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$$10 \left(\left(3 \tan^{2}{\left(x \right)} + 3\right) \cos{\left(x \right)} - 3 \sin{\left(x \right)} \tan{\left(x \right)}\right) \operatorname{sign}{\left(\cos{\left(x \right)} \tan{\left(x \right)} \right)} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = 1229.9335238804$$
$$x_{2} = -54.9778714378214$$
$$x_{3} = -26.7035375555132$$
$$x_{4} = 7.85398163397448$$
$$x_{5} = -262.322986574748$$
$$x_{6} = 73.8274273593601$$
$$x_{7} = 61.261056745001$$
$$x_{8} = 0$$
$$x_{9} = -26.7035375555132$$
$$x_{10} = -306.305283725005$$
$$x_{11} = -1.5707963267949$$
$$x_{12} = -95.8185759344887$$
$$x_{13} = -39.2699081698724$$
$$x_{14} = 14.1371669411541$$
$$x_{15} = 10.9955742875643$$
$$x_{16} = -42.4115008234622$$
$$x_{17} = 58.1194640914112$$
$$x_{18} = -29.845130209103$$
$$x_{19} = 70.6858347057703$$
$$x_{20} = 54.9778714378214$$
$$x_{21} = 54.9778714378214$$
$$x_{22} = -36.1283155162826$$
$$x_{23} = 23.5619449019235$$
$$x_{24} = 237.190245346029$$
$$x_{25} = 17.2787595947438$$
$$x_{26} = 80.1106126665397$$
$$x_{27} = -7.85398163397449$$
$$x_{28} = -92.6769832808989$$
$$x_{29} = -10.9955742875643$$
$$x_{30} = -86.3937979737193$$
$$x_{31} = -54.9778714378214$$
$$x_{32} = 39.2699081698724$$
$$x_{33} = -32.9867228626928$$
$$x_{34} = 36.1283155162826$$
$$x_{35} = 98.9601685880785$$
$$x_{36} = -61.261056745001$$
$$x_{37} = -67.5442420521806$$
$$x_{38} = 92.6769832808989$$
$$x_{39} = -58.1194640914112$$
$$x_{40} = 26.7035375555132$$
$$x_{41} = -4.71238898038469$$
$$x_{42} = -48.6946861306418$$
$$x_{43} = 51.8362787842316$$
$$x_{44} = -73.8274273593601$$
$$x_{45} = 86.3937979737193$$
$$x_{46} = -98.9601685880785$$
$$x_{47} = -89.5353906273091$$
$$x_{48} = -14.1371669411541$$
$$x_{49} = -64.4026493985908$$
$$x_{50} = 95.8185759344887$$
$$x_{51} = 1.5707963267949$$
$$x_{52} = 45.553093477052$$
$$x_{53} = -17.2787595947439$$
$$x_{54} = 4.71238898038469$$
$$x_{55} = 76.9690200129499$$
$$x_{56} = -45.553093477052$$
$$x_{57} = 20.4203522483337$$
$$x_{58} = -86.3937979737193$$
$$x_{59} = -83.2522053201295$$
$$x_{60} = -20.4203522483337$$
$$x_{61} = -80.1106126665397$$
$$x_{62} = 61.261056745001$$
$$x_{63} = 32.9867228626928$$
$$x_{64} = 64.4026493985908$$
$$x_{65} = -23.5619449019235$$
$$x_{66} = 32.9867228626928$$
$$x_{67} = 29.845130209103$$
$$x_{68} = 42.4115008234622$$
$$x_{69} = 89.5353906273091$$
$$x_{70} = -51.8362787842316$$
$$x_{71} = -70.6858347057703$$
$$x_{72} = 83.2522053201295$$
$$x_{73} = 67.5442420521806$$
$$x_{74} = -76.9690200129499$$
$$x_{75} = -2279.22547017939$$
Signos de extremos en los puntos:
(1229.933523880404, 30)
(-54.977871437821385, 30)
(-26.70353755551324, 30)
(7.853981633974483, 30)
(-262.32298657474774, 30)
(73.82742735936014, 30)
(61.261056745000964, 30)
(0, 0)
(-26.703537555513243, 30)
(-306.3052837250048, 30)
(-1.5707963267948966, 30)
(-95.81857593448869, 30)
(-39.269908169872416, 30)
(14.137166941154069, 30)
(10.995574287564276, 30)
(-42.41150082346223, 30)
(58.119464091411174, 30)
(-29.845130209103033, 30)
(70.68583470577035, 30)
(54.97787143782138, 30)
(54.977871437821385, 30)
(-36.12831551628262, 30)
(23.56194490192345, 30)
(237.1902453460294, 30)
(17.278759594743807, 30)
(80.11061266653974, 30)
(-7.853981633974486, 30)
(-92.6769832808989, 30)
(-10.995574287564276, 30)
(-86.39379797371932, 30)
(-54.97787143782138, 30)
(39.269908169872416, 30)
(-32.98672286269283, 30)
(36.12831551628262, 30)
(98.96016858807849, 30)
(-61.26105674500097, 30)
(-67.54424205218055, 30)
(92.67698328089891, 30)
(-58.119464091411174, 30)
(26.703537555513243, 30)
(-4.712388980384691, 30)
(-48.6946861306418, 30)
(51.83627878423159, 30)
(-73.82742735936013, 30)
(86.39379797371932, 30)
(-98.96016858807849, 30)
(-89.53539062730911, 30)
(-14.137166941154069, 30)
(-64.40264939859077, 30)
(95.81857593448869, 30)
(1.5707963267948966, 30)
(45.553093477052, 30)
(-17.278759594743864, 30)
(4.71238898038469, 30)
(76.96902001294994, 30)
(-45.553093477052, 30)
(20.420352248333657, 30)
(-86.3937979737193, 30)
(-83.25220532012952, 30)
(-20.420352248333657, 30)
(-80.11061266653972, 30)
(61.26105674500097, 30)
(32.98672286269282, 30)
(64.40264939859077, 30)
(-23.56194490192345, 30)
(32.98672286269283, 30)
(29.845130209103036, 30)
(42.411500823462205, 30)
(89.53539062730911, 30)
(-51.83627878423159, 30)
(-70.68583470577035, 30)
(83.25220532012952, 30)
(67.54424205218055, 30)
(-76.96902001294994, 30)
(-2279.225470179395, 30)
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:
$$x_{1} = 0$$
Puntos máximos de la función:
$$x_{1} = 1229.9335238804$$
$$x_{1} = -54.9778714378214$$
$$x_{1} = -26.7035375555132$$
$$x_{1} = 7.85398163397448$$
$$x_{1} = -262.322986574748$$
$$x_{1} = 73.8274273593601$$
$$x_{1} = 61.261056745001$$
$$x_{1} = -26.7035375555132$$
$$x_{1} = -306.305283725005$$
$$x_{1} = -1.5707963267949$$
$$x_{1} = -95.8185759344887$$
$$x_{1} = -39.2699081698724$$
$$x_{1} = 14.1371669411541$$
$$x_{1} = 10.9955742875643$$
$$x_{1} = -42.4115008234622$$
$$x_{1} = 58.1194640914112$$
$$x_{1} = -29.845130209103$$
$$x_{1} = 70.6858347057703$$
$$x_{1} = 54.9778714378214$$
$$x_{1} = 54.9778714378214$$
$$x_{1} = -36.1283155162826$$
$$x_{1} = 23.5619449019235$$
$$x_{1} = 237.190245346029$$
$$x_{1} = 17.2787595947438$$
$$x_{1} = 80.1106126665397$$
$$x_{1} = -7.85398163397449$$
$$x_{1} = -92.6769832808989$$
$$x_{1} = -10.9955742875643$$
$$x_{1} = -86.3937979737193$$
$$x_{1} = -54.9778714378214$$
$$x_{1} = 39.2699081698724$$
$$x_{1} = -32.9867228626928$$
$$x_{1} = 36.1283155162826$$
$$x_{1} = 98.9601685880785$$
$$x_{1} = -61.261056745001$$
$$x_{1} = -67.5442420521806$$
$$x_{1} = 92.6769832808989$$
$$x_{1} = -58.1194640914112$$
$$x_{1} = 26.7035375555132$$
$$x_{1} = -4.71238898038469$$
$$x_{1} = -48.6946861306418$$
$$x_{1} = 51.8362787842316$$
$$x_{1} = -73.8274273593601$$
$$x_{1} = 86.3937979737193$$
$$x_{1} = -98.9601685880785$$
$$x_{1} = -89.5353906273091$$
$$x_{1} = -14.1371669411541$$
$$x_{1} = -64.4026493985908$$
$$x_{1} = 95.8185759344887$$
$$x_{1} = 1.5707963267949$$
$$x_{1} = 45.553093477052$$
$$x_{1} = -17.2787595947439$$
$$x_{1} = 4.71238898038469$$
$$x_{1} = 76.9690200129499$$
$$x_{1} = -45.553093477052$$
$$x_{1} = 20.4203522483337$$
$$x_{1} = -86.3937979737193$$
$$x_{1} = -83.2522053201295$$
$$x_{1} = -20.4203522483337$$
$$x_{1} = -80.1106126665397$$
$$x_{1} = 61.261056745001$$
$$x_{1} = 32.9867228626928$$
$$x_{1} = 64.4026493985908$$
$$x_{1} = -23.5619449019235$$
$$x_{1} = 32.9867228626928$$
$$x_{1} = 29.845130209103$$
$$x_{1} = 42.4115008234622$$
$$x_{1} = 89.5353906273091$$
$$x_{1} = -51.8362787842316$$
$$x_{1} = -70.6858347057703$$
$$x_{1} = 83.2522053201295$$
$$x_{1} = 67.5442420521806$$
$$x_{1} = -76.9690200129499$$
$$x_{1} = -2279.22547017939$$
Decrece en los intervalos
$$\left(-\infty, -2279.22547017939\right] \cup \left[0, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, 0\right] \cup \left[1229.9335238804, \infty\right)$$