Para hallar los extremos hay que resolver la ecuación
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
$$\frac{d}{d x} f{\left(x \right)} = $$
primera derivada$$- \frac{\sin{\left(x \right)} \operatorname{acot}^{4}{\left(x \right)}}{5 \cos^{\frac{4}{5}}{\left(x \right)}} - \frac{4 \sqrt[5]{\cos{\left(x \right)}} \operatorname{acot}^{3}{\left(x \right)}}{x^{2} + 1} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = -87.7404973846353$$
$$x_{2} = 65.6778936140897$$
$$x_{3} = -21.2364630185626$$
$$x_{4} = 27.6485111882011$$
$$x_{5} = 90.8896086055559$$
$$x_{6} = 37.2060808189303$$
$$x_{7} = 97.1864295308975$$
$$x_{8} = -75.1381083055309$$
$$x_{9} = -34.026385463801$$
$$x_{10} = 100.334222727025$$
$$x_{11} = -49.8842688462602$$
$$x_{12} = 68.832301011785$$
$$x_{13} = -97.1864295308975$$
$$x_{14} = 8.24859586246451$$
$$x_{15} = 87.7404973846353$$
$$x_{16} = -81.4406200852641$$
$$x_{17} = -30.8409341533777$$
$$x_{18} = -90.8896086055559$$
$$x_{19} = -40.3810005938018$$
$$x_{20} = 84.5908537516106$$
$$x_{21} = -84.5908537516106$$
$$x_{22} = -46.7195181419734$$
$$x_{23} = 78.289730306746$$
$$x_{24} = 24.4476215123556$$
$$x_{25} = -68.832301011785$$
$$x_{26} = 30.8409341533777$$
$$x_{27} = 53.0466067646355$$
$$x_{28} = -53.0466067646355$$
$$x_{29} = 46.7195181419734$$
$$x_{30} = -56.2068767524321$$
$$x_{31} = 40.3810005938018$$
$$x_{32} = 34.026385463801$$
$$x_{33} = 62.5223031592894$$
$$x_{34} = 75.1381083055309$$
$$x_{35} = -37.2060808189303$$
$$x_{36} = 59.365363414655$$
$$x_{37} = -65.6778936140897$$
$$x_{38} = -8.24859586246451$$
$$x_{39} = -24.4476215123556$$
$$x_{40} = -59.365363414655$$
$$x_{41} = -11.5203391070543$$
$$x_{42} = -71.9856660103933$$
$$x_{43} = -62.5223031592894$$
$$x_{44} = 43.5519340864767$$
$$x_{45} = -94.0382378694922$$
$$x_{46} = 94.0382378694922$$
$$x_{47} = 18.012952547601$$
$$x_{48} = 81.4406200852641$$
$$x_{49} = 56.2068767524321$$
$$x_{50} = -100.334222727025$$
$$x_{51} = -27.6485111882011$$
$$x_{52} = -78.289730306746$$
$$x_{53} = 49.8842688462602$$
$$x_{54} = -4.96183907917208$$
$$x_{55} = -43.5519340864767$$
$$x_{56} = 71.9856660103933$$
$$x_{57} = -18.012952547601$$
$$x_{58} = 14.7748819055267$$
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:
$$x_{58} = -87.7404973846353$$
$$x_{58} = 65.6778936140897$$
$$x_{58} = 27.6485111882011$$
$$x_{58} = 90.8896086055559$$
$$x_{58} = 37.2060808189303$$
$$x_{58} = 97.1864295308975$$
$$x_{58} = -75.1381083055309$$
$$x_{58} = -34.026385463801$$
$$x_{58} = 100.334222727025$$
$$x_{58} = -49.8842688462602$$
$$x_{58} = 68.832301011785$$
$$x_{58} = -97.1864295308975$$
$$x_{58} = 87.7404973846353$$
$$x_{58} = -81.4406200852641$$
$$x_{58} = -30.8409341533777$$
$$x_{58} = -90.8896086055559$$
$$x_{58} = -40.3810005938018$$
$$x_{58} = 84.5908537516106$$
$$x_{58} = -84.5908537516106$$
$$x_{58} = -46.7195181419734$$
$$x_{58} = 78.289730306746$$
$$x_{58} = 24.4476215123556$$
$$x_{58} = -68.832301011785$$
$$x_{58} = 30.8409341533777$$
$$x_{58} = 53.0466067646355$$
$$x_{58} = -53.0466067646355$$
$$x_{58} = 46.7195181419734$$
$$x_{58} = -56.2068767524321$$
$$x_{58} = 40.3810005938018$$
$$x_{58} = 34.026385463801$$
$$x_{58} = 62.5223031592894$$
$$x_{58} = 75.1381083055309$$
$$x_{58} = -37.2060808189303$$
$$x_{58} = 59.365363414655$$
$$x_{58} = -65.6778936140897$$
$$x_{58} = -24.4476215123556$$
$$x_{58} = -59.365363414655$$
$$x_{58} = -11.5203391070543$$
$$x_{58} = -71.9856660103933$$
$$x_{58} = -62.5223031592894$$
$$x_{58} = 43.5519340864767$$
$$x_{58} = -94.0382378694922$$
$$x_{58} = 94.0382378694922$$
$$x_{58} = 18.012952547601$$
$$x_{58} = 81.4406200852641$$
$$x_{58} = 56.2068767524321$$
$$x_{58} = -100.334222727025$$
$$x_{58} = -27.6485111882011$$
$$x_{58} = -78.289730306746$$
$$x_{58} = 49.8842688462602$$
$$x_{58} = -4.96183907917208$$
$$x_{58} = -43.5519340864767$$
$$x_{58} = 71.9856660103933$$
$$x_{58} = -18.012952547601$$
Decrece en los intervalos
$$\left(-\infty, -100.334222727025\right]$$
Crece en los intervalos
$$\left[100.334222727025, \infty\right)$$