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$$- 6 \sin{\left(2 x \right)} \operatorname{sign}{\left(3 \cos{\left(2 x \right)} - 2 \right)} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = 1.5707963267949$$
$$x_{2} = -50.2654824574367$$
$$x_{3} = 14.1371669411541$$
$$x_{4} = 61.261056745001$$
$$x_{5} = -36.1283155162826$$
$$x_{6} = 7.85398163397448$$
$$x_{7} = 72.2566310325652$$
$$x_{8} = 42.4115008234622$$
$$x_{9} = -43.9822971502571$$
$$x_{10} = 40.8407044966673$$
$$x_{11} = -37.6991118430775$$
$$x_{12} = 83.2522053201295$$
$$x_{13} = -89.5353906273091$$
$$x_{14} = 34.5575191894877$$
$$x_{15} = -94.2477796076938$$
$$x_{16} = -67.5442420521806$$
$$x_{17} = -51.8362787842316$$
$$x_{18} = -83.2522053201295$$
$$x_{19} = -14.1371669411541$$
$$x_{20} = -21.9911485751286$$
$$x_{21} = 92.6769832808989$$
$$x_{22} = -15.707963267949$$
$$x_{23} = 50.2654824574367$$
$$x_{24} = 64.4026493985908$$
$$x_{25} = 48.6946861306418$$
$$x_{26} = 67.5442420521806$$
$$x_{27} = 21.9911485751286$$
$$x_{28} = -87.9645943005142$$
$$x_{29} = -72.2566310325652$$
$$x_{30} = 23.5619449019235$$
$$x_{31} = -117.809724509617$$
$$x_{32} = -80.1106126665397$$
$$x_{33} = -9.42477796076938$$
$$x_{34} = -64.4026493985908$$
$$x_{35} = -29.845130209103$$
$$x_{36} = -86.3937979737193$$
$$x_{37} = 4.71238898038469$$
$$x_{38} = 20.4203522483337$$
$$x_{39} = -65.9734457253857$$
$$x_{40} = -76.9690200129499$$
$$x_{41} = 15.707963267949$$
$$x_{42} = 28.2743338823081$$
$$x_{43} = 94.2477796076938$$
$$x_{44} = 37.6991118430775$$
$$x_{45} = -1.5707963267949$$
$$x_{46} = 6.28318530717959$$
$$x_{47} = 86.3937979737193$$
$$x_{48} = -53.4070751110265$$
$$x_{49} = -39.2699081698724$$
$$x_{50} = -6.28318530717959$$
$$x_{51} = 36.1283155162826$$
$$x_{52} = -75.398223686155$$
$$x_{53} = 51.8362787842316$$
$$x_{54} = 80.1106126665397$$
$$x_{55} = 0$$
$$x_{56} = 89.5353906273091$$
$$x_{57} = 58.1194640914112$$
$$x_{58} = 73.8274273593601$$
$$x_{59} = -58.1194640914112$$
$$x_{60} = -20.4203522483337$$
$$x_{61} = -97.3893722612836$$
$$x_{62} = -95.8185759344887$$
$$x_{63} = 29.845130209103$$
$$x_{64} = -28.2743338823081$$
$$x_{65} = 26.7035375555132$$
$$x_{66} = 100.530964914873$$
$$x_{67} = -23.5619449019235$$
$$x_{68} = -42.4115008234622$$
$$x_{69} = -61.261056745001$$
$$x_{70} = 78.5398163397448$$
$$x_{71} = 70.6858347057703$$
$$x_{72} = -98.9601685880785$$
$$x_{73} = 12.5663706143592$$
$$x_{74} = 87.9645943005142$$
$$x_{75} = 43.9822971502571$$
$$x_{76} = -31.4159265358979$$
$$x_{77} = 45.553093477052$$
$$x_{78} = 56.5486677646163$$
$$x_{79} = -81.6814089933346$$
$$x_{80} = -73.8274273593601$$
$$x_{81} = -59.6902604182061$$
$$x_{82} = 65.9734457253857$$
$$x_{83} = 174.358392274234$$
$$x_{84} = -45.553093477052$$
$$x_{85} = -7.85398163397448$$
$$x_{86} = -17.2787595947439$$
$$x_{87} = 95.8185759344887$$
Signos de extremos en los puntos:
(1.5707963267948966, 5)
(-50.26548245743669, 1)
(14.137166941154069, 5)
(61.26105674500097, 5)
(-36.12831551628262, 5)
(7.853981633974483, 5)
(72.25663103256524, 1)
(42.411500823462205, 5)
(-43.982297150257104, 1)
(40.840704496667314, 1)
(-37.69911184307752, 1)
(83.25220532012952, 5)
(-89.53539062730911, 5)
(34.55751918948773, 1)
(-94.2477796076938, 1)
(-67.54424205218055, 5)
(-51.83627878423159, 5)
(-83.25220532012952, 5)
(-14.137166941154069, 5)
(-21.991148575128552, 1)
(92.6769832808989, 5)
(-15.707963267948966, 1)
(50.26548245743669, 1)
(64.40264939859077, 5)
(48.6946861306418, 5)
(67.54424205218055, 5)
(21.991148575128552, 1)
(-87.96459430051421, 1)
(-72.25663103256524, 1)
(23.56194490192345, 5)
(-117.80972450961724, 5)
(-80.11061266653972, 5)
(-9.42477796076938, 1)
(-64.40264939859077, 5)
(-29.845130209103036, 5)
(-86.39379797371932, 5)
(4.71238898038469, 5)
(20.420352248333657, 5)
(-65.97344572538566, 1)
(-76.96902001294994, 5)
(15.707963267948966, 1)
(28.274333882308138, 1)
(94.2477796076938, 1)
(37.69911184307752, 1)
(-1.5707963267948966, 5)
(6.283185307179586, 1)
(86.39379797371932, 5)
(-53.40707511102649, 1)
(-39.269908169872416, 5)
(-6.283185307179586, 1)
(36.12831551628262, 5)
(-75.39822368615503, 1)
(51.83627878423159, 5)
(80.11061266653972, 5)
(0, 1)
(89.53539062730911, 5)
(58.119464091411174, 5)
(73.82742735936014, 5)
(-58.119464091411174, 5)
(-20.420352248333657, 5)
(-97.3893722612836, 1)
(-95.81857593448869, 5)
(29.845130209103036, 5)
(-28.274333882308138, 1)
(26.703537555513243, 5)
(100.53096491487338, 1)
(-23.56194490192345, 5)
(-42.411500823462205, 5)
(-61.26105674500097, 5)
(78.53981633974483, 1)
(70.68583470577035, 5)
(-98.96016858807849, 5)
(12.566370614359172, 1)
(87.96459430051421, 1)
(43.982297150257104, 1)
(-31.41592653589793, 1)
(45.553093477052, 5)
(56.548667764616276, 1)
(-81.68140899333463, 1)
(-73.82742735936014, 5)
(-59.69026041820607, 1)
(65.97344572538566, 1)
(174.35839227423352, 5)
(-45.553093477052, 5)
(-7.853981633974483, 5)
(-17.278759594743864, 5)
(95.81857593448869, 5)
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:
$$x_{87} = 1.5707963267949$$
$$x_{87} = -50.2654824574367$$
$$x_{87} = 14.1371669411541$$
$$x_{87} = 61.261056745001$$
$$x_{87} = -36.1283155162826$$
$$x_{87} = 7.85398163397448$$
$$x_{87} = 72.2566310325652$$
$$x_{87} = 42.4115008234622$$
$$x_{87} = -43.9822971502571$$
$$x_{87} = 40.8407044966673$$
$$x_{87} = -37.6991118430775$$
$$x_{87} = 83.2522053201295$$
$$x_{87} = -89.5353906273091$$
$$x_{87} = 34.5575191894877$$
$$x_{87} = -94.2477796076938$$
$$x_{87} = -67.5442420521806$$
$$x_{87} = -51.8362787842316$$
$$x_{87} = -83.2522053201295$$
$$x_{87} = -14.1371669411541$$
$$x_{87} = -21.9911485751286$$
$$x_{87} = 92.6769832808989$$
$$x_{87} = -15.707963267949$$
$$x_{87} = 50.2654824574367$$
$$x_{87} = 64.4026493985908$$
$$x_{87} = 48.6946861306418$$
$$x_{87} = 67.5442420521806$$
$$x_{87} = 21.9911485751286$$
$$x_{87} = -87.9645943005142$$
$$x_{87} = -72.2566310325652$$
$$x_{87} = 23.5619449019235$$
$$x_{87} = -117.809724509617$$
$$x_{87} = -80.1106126665397$$
$$x_{87} = -9.42477796076938$$
$$x_{87} = -64.4026493985908$$
$$x_{87} = -29.845130209103$$
$$x_{87} = -86.3937979737193$$
$$x_{87} = 4.71238898038469$$
$$x_{87} = 20.4203522483337$$
$$x_{87} = -65.9734457253857$$
$$x_{87} = -76.9690200129499$$
$$x_{87} = 15.707963267949$$
$$x_{87} = 28.2743338823081$$
$$x_{87} = 94.2477796076938$$
$$x_{87} = 37.6991118430775$$
$$x_{87} = -1.5707963267949$$
$$x_{87} = 6.28318530717959$$
$$x_{87} = 86.3937979737193$$
$$x_{87} = -53.4070751110265$$
$$x_{87} = -39.2699081698724$$
$$x_{87} = -6.28318530717959$$
$$x_{87} = 36.1283155162826$$
$$x_{87} = -75.398223686155$$
$$x_{87} = 51.8362787842316$$
$$x_{87} = 80.1106126665397$$
$$x_{87} = 0$$
$$x_{87} = 89.5353906273091$$
$$x_{87} = 58.1194640914112$$
$$x_{87} = 73.8274273593601$$
$$x_{87} = -58.1194640914112$$
$$x_{87} = -20.4203522483337$$
$$x_{87} = -97.3893722612836$$
$$x_{87} = -95.8185759344887$$
$$x_{87} = 29.845130209103$$
$$x_{87} = -28.2743338823081$$
$$x_{87} = 26.7035375555132$$
$$x_{87} = 100.530964914873$$
$$x_{87} = -23.5619449019235$$
$$x_{87} = -42.4115008234622$$
$$x_{87} = -61.261056745001$$
$$x_{87} = 78.5398163397448$$
$$x_{87} = 70.6858347057703$$
$$x_{87} = -98.9601685880785$$
$$x_{87} = 12.5663706143592$$
$$x_{87} = 87.9645943005142$$
$$x_{87} = 43.9822971502571$$
$$x_{87} = -31.4159265358979$$
$$x_{87} = 45.553093477052$$
$$x_{87} = 56.5486677646163$$
$$x_{87} = -81.6814089933346$$
$$x_{87} = -73.8274273593601$$
$$x_{87} = -59.6902604182061$$
$$x_{87} = 65.9734457253857$$
$$x_{87} = 174.358392274234$$
$$x_{87} = -45.553093477052$$
$$x_{87} = -7.85398163397448$$
$$x_{87} = -17.2787595947439$$
$$x_{87} = 95.8185759344887$$
Decrece en los intervalos
$$\left(-\infty, -117.809724509617\right]$$
Crece en los intervalos
$$\left[174.358392274234, \infty\right)$$