In order to find the extrema, we need to solve the equation
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative$$\left(- 2 \sin{\left(2 x \right)} - \cos{\left(x \right)}\right) \operatorname{sign}{\left(- \sin{\left(x \right)} + \cos{\left(2 x \right)} \right)} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = 23.5619449019235$$
$$x_{2} = -58.1194640914112$$
$$x_{3} = -78.2871360846027$$
$$x_{4} = -6.53586556232167$$
$$x_{5} = -17.2787595947439$$
$$x_{6} = 45.553093477052$$
$$x_{7} = 56.2959875094742$$
$$x_{8} = 36.1283155162826$$
$$x_{9} = -102.101761241668$$
$$x_{10} = 73.8274273593601$$
$$x_{11} = -51.8362787842316$$
$$x_{12} = -25.3854214838604$$
$$x_{13} = -53.1543948558844$$
$$x_{14} = -67.5442420521806$$
$$x_{15} = -9.1720977056273$$
$$x_{16} = 1.5707963267949$$
$$x_{17} = 87.7119140453721$$
$$x_{18} = -72.0039507774232$$
$$x_{19} = -117.809724509617$$
$$x_{20} = -83.2522053201295$$
$$x_{21} = 66.2261259805277$$
$$x_{22} = -64.4026493985908$$
$$x_{23} = -75.6509039412971$$
$$x_{24} = 97.6420525164257$$
$$x_{25} = -45.553093477052$$
$$x_{26} = 6.03050505203751$$
$$x_{27} = 48.6946861306418$$
$$x_{28} = -65.7207654702436$$
$$x_{29} = 34.8101994446298$$
$$x_{30} = 64.4026493985908$$
$$x_{31} = -81.9340892484767$$
$$x_{32} = 58.1194640914112$$
$$x_{33} = 43.729616895115$$
$$x_{34} = 80.1106126665397$$
$$x_{35} = -28.0216536271661$$
$$x_{36} = 51.8362787842316$$
$$x_{37} = -15.4552830128069$$
$$x_{38} = 86.3937979737193$$
$$x_{39} = -59.437580163064$$
$$x_{40} = 28.5270141374502$$
$$x_{41} = -80.1106126665397$$
$$x_{42} = -50.5181627125788$$
$$x_{43} = -88.2172745556563$$
$$x_{44} = 22.2438288302706$$
$$x_{45} = 37.4464315879354$$
$$x_{46} = 14.1371669411541$$
$$x_{47} = -21.7384683199865$$
$$x_{48} = 95.8185759344887$$
$$x_{49} = 100.278284659731$$
$$x_{50} = 92.6769832808989$$
$$x_{51} = -44.2349774053992$$
$$x_{52} = 53.6597553661686$$
$$x_{53} = -86.3937979737193$$
$$x_{54} = 7.85398163397448$$
$$x_{55} = 50.0128022022946$$
$$x_{56} = -20.4203522483337$$
$$x_{57} = 59.9429406733481$$
$$x_{58} = -0.252680255142079$$
$$x_{59} = -89.5353906273091$$
$$x_{60} = -95.8185759344887$$
$$x_{61} = -36.1283155162826$$
$$x_{62} = 89.5353906273091$$
$$x_{63} = -42.4115008234622$$
$$x_{64} = -14.1371669411541$$
$$x_{65} = 67.5442420521806$$
$$x_{66} = -31.66860679104$$
$$x_{67} = -34.3048389343456$$
$$x_{68} = -103.419877313321$$
$$x_{69} = -23.5619449019235$$
$$x_{70} = 9.67745821591146$$
$$x_{71} = 93.9950993525517$$
$$x_{72} = -73.8274273593601$$
$$x_{73} = 81.4287287381925$$
$$x_{74} = 12.3136903592171$$
$$x_{75} = 29.845130209103$$
$$x_{76} = 18.5968756663967$$
$$x_{77} = 62.5791728166538$$
$$x_{78} = 78.7924965948869$$
$$x_{79} = -94.5004598628359$$
$$x_{80} = -1.5707963267949$$
$$x_{81} = 42.4115008234622$$
$$x_{82} = -97.1366920061415$$
$$x_{83} = 20.4203522483337$$
$$x_{84} = -29.845130209103$$
$$x_{85} = -37.9517920982196$$
$$x_{86} = -61.261056745001$$
$$x_{87} = 26.7035375555132$$
$$x_{88} = -70.6858347057703$$
$$x_{89} = -7.85398163397448$$
$$x_{90} = 70.6858347057703$$
$$x_{91} = 15.960643523091$$
$$x_{92} = 72.5093112877073$$
The values of the extrema at the points:
(23.5619449019235, 0)
(-58.1194640914112, 0)
(-78.2871360846027, 1.125)
(-6.53586556232167, 1.125)
(-17.2787595947439, 2)
(45.553093477052, 2)
(56.2959875094742, 1.125)
(36.1283155162826, 0)
(-102.101761241668, 0)
(73.8274273593601, 0)
(-51.8362787842316, 0)
(-25.3854214838604, 1.125)
(-53.1543948558844, 1.125)
(-67.5442420521806, 2)
(-9.1720977056273, 1.125)
(1.5707963267949, 2)
(87.7119140453721, 1.125)
(-72.0039507774232, 1.125)
(-117.809724509617, 2)
(-83.2522053201295, 0)
(66.2261259805277, 1.125)
(-64.4026493985908, 0)
(-75.6509039412971, 1.125)
(97.6420525164257, 1.125)
(-45.553093477052, 0)
(6.03050505203751, 1.125)
(48.6946861306418, 0)
(-65.7207654702436, 1.125)
(34.8101994446298, 1.125)
(64.4026493985908, 2)
(-81.9340892484767, 1.125)
(58.1194640914112, 2)
(43.729616895115, 1.125)
(80.1106126665397, 0)
(-28.0216536271661, 1.125)
(51.8362787842316, 2)
(-15.4552830128069, 1.125)
(86.3937979737193, 0)
(-59.437580163064, 1.125)
(28.5270141374502, 1.125)
(-80.1106126665397, 2)
(-50.5181627125788, 1.125)
(-88.2172745556563, 1.125)
(22.2438288302706, 1.125)
(37.4464315879354, 1.125)
(14.1371669411541, 2)
(-21.7384683199865, 1.125)
(95.8185759344887, 2)
(100.278284659731, 1.125)
(92.6769832808989, 0)
(-44.2349774053992, 1.125)
(53.6597553661686, 1.125)
(-86.3937979737193, 2)
(7.85398163397448, 2)
(50.0128022022946, 1.125)
(-20.4203522483337, 0)
(59.9429406733481, 1.125)
(-0.252680255142079, 1.125)
(-89.5353906273091, 0)
(-95.8185759344887, 0)
(-36.1283155162826, 2)
(89.5353906273091, 2)
(-42.4115008234622, 2)
(-14.1371669411541, 0)
(67.5442420521806, 0)
(-31.66860679104, 1.125)
(-34.3048389343456, 1.125)
(-103.419877313321, 1.125)
(-23.5619449019235, 2)
(9.67745821591146, 1.125)
(93.9950993525517, 1.125)
(-73.8274273593601, 2)
(81.4287287381925, 1.125)
(12.3136903592171, 1.125)
(29.845130209103, 0)
(18.5968756663967, 1.125)
(62.5791728166538, 1.125)
(78.7924965948869, 1.125)
(-94.5004598628359, 1.125)
(-1.5707963267949, 0)
(42.4115008234622, 0)
(-97.1366920061415, 1.125)
(20.4203522483337, 2)
(-29.845130209103, 2)
(-37.9517920982196, 1.125)
(-61.261056745001, 2)
(26.7035375555132, 2)
(-70.6858347057703, 0)
(-7.85398163397448, 0)
(70.6858347057703, 2)
(15.960643523091, 1.125)
(72.5093112877073, 1.125)
Intervals of increase and decrease of the function:Let's find intervals where the function increases and decreases, as well as minima and maxima of the function, for this let's look how the function behaves itself in the extremas and at the slightest deviation from:
Minima of the function at points:
$$x_{1} = 23.5619449019235$$
$$x_{2} = -58.1194640914112$$
$$x_{3} = 36.1283155162826$$
$$x_{4} = -102.101761241668$$
$$x_{5} = 73.8274273593601$$
$$x_{6} = -51.8362787842316$$
$$x_{7} = -83.2522053201295$$
$$x_{8} = -64.4026493985908$$
$$x_{9} = -45.553093477052$$
$$x_{10} = 48.6946861306418$$
$$x_{11} = 80.1106126665397$$
$$x_{12} = 86.3937979737193$$
$$x_{13} = 92.6769832808989$$
$$x_{14} = -20.4203522483337$$
$$x_{15} = -89.5353906273091$$
$$x_{16} = -95.8185759344887$$
$$x_{17} = -14.1371669411541$$
$$x_{18} = 67.5442420521806$$
$$x_{19} = 29.845130209103$$
$$x_{20} = -1.5707963267949$$
$$x_{21} = 42.4115008234622$$
$$x_{22} = -70.6858347057703$$
$$x_{23} = -7.85398163397448$$
Maxima of the function at points:
$$x_{23} = -78.2871360846027$$
$$x_{23} = -6.53586556232167$$
$$x_{23} = -17.2787595947439$$
$$x_{23} = 45.553093477052$$
$$x_{23} = 56.2959875094742$$
$$x_{23} = -25.3854214838604$$
$$x_{23} = -53.1543948558844$$
$$x_{23} = -67.5442420521806$$
$$x_{23} = -9.1720977056273$$
$$x_{23} = 1.5707963267949$$
$$x_{23} = 87.7119140453721$$
$$x_{23} = -72.0039507774232$$
$$x_{23} = -117.809724509617$$
$$x_{23} = 66.2261259805277$$
$$x_{23} = -75.6509039412971$$
$$x_{23} = 97.6420525164257$$
$$x_{23} = 6.03050505203751$$
$$x_{23} = -65.7207654702436$$
$$x_{23} = 34.8101994446298$$
$$x_{23} = 64.4026493985908$$
$$x_{23} = -81.9340892484767$$
$$x_{23} = 58.1194640914112$$
$$x_{23} = 43.729616895115$$
$$x_{23} = -28.0216536271661$$
$$x_{23} = 51.8362787842316$$
$$x_{23} = -15.4552830128069$$
$$x_{23} = -59.437580163064$$
$$x_{23} = 28.5270141374502$$
$$x_{23} = -80.1106126665397$$
$$x_{23} = -50.5181627125788$$
$$x_{23} = -88.2172745556563$$
$$x_{23} = 22.2438288302706$$
$$x_{23} = 37.4464315879354$$
$$x_{23} = 14.1371669411541$$
$$x_{23} = -21.7384683199865$$
$$x_{23} = 95.8185759344887$$
$$x_{23} = 100.278284659731$$
$$x_{23} = -44.2349774053992$$
$$x_{23} = 53.6597553661686$$
$$x_{23} = -86.3937979737193$$
$$x_{23} = 7.85398163397448$$
$$x_{23} = 50.0128022022946$$
$$x_{23} = 59.9429406733481$$
$$x_{23} = -0.252680255142079$$
$$x_{23} = -36.1283155162826$$
$$x_{23} = 89.5353906273091$$
$$x_{23} = -42.4115008234622$$
$$x_{23} = -31.66860679104$$
$$x_{23} = -34.3048389343456$$
$$x_{23} = -103.419877313321$$
$$x_{23} = -23.5619449019235$$
$$x_{23} = 9.67745821591146$$
$$x_{23} = 93.9950993525517$$
$$x_{23} = -73.8274273593601$$
$$x_{23} = 81.4287287381925$$
$$x_{23} = 12.3136903592171$$
$$x_{23} = 18.5968756663967$$
$$x_{23} = 62.5791728166538$$
$$x_{23} = 78.7924965948869$$
$$x_{23} = -94.5004598628359$$
$$x_{23} = -97.1366920061415$$
$$x_{23} = 20.4203522483337$$
$$x_{23} = -29.845130209103$$
$$x_{23} = -37.9517920982196$$
$$x_{23} = -61.261056745001$$
$$x_{23} = 26.7035375555132$$
$$x_{23} = 70.6858347057703$$
$$x_{23} = 15.960643523091$$
$$x_{23} = 72.5093112877073$$
Decreasing at intervals
$$\left[92.6769832808989, \infty\right)$$
Increasing at intervals
$$\left(-\infty, -102.101761241668\right]$$