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Trazado el gráfico de la función/ cos(x)^(1/5)*acot(x)^4

Gráfico de la función y = cos(x)^(1/5)*acot(x)^4

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       5 ________     4   
f(x) = \/ cos(x) *acot (x)
f(x)=cos⁡(x)5acot⁡4(x)f{\left(x \right)} = \sqrt[5]{\cos{\left(x \right)}} \operatorname{acot}^{4}{\left(x \right)} 
f = cos(x)^(1/5)*acot(x)^4
Gráfico de la función
02468-8-6-4-2-1010010
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:
cos⁡(x)5acot⁡4(x)=0\sqrt[5]{\cos{\left(x \right)}} \operatorname{acot}^{4}{\left(x \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=π2x_{1} = \frac{\pi}{2}
x2=3π2x_{2} = \frac{3 \pi}{2}
Solución numérica
x1=1.5707963267949x_{1} = 1.5707963267949
x2=4.71238898038469x_{2} = 4.71238898038469
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 cos(x)^(1/5)*acot(x)^4.
cos⁡(0)5acot⁡4(0)\sqrt[5]{\cos{\left(0 \right)}} \operatorname{acot}^{4}{\left(0 \right)}
Resultado:
f(0)=π416f{\left(0 \right)} = \frac{\pi^{4}}{16}
Punto:
(0, pi^4/16)
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
−sin⁡(x)acot⁡4(x)5cos⁡45(x)−4cos⁡(x)5acot⁡3(x)x2+1=0- \frac{\sin{\left(x \right)} \operatorname{acot}^{4}{\left(x \right)}}{5 \cos^{\frac{4}{5}}{\left(x \right)}} - \frac{4 \sqrt[5]{\cos{\left(x \right)}} \operatorname{acot}^{3}{\left(x \right)}}{x^{2} + 1} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−87.7404973846353x_{1} = -87.7404973846353
x2=65.6778936140897x_{2} = 65.6778936140897
x3=−21.2364630185626x_{3} = -21.2364630185626
x4=27.6485111882011x_{4} = 27.6485111882011
x5=90.8896086055559x_{5} = 90.8896086055559
x6=37.2060808189303x_{6} = 37.2060808189303
x7=97.1864295308975x_{7} = 97.1864295308975
x8=−75.1381083055309x_{8} = -75.1381083055309
x9=−34.026385463801x_{9} = -34.026385463801
x10=100.334222727025x_{10} = 100.334222727025
x11=−49.8842688462602x_{11} = -49.8842688462602
x12=68.832301011785x_{12} = 68.832301011785
x13=−97.1864295308975x_{13} = -97.1864295308975
x14=8.24859586246451x_{14} = 8.24859586246451
x15=87.7404973846353x_{15} = 87.7404973846353
x16=−81.4406200852641x_{16} = -81.4406200852641
x17=−30.8409341533777x_{17} = -30.8409341533777
x18=−90.8896086055559x_{18} = -90.8896086055559
x19=−40.3810005938018x_{19} = -40.3810005938018
x20=84.5908537516106x_{20} = 84.5908537516106
x21=−84.5908537516106x_{21} = -84.5908537516106
x22=−46.7195181419734x_{22} = -46.7195181419734
x23=78.289730306746x_{23} = 78.289730306746
x24=24.4476215123556x_{24} = 24.4476215123556
x25=−68.832301011785x_{25} = -68.832301011785
x26=30.8409341533777x_{26} = 30.8409341533777
x27=53.0466067646355x_{27} = 53.0466067646355
x28=−53.0466067646355x_{28} = -53.0466067646355
x29=46.7195181419734x_{29} = 46.7195181419734
x30=−56.2068767524321x_{30} = -56.2068767524321
x31=40.3810005938018x_{31} = 40.3810005938018
x32=34.026385463801x_{32} = 34.026385463801
x33=62.5223031592894x_{33} = 62.5223031592894
x34=75.1381083055309x_{34} = 75.1381083055309
x35=−37.2060808189303x_{35} = -37.2060808189303
x36=59.365363414655x_{36} = 59.365363414655
x37=−65.6778936140897x_{37} = -65.6778936140897
x38=−8.24859586246451x_{38} = -8.24859586246451
x39=−24.4476215123556x_{39} = -24.4476215123556
x40=−59.365363414655x_{40} = -59.365363414655
x41=−11.5203391070543x_{41} = -11.5203391070543
x42=−71.9856660103933x_{42} = -71.9856660103933
x43=−62.5223031592894x_{43} = -62.5223031592894
x44=43.5519340864767x_{44} = 43.5519340864767
x45=−94.0382378694922x_{45} = -94.0382378694922
x46=94.0382378694922x_{46} = 94.0382378694922
x47=18.012952547601x_{47} = 18.012952547601
x48=81.4406200852641x_{48} = 81.4406200852641
x49=56.2068767524321x_{49} = 56.2068767524321
x50=−100.334222727025x_{50} = -100.334222727025
x51=−27.6485111882011x_{51} = -27.6485111882011
x52=−78.289730306746x_{52} = -78.289730306746
x53=49.8842688462602x_{53} = 49.8842688462602
x54=−4.96183907917208x_{54} = -4.96183907917208
x55=−43.5519340864767x_{55} = -43.5519340864767
x56=71.9856660103933x_{56} = 71.9856660103933
x57=−18.012952547601x_{57} = -18.012952547601
x58=14.7748819055267x_{58} = 14.7748819055267
Signos de extremos en los puntos:
(-87.74049738463532, 1.67851378747598e-8)

                                        5 ____ 
(65.67789361408974, 5.32523466428789e-8*\/ -1 )

                                          5 ____ 
(-21.236463018562645, 4.60121861638581e-6*\/ -1 )

                                         5 ____ 
(27.648511188201073, 1.63796475526303e-6*\/ -1 )

                                        5 ____ 
(90.88960860555594, 1.45820887574644e-8*\/ -1 )

(37.206080818930346, 5.08290128178658e-7)

                                        5 ____ 
(97.18642953089754, 1.11613081318599e-8*\/ -1 )

(-75.13810830553089, 3.11518778362253e-8)

                                          5 ____ 
(-34.026385463800956, 7.23372081977916e-7*\/ -1 )

(100.33422272702532, 9.82775036132693e-9)

(-49.88426884626023, 1.59016613503664e-7)

(68.83230101178502, 4.41763271860867e-8)

                                         5 ____ 
(-97.18642953089754, 1.11613081318599e-8*\/ -1 )

                                         5 ____ 
(8.248595862464514, 0.000174981206628228*\/ -1 )

(87.74049738463532, 1.67851378747598e-8)

(-81.44062008526413, 2.25946537071482e-8)

(-30.840934153377656, 1.06574699602049e-6)

                                         5 ____ 
(-90.88960860555594, 1.45820887574644e-8*\/ -1 )

                                        5 ____ 
(-40.38100059380177, 3.6763436752227e-7*\/ -1 )

                                        5 ____ 
(84.59085375161057, 1.94206423536575e-8*\/ -1 )

                                         5 ____ 
(-84.59085375161057, 1.94206423536575e-8*\/ -1 )

                                          5 ____ 
(-46.719518141973374, 2.06270678099626e-7*\/ -1 )

                                        5 ____ 
(78.28973030674597, 2.64448641719372e-8*\/ -1 )

(24.447621512355592, 2.6538240295827e-6)

(-68.83230101178502, 4.41763271860867e-8)

(30.840934153377656, 1.06574699602049e-6)

                                        5 ____ 
(53.04660676463553, 1.24564441983448e-7*\/ -1 )

                                         5 ____ 
(-53.04660676463553, 1.24564441983448e-7*\/ -1 )

                                         5 ____ 
(46.719518141973374, 2.06270678099626e-7*\/ -1 )

(-56.20687675243206, 9.89654197123042e-8)

                                       5 ____ 
(40.38100059380177, 3.6763436752227e-7*\/ -1 )

                                         5 ____ 
(34.026385463800956, 7.23372081977916e-7*\/ -1 )

(62.52230315928937, 6.47861755097637e-8)

(75.13810830553089, 3.11518778362253e-8)

(-37.206080818930346, 5.08290128178658e-7)

                                        5 ____ 
(59.36536341465497, 7.96225099475449e-8*\/ -1 )

                                         5 ____ 
(-65.67789361408974, 5.32523466428789e-8*\/ -1 )

                                          5 ____ 
(-8.248595862464514, 0.000174981206628228*\/ -1 )

(-24.447621512355592, 2.6538240295827e-6)

                                         5 ____ 
(-59.36536341465497, 7.96225099475449e-8*\/ -1 )

(-11.520339107054326, 4.89507380632813e-5)

                                         5 ____ 
(-71.98566601039333, 3.69552100485899e-8*\/ -1 )

(-62.52230315928937, 6.47861755097637e-8)

(43.55193408647669, 2.72495937498819e-7)

(-94.03823786949218, 1.27290380074148e-8)

(94.03823786949218, 1.27290380074148e-8)

(18.01295254760103, 8.73155401666895e-6)

(81.44062008526413, 2.25946537071482e-8)

(56.20687675243206, 9.89654197123042e-8)

(-100.33422272702532, 9.82775036132693e-9)

                                          5 ____ 
(-27.648511188201073, 1.63796475526303e-6*\/ -1 )

                                         5 ____ 
(-78.28973030674597, 2.64448641719372e-8*\/ -1 )

(49.88426884626023, 1.59016613503664e-7)

(-4.9618390791720755, 0.00118251672094245)

(-43.55193408647669, 2.72495937498819e-7)

                                        5 ____ 
(71.98566601039333, 3.69552100485899e-8*\/ -1 )

(-18.01295254760103, 8.73155401666895e-6)

                                         5 ____ 
(14.774881905526708, 1.88023721386445e-5*\/ -1 )


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
Puntos máximos de la función:
x58=−87.7404973846353x_{58} = -87.7404973846353
x58=65.6778936140897x_{58} = 65.6778936140897
x58=27.6485111882011x_{58} = 27.6485111882011
x58=90.8896086055559x_{58} = 90.8896086055559
x58=37.2060808189303x_{58} = 37.2060808189303
x58=97.1864295308975x_{58} = 97.1864295308975
x58=−75.1381083055309x_{58} = -75.1381083055309
x58=−34.026385463801x_{58} = -34.026385463801
x58=100.334222727025x_{58} = 100.334222727025
x58=−49.8842688462602x_{58} = -49.8842688462602
x58=68.832301011785x_{58} = 68.832301011785
x58=−97.1864295308975x_{58} = -97.1864295308975
x58=87.7404973846353x_{58} = 87.7404973846353
x58=−81.4406200852641x_{58} = -81.4406200852641
x58=−30.8409341533777x_{58} = -30.8409341533777
x58=−90.8896086055559x_{58} = -90.8896086055559
x58=−40.3810005938018x_{58} = -40.3810005938018
x58=84.5908537516106x_{58} = 84.5908537516106
x58=−84.5908537516106x_{58} = -84.5908537516106
x58=−46.7195181419734x_{58} = -46.7195181419734
x58=78.289730306746x_{58} = 78.289730306746
x58=24.4476215123556x_{58} = 24.4476215123556
x58=−68.832301011785x_{58} = -68.832301011785
x58=30.8409341533777x_{58} = 30.8409341533777
x58=53.0466067646355x_{58} = 53.0466067646355
x58=−53.0466067646355x_{58} = -53.0466067646355
x58=46.7195181419734x_{58} = 46.7195181419734
x58=−56.2068767524321x_{58} = -56.2068767524321
x58=40.3810005938018x_{58} = 40.3810005938018
x58=34.026385463801x_{58} = 34.026385463801
x58=62.5223031592894x_{58} = 62.5223031592894
x58=75.1381083055309x_{58} = 75.1381083055309
x58=−37.2060808189303x_{58} = -37.2060808189303
x58=59.365363414655x_{58} = 59.365363414655
x58=−65.6778936140897x_{58} = -65.6778936140897
x58=−24.4476215123556x_{58} = -24.4476215123556
x58=−59.365363414655x_{58} = -59.365363414655
x58=−11.5203391070543x_{58} = -11.5203391070543
x58=−71.9856660103933x_{58} = -71.9856660103933
x58=−62.5223031592894x_{58} = -62.5223031592894
x58=43.5519340864767x_{58} = 43.5519340864767
x58=−94.0382378694922x_{58} = -94.0382378694922
x58=94.0382378694922x_{58} = 94.0382378694922
x58=18.012952547601x_{58} = 18.012952547601
x58=81.4406200852641x_{58} = 81.4406200852641
x58=56.2068767524321x_{58} = 56.2068767524321
x58=−100.334222727025x_{58} = -100.334222727025
x58=−27.6485111882011x_{58} = -27.6485111882011
x58=−78.289730306746x_{58} = -78.289730306746
x58=49.8842688462602x_{58} = 49.8842688462602
x58=−4.96183907917208x_{58} = -4.96183907917208
x58=−43.5519340864767x_{58} = -43.5519340864767
x58=71.9856660103933x_{58} = 71.9856660103933
x58=−18.012952547601x_{58} = -18.012952547601
Decrece en los intervalos
(−∞,−100.334222727025]\left(-\infty, -100.334222727025\right]
Crece en los intervalos
[100.334222727025,∞)\left[100.334222727025, \infty\right)
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(cos⁡(x)5acot⁡4(x))=0\lim_{x \to -\infty}\left(\sqrt[5]{\cos{\left(x \right)}} \operatorname{acot}^{4}{\left(x \right)}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞(cos⁡(x)5acot⁡4(x))=0\lim_{x \to \infty}\left(\sqrt[5]{\cos{\left(x \right)}} \operatorname{acot}^{4}{\left(x \right)}\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 cos(x)^(1/5)*acot(x)^4, dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(cos⁡(x)5acot⁡4(x)x)=0\lim_{x \to -\infty}\left(\frac{\sqrt[5]{\cos{\left(x \right)}} \operatorname{acot}^{4}{\left(x \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(cos⁡(x)5acot⁡4(x)x)=0\lim_{x \to \infty}\left(\frac{\sqrt[5]{\cos{\left(x \right)}} \operatorname{acot}^{4}{\left(x \right)}}{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:
cos⁡(x)5acot⁡4(x)=cos⁡(x)5acot⁡4(x)\sqrt[5]{\cos{\left(x \right)}} \operatorname{acot}^{4}{\left(x \right)} = \sqrt[5]{\cos{\left(x \right)}} \operatorname{acot}^{4}{\left(x \right)}
- Sí
cos⁡(x)5acot⁡4(x)=−cos⁡(x)5acot⁡4(x)\sqrt[5]{\cos{\left(x \right)}} \operatorname{acot}^{4}{\left(x \right)} = - \sqrt[5]{\cos{\left(x \right)}} \operatorname{acot}^{4}{\left(x \right)}
- No
es decir, función
es
par
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