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$$\frac{\frac{2 \sin^{3}{\left(x \right)}}{\cos^{3}{\left(x \right)}} + \frac{2 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{2 \sin{\left(x \right)} \cos{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 2}{x^{2}} - \frac{2 \left(\left(\frac{\sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) + \frac{2 \sin{\left(x \right)}}{\cos{\left(x \right)}}\right)}{x^{3}} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = 21.2057504117311$$
$$x_{2} = 68.329640215578$$
$$x_{3} = 84.037603483527$$
$$x_{4} = 62.0464549083984$$
$$x_{5} = -63.6172512351933$$
$$x_{6} = -73.0420291959627$$
$$x_{7} = -35.3429173528852$$
$$x_{8} = -98.174770424681$$
$$x_{9} = -38.484510006475$$
$$x_{10} = -91.8915851175014$$
$$x_{11} = 8.63937979737193$$
$$x_{12} = 74.6128255227576$$
$$x_{13} = -51.0508806208341$$
$$x_{14} = -10.2101761241668$$
$$x_{15} = 58.9048622548086$$
$$x_{16} = 80.8960108299372$$
$$x_{17} = -54.1924732744239$$
$$x_{18} = -85.6083998103219$$
$$x_{19} = 18.0641577581413$$
$$x_{20} = -13.3517687777566$$
$$x_{21} = -47.9092879672443$$
$$x_{22} = 55.7632696012188$$
$$x_{23} = 46.3384916404494$$
$$x_{24} = 40.0553063332699$$
$$x_{25} = -79.3252145031423$$
$$x_{26} = 33.7721210260903$$
$$x_{27} = -101.316363078271$$
$$x_{28} = -16.4933614313464$$
$$x_{29} = -66.7588438887831$$
$$x_{30} = -57.3340659280137$$
$$x_{31} = 65.1880475619882$$
$$x_{32} = 96.6039740978861$$
$$x_{33} = -41.6261026600648$$
$$x_{34} = -29.0597320457056$$
$$x_{35} = -44.7676953136546$$
$$x_{36} = -22.776546738526$$
$$x_{37} = -69.9004365423729$$
$$x_{38} = 52.621676947629$$
$$x_{39} = -76.1836218495525$$
$$x_{40} = -25.9181393921158$$
$$x_{41} = 36.9137136796801$$
$$x_{42} = 30.6305283725005$$
$$x_{43} = 14.9225651045515$$
$$x_{44} = -19.6349540849362$$
$$x_{45} = 99.7455667514759$$
$$x_{46} = 2.35619449019234$$
$$x_{47} = -258.39599575776$$
$$x_{48} = 87.1791961371168$$
$$x_{49} = -82.4668071567321$$
$$x_{50} = -60.4756585816035$$
$$x_{51} = -3.92699081698724$$
$$x_{52} = -88.7499924639117$$
$$x_{53} = 90.3207887907066$$
$$x_{54} = -32.2013246992954$$
$$x_{55} = 43.1968989868597$$
$$x_{56} = -95.0331777710912$$
$$x_{57} = 77.7544181763474$$
$$x_{58} = -7.06858347057703$$
$$x_{59} = 24.3473430653209$$
Signos de extremos en los puntos:
(21.205750411731103, 9.875587244949e-19)
(68.329640215578, 0)
(84.03760348352696, 6.28815014548678e-20)
(62.04645490839842, 0)
(-63.617251235193315, 0)
(-73.0420291959627, 0)
(-35.34291735288517, 0)
(-98.17477042468104, 4.6075539850034e-20)
(-38.48451000647497, 0)
(-91.89158511750145, 0)
(8.639379797371932, 0)
(74.61282552275759, 0)
(-51.05088062083414, 0)
(-10.210176124166829, 0)
(58.90486225480862, 0)
(80.89601082993718, 0)
(-54.19247327442393, -7.56070479055642e-20)
(-85.60839981032187, 0)
(18.06415775814131, 0)
(-13.351768777756622, 2.49110833964284e-18)
(-47.909287967244346, 9.6738821574413e-20)
(55.76326960121883, 0)
(46.33849164044945, 0)
(40.05530633326986, 0)
(-79.32521450314228, 3.52872419447496e-20)
(33.772121026090275, 0)
(-101.31636307827083, 0)
(-16.493361431346415, 0)
(-66.7588438887831, 0)
(-57.33406592801373, 0)
(65.18804756198821, 5.22521636055147e-20)
(96.60397409788614, 0)
(-41.62610266006476, 1.28147082619577e-19)
(-29.059732045705587, 0)
(-44.767695313654556, 0)
(-22.776546738526, 0)
(-69.9004365423729, 9.08888158258783e-20)
(52.621676947629034, 0)
(-76.18362184955248, 3.82575358782433e-20)
(-25.918139392115794, 0)
(36.91371367968007, -1.62953895463282e-19)
(30.630528372500486, 2.36663481313867e-19)
(14.922565104551518, 0)
(-19.634954084936208, 0)
(99.74556675147593, 0)
(2.356194490192345, 0)
(-258.3959957577605, 3.3255897033323e-21)
(87.17919613711676, 0)
(-82.46680715673207, 0)
(-60.47565858160352, 6.07126252451326e-20)
(-3.9269908169872414, 0)
(-88.74999246391165, 0)
(90.32078879070656, 0)
(-32.201324699295384, 0)
(43.19689898685966, 1.1899674548046e-19)
(-95.03317777109125, 0)
(77.75441817634739, 0)
(-7.0685834705770345, 0)
(24.3473430653209, 0)
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:
$$x_{1} = 21.2057504117311$$
$$x_{2} = 68.329640215578$$
$$x_{3} = 84.037603483527$$
$$x_{4} = 62.0464549083984$$
$$x_{5} = -63.6172512351933$$
$$x_{6} = -73.0420291959627$$
$$x_{7} = -35.3429173528852$$
$$x_{8} = -98.174770424681$$
$$x_{9} = -38.484510006475$$
$$x_{10} = -91.8915851175014$$
$$x_{11} = 8.63937979737193$$
$$x_{12} = 74.6128255227576$$
$$x_{13} = -51.0508806208341$$
$$x_{14} = -10.2101761241668$$
$$x_{15} = 58.9048622548086$$
$$x_{16} = 80.8960108299372$$
$$x_{17} = -54.1924732744239$$
$$x_{18} = -85.6083998103219$$
$$x_{19} = 18.0641577581413$$
$$x_{20} = -13.3517687777566$$
$$x_{21} = -47.9092879672443$$
$$x_{22} = 55.7632696012188$$
$$x_{23} = 46.3384916404494$$
$$x_{24} = 40.0553063332699$$
$$x_{25} = -79.3252145031423$$
$$x_{26} = 33.7721210260903$$
$$x_{27} = -101.316363078271$$
$$x_{28} = -16.4933614313464$$
$$x_{29} = -66.7588438887831$$
$$x_{30} = -57.3340659280137$$
$$x_{31} = 65.1880475619882$$
$$x_{32} = 96.6039740978861$$
$$x_{33} = -41.6261026600648$$
$$x_{34} = -29.0597320457056$$
$$x_{35} = -44.7676953136546$$
$$x_{36} = -22.776546738526$$
$$x_{37} = -69.9004365423729$$
$$x_{38} = 52.621676947629$$
$$x_{39} = -76.1836218495525$$
$$x_{40} = -25.9181393921158$$
$$x_{41} = 36.9137136796801$$
$$x_{42} = 30.6305283725005$$
$$x_{43} = 14.9225651045515$$
$$x_{44} = -19.6349540849362$$
$$x_{45} = 99.7455667514759$$
$$x_{46} = 2.35619449019234$$
$$x_{47} = -258.39599575776$$
$$x_{48} = 87.1791961371168$$
$$x_{49} = -82.4668071567321$$
$$x_{50} = -60.4756585816035$$
$$x_{51} = -3.92699081698724$$
$$x_{52} = -88.7499924639117$$
$$x_{53} = 90.3207887907066$$
$$x_{54} = -32.2013246992954$$
$$x_{55} = 43.1968989868597$$
$$x_{56} = -95.0331777710912$$
$$x_{57} = 77.7544181763474$$
$$x_{58} = -7.06858347057703$$
$$x_{59} = 24.3473430653209$$
La función no tiene puntos máximos
Decrece en los intervalos
$$\left[99.7455667514759, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -258.39599575776\right]$$