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Trazado el gráfico de la función/ (2-cos(x))^(x/3)

Gráfico de la función y = (2-cos(x))^(x/3)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
                   x
                   -
                   3
f(x) = (2 - cos(x))
f(x)=(2−cos⁡(x))x3f{\left(x \right)} = \left(2 - \cos{\left(x \right)}\right)^{\frac{x}{3}} 
f = (2 - cos(x))^(x/3)
Gráfico de la función
02468-8-6-4-2-1010050
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:
(2−cos⁡(x))x3=0\left(2 - \cos{\left(x \right)}\right)^{\frac{x}{3}} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=−97.4314646436932x_{1} = -97.4314646436932
x2=−209.229112917807x_{2} = -209.229112917807
x3=−97.4918426847792x_{3} = -97.4918426847792
x4=−98.557336816799x_{4} = -98.557336816799
x5=−203.978621505412x_{5} = -203.978621505412
x6=−97.2336266312007x_{6} = -97.2336266312007
x7=−91.4777702500747x_{7} = -91.4777702500747
x8=−78.332530414984x_{8} = -78.332530414984
x9=−85.1138011294668x_{9} = -85.1138011294668
x10=−97.9417659402468x_{10} = -97.9417659402468
x11=−280.303024050475x_{11} = -280.303024050475
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 (2 - cos(x))^(x/3).
(2−cos⁡(0))03\left(2 - \cos{\left(0 \right)}\right)^{\frac{0}{3}}
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
(2−cos⁡(x))x3(xsin⁡(x)3(2−cos⁡(x))+log⁡(2−cos⁡(x))3)=0\left(2 - \cos{\left(x \right)}\right)^{\frac{x}{3}} \left(\frac{x \sin{\left(x \right)}}{3 \left(2 - \cos{\left(x \right)}\right)} + \frac{\log{\left(2 - \cos{\left(x \right)} \right)}}{3}\right) = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=31.4159265358979x_{1} = 31.4159265358979
x2=−72.3022008575693x_{2} = -72.3022008575693
x3=75.398223686155x_{3} = 75.398223686155
x4=−53.4686801390812x_{4} = -53.4686801390812
x5=2.64255930932882⋅10−7x_{5} = 2.64255930932882 \cdot 10^{-7}
x6=−66.0233461340613x_{6} = -66.0233461340613
x7=3.91047799019543x_{7} = 3.91047799019543
x8=−6.28318530717959x_{8} = -6.28318530717959
x9=−3.91047799019543x_{9} = -3.91047799019543
x10=9.75694089241645x_{10} = 9.75694089241645
x11=6.28318530717959x_{11} = 6.28318530717959
x12=62.8318530717959x_{12} = 62.8318530717959
x13=−28.3901883107888x_{13} = -28.3901883107888
x14=28.3901883107888x_{14} = 28.3901883107888
x15=94.2477796076938x_{15} = 94.2477796076938
x16=−78.5817466605474x_{16} = -78.5817466605474
x17=−37.6991118430775x_{17} = -37.6991118430775
x18=−22.1395185101973x_{18} = -22.1395185101973
x19=−40.9211665372733x_{19} = -40.9211665372733
x20=−43.9822971502571x_{20} = -43.9822971502571
x21=25.1327412287183x_{21} = 25.1327412287183
x22=−34.6525001476063x_{22} = -34.6525001476063
x23=87.9645943005142x_{23} = 87.9645943005142
x24=−59.7453996738558x_{24} = -59.7453996738558
x25=22.1395185101973x_{25} = 22.1395185101973
x26=66.0233461340601x_{26} = 66.0233461340601
x27=−78.8514291440376x_{27} = -78.8514291440376
x28=43.9822971502571x_{28} = 43.9822971502571
x29=−9.75694089241645x_{29} = -9.75694089241645
x30=18.8495559215388x_{30} = 18.8495559215388
x31=−91.1423412078634x_{31} = -91.1423412078634
x32=−15.9137412421555x_{32} = -15.9137412421555
x33=12.5663706143592x_{33} = 12.5663706143592
x34=−66.0233461340601x_{34} = -66.0233461340601
x35=81.6814089933346x_{35} = 81.6814089933346
x36=−84.8618304472507x_{36} = -84.8618304472507
x37=50.2654824574367x_{37} = 50.2654824574367
x38=−47.1936746291492x_{38} = -47.1936746291492
x39=−87.9645943005142x_{39} = -87.9645943005142
x40=56.5486677646163x_{40} = 56.5486677646163
x41=72.3022008575693x_{41} = 72.3022008575693
x42=−97.4231964937992x_{42} = -97.4231964937992
x43=37.6991118430775x_{43} = 37.6991118430775
x44=100.530964914873x_{44} = 100.530964914873
x45=69.1150383789755x_{45} = 69.1150383789755
x46=0x_{46} = 0
x47=15.9137412421555x_{47} = 15.9137412421555
x48=−53.4686801390811x_{48} = -53.4686801390811
Signos de extremos en los puntos:
(31.41592653589793, 1)

(-72.30220085756935, 3.19632047214314e-12)

(75.39822368615503, 1)

(-53.46868013908118, 3.17113530230906e-9)

(2.6425593093288245e-07, 1)

(-66.02334613406126, 3.1884969749552e-11)

(3.9104779901954276, 3.68284866652983)

(-6.283185307179586, 1)

(-3.9104779901954276, 0.271528941465372)

(9.756940892416452, 33.5563882845969)

(6.283185307179586, 1)

(62.83185307179586, 1)

(-28.39018831078879, 3.11893692669164e-5)

(28.39018831078879, 32062.2065628218)

(94.2477796076938, 1)

(-78.58174666054741, 3.20375696729864e-13)

(-37.69911184307752, 1)

(-22.139518510197316, 0.00030951431783095)

(-40.92116653727331, 3.14972497482264e-7)

(-43.982297150257104, 1)

(25.132741228718345, 1)

(-34.652500147606325, 3.13613287080351e-6)

(87.96459430051421, 1)

(-59.745399673855765, 3.18016249065744e-10)

(22.139518510197316, 3230.86830686191)

(66.02334613406013, 31362739493.0821)

(-78.85142914403762, 4.40725986756905e-13)

(43.982297150257104, 1)

(-9.756940892416452, 0.0298005849592288)

(18.84955592153876, 1)

(-91.14234120786342, 3.21779489738191e-15)

(-15.913741242155513, 0.00305724635040484)

(12.566370614359172, 1)

(-66.02334613406013, 3.1884969749552e-11)

(81.68140899333463, 1)

(-84.8618304472507, 3.21089394514855e-14)

(50.26548245743669, 1)

(-47.193674629149186, 3.16113798414191e-8)

(-87.96459430051421, 1)

(56.548667764616276, 1)

(72.30220085756935, 312859742543.118)

(-97.42319649379918, 3.22450701859755e-16)

(37.69911184307752, 1)

(100.53096491487338, 1)

(69.11503837897546, 1)

(0, 1)

(15.913741242155513, 327.091730722838)

(-53.46868013908108, 3.17113530230905e-9)


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=31.4159265358979x_{1} = 31.4159265358979
x2=−72.3022008575693x_{2} = -72.3022008575693
x3=75.398223686155x_{3} = 75.398223686155
x4=−53.4686801390812x_{4} = -53.4686801390812
x5=−66.0233461340613x_{5} = -66.0233461340613
x6=−3.91047799019543x_{6} = -3.91047799019543
x7=6.28318530717959x_{7} = 6.28318530717959
x8=62.8318530717959x_{8} = 62.8318530717959
x9=−28.3901883107888x_{9} = -28.3901883107888
x10=94.2477796076938x_{10} = 94.2477796076938
x11=−78.5817466605474x_{11} = -78.5817466605474
x12=−22.1395185101973x_{12} = -22.1395185101973
x13=−40.9211665372733x_{13} = -40.9211665372733
x14=25.1327412287183x_{14} = 25.1327412287183
x15=−34.6525001476063x_{15} = -34.6525001476063
x16=87.9645943005142x_{16} = 87.9645943005142
x17=−59.7453996738558x_{17} = -59.7453996738558
x18=43.9822971502571x_{18} = 43.9822971502571
x19=−9.75694089241645x_{19} = -9.75694089241645
x20=18.8495559215388x_{20} = 18.8495559215388
x21=−91.1423412078634x_{21} = -91.1423412078634
x22=−15.9137412421555x_{22} = -15.9137412421555
x23=12.5663706143592x_{23} = 12.5663706143592
x24=−66.0233461340601x_{24} = -66.0233461340601
x25=81.6814089933346x_{25} = 81.6814089933346
x26=−84.8618304472507x_{26} = -84.8618304472507
x27=50.2654824574367x_{27} = 50.2654824574367
x28=−47.1936746291492x_{28} = -47.1936746291492
x29=56.5486677646163x_{29} = 56.5486677646163
x30=−97.4231964937992x_{30} = -97.4231964937992
x31=37.6991118430775x_{31} = 37.6991118430775
x32=100.530964914873x_{32} = 100.530964914873
x33=69.1150383789755x_{33} = 69.1150383789755
x34=−53.4686801390811x_{34} = -53.4686801390811
Puntos máximos de la función:
x34=3.91047799019543x_{34} = 3.91047799019543
x34=−6.28318530717959x_{34} = -6.28318530717959
x34=9.75694089241645x_{34} = 9.75694089241645
x34=28.3901883107888x_{34} = 28.3901883107888
x34=−37.6991118430775x_{34} = -37.6991118430775
x34=−43.9822971502571x_{34} = -43.9822971502571
x34=22.1395185101973x_{34} = 22.1395185101973
x34=66.0233461340601x_{34} = 66.0233461340601
x34=−87.9645943005142x_{34} = -87.9645943005142
x34=72.3022008575693x_{34} = 72.3022008575693
x34=15.9137412421555x_{34} = 15.9137412421555
Decrece en los intervalos
[100.530964914873,∞)\left[100.530964914873, \infty\right)
Crece en los intervalos
(−∞,−97.4231964937992]\left(-\infty, -97.4231964937992\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
(2−cos⁡(x))x3((xsin⁡(x)cos⁡(x)−2−log⁡(2−cos⁡(x)))2−3(xcos⁡(x)+xsin⁡2(x)cos⁡(x)−2+2sin⁡(x))cos⁡(x)−2)9=0\frac{\left(2 - \cos{\left(x \right)}\right)^{\frac{x}{3}} \left(\left(\frac{x \sin{\left(x \right)}}{\cos{\left(x \right)} - 2} - \log{\left(2 - \cos{\left(x \right)} \right)}\right)^{2} - \frac{3 \left(x \cos{\left(x \right)} + \frac{x \sin^{2}{\left(x \right)}}{\cos{\left(x \right)} - 2} + 2 \sin{\left(x \right)}\right)}{\cos{\left(x \right)} - 2}\right)}{9} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−121.610735760973x_{1} = -121.610735760973
x2=8.83980914554789x_{2} = 8.83980914554789
x3=60.1330316888051x_{3} = 60.1330316888051
x4=−91.1649242745032x_{4} = -91.1649242745032
x5=−43.7270981123841x_{5} = -43.7270981123841
x6=27.8299149564093x_{6} = 27.8299149564093
x7=−12.1194500881031x_{7} = -12.1194500881031
x8=−1.50216216202101x_{8} = -1.50216216202101
x9=−31.7156015160862x_{9} = -31.7156015160862
x10=−97.6530576077939x_{10} = -97.6530576077939
x11=−81.870745512552x_{11} = -81.870745512552
x12=71.9496479762765x_{12} = 71.9496479762765
x13=34.1445368219039x_{13} = 34.1445368219039
x14=−84.8431686016826x_{14} = -84.8431686016826
x15=53.8782982466425x_{15} = 53.8782982466425
x16=−5.72363505593024x_{16} = -5.72363505593024
x17=−37.9740248058154x_{17} = -37.9740248058154
x18=5.03264783880843x_{18} = 5.03264783880843
x19=10.6644557223496x_{19} = 10.6644557223496
x20=−252.984045439947x_{20} = -252.984045439947
x21=−91.1075173083713x_{21} = -91.1075173083713
x22=−605.810241192968x_{22} = -605.810241192968
x23=−97.0735610941528x_{23} = -97.0735610941528
x24=2.76740685058427x_{24} = 2.76740685058427
x25=16.6504811513294x_{25} = 16.6504811513294
x26=−50.0260588504615x_{26} = -50.0260588504615
x27=66.3921760681942x_{27} = 66.3921760681942
x28=−84.8816814335199x_{28} = -84.8816814335199
x29=−91.5102826400197x_{29} = -91.5102826400197
x30=0x_{30} = 0
x31=−87.7820187684829x_{31} = -87.7820187684829

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
[66.3921760681942,∞)\left[66.3921760681942, \infty\right)
Convexa en los intervalos
(−∞,−87.7820187684829]\left(-\infty, -87.7820187684829\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(2−cos⁡(x))x3=⟨0,∞⟩\lim_{x \to -\infty} \left(2 - \cos{\left(x \right)}\right)^{\frac{x}{3}} = \left\langle 0, \infty\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=⟨0,∞⟩y = \left\langle 0, \infty\right\rangle
lim⁡x→∞(2−cos⁡(x))x3=⟨0,∞⟩\lim_{x \to \infty} \left(2 - \cos{\left(x \right)}\right)^{\frac{x}{3}} = \left\langle 0, \infty\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=⟨0,∞⟩y = \left\langle 0, \infty\right\rangle
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (2 - cos(x))^(x/3), dividida por x con x->+oo y x ->-oo
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
y=xlim⁡x→−∞((2−cos⁡(x))x3x)y = x \lim_{x \to -\infty}\left(\frac{\left(2 - \cos{\left(x \right)}\right)^{\frac{x}{3}}}{x}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞((2−cos⁡(x))x3x)y = x \lim_{x \to \infty}\left(\frac{\left(2 - \cos{\left(x \right)}\right)^{\frac{x}{3}}}{x}\right)
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:
(2−cos⁡(x))x3=(2−cos⁡(x))−x3\left(2 - \cos{\left(x \right)}\right)^{\frac{x}{3}} = \left(2 - \cos{\left(x \right)}\right)^{- \frac{x}{3}}
- No
(2−cos⁡(x))x3=−(2−cos⁡(x))−x3\left(2 - \cos{\left(x \right)}\right)^{\frac{x}{3}} = - \left(2 - \cos{\left(x \right)}\right)^{- \frac{x}{3}}
- No
es decir, función
no es
par ni impar
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