x_{20} = 68.25
x21=94.25x_{21} = 94.25
x22=−58x_{22} = -58
x23=−42x_{23} = -42
x24=88.25x_{24} = 88.25
x25=34.25x_{25} = 34.25
x26=42.25x_{26} = 42.25
x27=−76x_{27} = -76
x28=−44x_{28} = -44
x29=74.25x_{29} = 74.25
x30=76.25x_{30} = 76.25
x31=64.25x_{31} = 64.25
x32=12.5503750619372x_{32} = 12.5503750619372
x33=−94x_{33} = -94
x34=8.77782887226638x_{34} = 8.77782887226638
x35=7.03874975320766x_{35} = 7.03874975320766
x36=−32.0011367361208x_{36} = -32.0011367361208
x37=−36.0000000382316x_{37} = -36.0000000382316
x38=−100x_{38} = -100
x39=−96x_{39} = -96
x40=38.25x_{40} = 38.25
x41=10.6364816145142x_{41} = 10.6364816145142
x42=−78x_{42} = -78
x43=−66x_{43} = -66
x44=−34.0000055400673x_{44} = -34.0000055400673
x45=−86x_{45} = -86
x46=−50x_{46} = -50
x47=72.25x_{47} = 72.25
x48=−54x_{48} = -54
x49=60.25x_{49} = 60.25
x50=18.4235826876456x_{50} = 18.4235826876456
x51=−64x_{51} = -64
x52=−60x_{52} = -60
x53=100.25x_{53} = 100.25
x54=90.25x_{54} = 90.25
x55=−84x_{55} = -84
x56=32.25x_{56} = 32.25
x57=26.3571549429987x_{57} = 26.3571549429987
x58=−52x_{58} = -52
x59=28.2548020806379x_{59} = 28.2548020806379
x60=58.25x_{60} = 58.25
x61=−46x_{61} = -46
x62=−82x_{62} = -82
x63=20.4009642128405x_{63} = 20.4009642128405
x64=86.25x_{64} = 86.25
x65=84.25x_{65} = 84.25
x66=−68x_{66} = -68
x67=−38.0000000001966x_{67} = -38.0000000001966
x68=0x_{68} = 0
x69=66.25x_{69} = 66.25
x70=36.25x_{70} = 36.25
x71=5.60109197850222x_{71} = 5.60109197850222
x72=70.25x_{72} = 70.25
x73=−88x_{73} = -88
x74=40.25x_{74} = 40.25
x75=46.25x_{75} = 46.25
x76=−92x_{76} = -92
x77=98.25x_{77} = 98.25
x78=44.25x_{78} = 44.25
x79=56.25x_{79} = 56.25
x80=30.2547609196464x_{80} = 30.2547609196464
x81=82.25x_{81} = 82.25
x82=−40.0000000000017x_{82} = -40.0000000000017
x83=48.25x_{83} = 48.25
x84=54.25x_{84} = 54.25
x85=−90x_{85} = -90
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 x/((x^((x - 1)^2)/(x + 2))*2 - 1).
0−1+20(−1)22\frac{0}{-1 + 2 \frac{0^{\left(-1\right)^{2}}}{2}}
Resultado:
f(0)=0f{\left(0 \right)} = 0
Punto:
(0, 0)
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(−2x(x−1)2((2x−2)log⁡(x)+(x−1)2x)x+2+2x(x−1)2(x+2)2)(2x(x−1)2x+2−1)2+12x(x−1)2x+2−1=0\frac{x \left(- \frac{2 x^{\left(x - 1\right)^{2}} \left(\left(2 x - 2\right) \log{\left(x \right)} + \frac{\left(x - 1\right)^{2}}{x}\right)}{x + 2} + \frac{2 x^{\left(x - 1\right)^{2}}}{\left(x + 2\right)^{2}}\right)}{\left(2 \frac{x^{\left(x - 1\right)^{2}}}{x + 2} - 1\right)^{2}} + \frac{1}{2 \frac{x^{\left(x - 1\right)^{2}}}{x + 2} - 1} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=24.3689202107084x_{1} = 24.3689202107084
x2=18.4236923269034x_{2} = 18.4236923269034
x3=8.78024047675981x_{3} = 8.78024047675981
x4=12.5508840955655x_{4} = 12.5508840955655
x5=−80x_{5} = -80
x6=92.25x_{6} = 92.25
x7=30.2547621862745x_{7} = 30.2547621862745
x8=50.25x_{8} = 50.25
x9=−96x_{9} = -96
x10=−72x_{10} = -72
x11=−90x_{11} = -90
x12=80.25x_{12} = 80.25
x13=62.25x_{13} = 62.25
x14=−34x_{14} = -34
x15=−94x_{15} = -94
x16=−42x_{16} = -42
x17=96.25x_{17} = 96.25
x18=−78x_{18} = -78
x19=78.25x_{19} = 78.25
x20=68.25x_{20} = 68.25
x21=94.25x_{21} = 94.25
x22=−46x_{22} = -46
x23=88.25x_{23} = 88.25
x24=34.25x_{24} = 34.25
x25=−88x_{25} = -88
x26=−98x_{26} = -98
x27=−62x_{27} = -62
x28=−56x_{28} = -56
x29=42.25x_{29} = 42.25
x30=74.25x_{30} = 74.25
x31=76.25x_{31} = 76.25
x32=64.25x_{32} = 64.25
x33=−86x_{33} = -86
x34=−60x_{34} = -60
x35=10.6375059265676x_{35} = 10.6375059265676
x36=−48x_{36} = -48
x37=38.25x_{37} = 38.25
x38=−92x_{38} = -92
x39=−52x_{39} = -52
x40=26.3571830309308x_{40} = 26.3571830309308
x41=−54x_{41} = -54
x42=−38x_{42} = -38
x43=72.25x_{43} = 72.25
x44=60.25x_{44} = 60.25
x45=−58x_{45} = -58
x46=22.3832367976476x_{46} = 22.3832367976476
x47=100.25x_{47} = 100.25
x48=90.25x_{48} = 90.25
x49=−68x_{49} = -68
x50=−70x_{50} = -70
x51=32.25x_{51} = 32.25
x52=−64x_{52} = -64
x53=28.2548039799089x_{53} = 28.2548039799089
x54=−76x_{54} = -76
x55=14.4936250517724x_{55} = 14.4936250517724
x56=−100x_{56} = -100
x57=58.25x_{57} = 58.25
x58=5.62988874456936x_{58} = 5.62988874456936
x59=−66x_{59} = -66
x60=86.25x_{60} = 86.25
x61=84.25x_{61} = 84.25
x62=66.25x_{62} = 66.25
x63=36.25x_{63} = 36.25
x64=70.25x_{64} = 70.25
x65=40.25x_{65} = 40.25
x66=16.4533573770923x_{66} = 16.4533573770923
x67=46.25x_{67} = 46.25
x68=−50x_{68} = -50
x69=7.04587955082479x_{69} = 7.04587955082479
x70=−82x_{70} = -82
x71=20.4010382169648x_{71} = 20.4010382169648
x72=−74x_{72} = -74
x73=98.25x_{73} = 98.25
x74=−84x_{74} = -84
x75=44.25x_{75} = 44.25
x76=56.25x_{76} = 56.25
x77=82.25x_{77} = 82.25
x78=48.25x_{78} = 48.25
x79=54.25x_{79} = 54.25
x80=52.25x_{80} = 52.25
Signos de extremos en los puntos:
(24.368920210708364, 1.40179010335888e-755)

(18.423692326903414, 1.3336293498273e-382)

(8.780240476759813, 3.65292663698181e-56)

(12.550884095565545, 2.35610267646134e-145)

(-80, -2.09290704497434e-12483)

(92.25, 1.67065705294236e-16358)

(30.25476218627452, 2.31310616505108e-1265)

(50.25, 8.06014869203219e-4124)

(-96, -2.91297945553796e-18648)

(-72, -4.74817873366383e-9895)

(-90, -3.27698549362478e-16180)

(80.25, 3.40755326675673e-11958)

(62.25, 2.98576497773342e-6728)

(-34, -4.71982094218372e-1874)

(-94, -1.43551239901873e-17804)

(-42, -3.43828078629602e-2999)

(96.25, 1.40457072556546e-17991)

(-78, -8.11825723070333e-11806)

(78.25, 1.02499220158807e-11296)

(68.25, 3.43421207997117e-8292)

(94.25, 1.47809856134539e-17164)

(-46, -9.40005355479421e-3671)

(88.25, 5.31207231578783e-14809)

(34.25, 1.34820435830336e-1694)

(-88, -2.1414356393293e-15399)

(-98, -4.63126306965719e-19513)

(-62, -1.84883147316621e-7111)

(-56, -2.07325178525103e-5677)

(42.25, 3.34921607841906e-2764)

(74.25, 1.29760479109698e-10034)

(76.25, 1.49250158662435e-10655)

(64.25, 6.58525154453729e-7230)

(-86, -2.18258054193439e-14639)

(-60, -5.49220286990291e-6614)

(10.637505926567604, 2.83810118615477e-94)

(-48, -2.41411600102927e-4034)

(38.25, 7.69716101744558e-2194)

(-92, -6.54759230176509e-16982)

(-52, -7.25355790155216e-4818)

(26.357183030930827, 8.9811816500664e-912)

(-54, -4.53151731015615e-5238)

(-38, -9.64285524967819e-2401)

(72.25, 8.35553323161847e-9434)

(60.25, 4.19363562912984e-6246)

(-58, -5.22084058270641e-6136)

(22.383236797647573, 1.56113717809856e-615)

(100.25, 7.99745876372152e-19709)

(90.25, 2.41219427532564e-15573)

(-68, -5.97158279624976e-8722)

(-70, -1.72725812862493e-9298)

(32.25, 3.7034511135961e-1471)

(-64, -1.5386916106158e-7628)

(28.25480397990893, 5.29880109711651e-1076)

(-76, -1.27416796981098e-11148)

(14.493625051772359, 4.49541374058221e-210)

(-100, -4.9e-20399)

(58.25, 2.37948189284444e-5783)

(5.6298887445693575, 1.75538249917869e-15)

(-66, -2.46585347424553e-8165)

(86.25, 2.13881100036564e-14065)

(84.25, 1.8953647873386e-13342)

(66.25, 3.47967928570283e-7751)

(36.25, 2.01024574755113e-1935)

(70.25, 4.97244624946093e-8853)

(40.25, 4.86877099968678e-2470)

(16.453357377092285, 5.39648790343141e-289)

(46.25, 4.2386534892034e-3407)

(-50, -1.14340468683616e-4416)

(7.045879550824791, 3.22900564197826e-30)

(-82, -1.78749473916041e-13181)

(20.401038216964768, 2.54333993491323e-491)

(-74, -9.93369299376363e-10512)

(98.25, 1.20973144156964e-18839)

(-84, -4.1612264055075e-13900)

(44.25, 1.68400930144634e-3076)

(56.25, 7.17782385658809e-5340)

(82.25, 4.46984291771843e-12640)

(48.25, 3.72068656061829e-3756)

(54.25, 1.52975673056248e-4915)

(52.25, 3.09330255593455e-4510)


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
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(x2x(x−1)2x+2−1)=∞\lim_{x \to -\infty}\left(\frac{x}{2 \frac{x^{\left(x - 1\right)^{2}}}{x + 2} - 1}\right) = \infty
Tomamos como el límite
es decir,
no hay asíntota horizontal a la izquierda
lim⁡x→∞(x2x(x−1)2x+2−1)=0\lim_{x \to \infty}\left(\frac{x}{2 \frac{x^{\left(x - 1\right)^{2}}}{x + 2} - 1}\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 x/((x^((x - 1)^2)/(x + 2))*2 - 1), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞12x(x−1)2x+2−1=−1\lim_{x \to -\infty} \frac{1}{2 \frac{x^{\left(x - 1\right)^{2}}}{x + 2} - 1} = -1
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
y=−xy = - x
lim⁡x→∞12x(x−1)2x+2−1=0\lim_{x \to \infty} \frac{1}{2 \frac{x^{\left(x - 1\right)^{2}}}{x + 2} - 1} = 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:
x2x(x−1)2x+2−1=−x2(−x)(−x−1)22−x−1\frac{x}{2 \frac{x^{\left(x - 1\right)^{2}}}{x + 2} - 1} = - \frac{x}{\frac{2 \left(- x\right)^{\left(- x - 1\right)^{2}}}{2 - x} - 1}
- No
x2x(x−1)2x+2−1=x2(−x)(−x−1)22−x−1\frac{x}{2 \frac{x^{\left(x - 1\right)^{2}}}{x + 2} - 1} = \frac{x}{\frac{2 \left(- x\right)^{\left(- x - 1\right)^{2}}}{2 - x} - 1}
- No
es decir, función
no es
par ni impar
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