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$$4 \cos{\left(2 x \right)} \operatorname{sign}{\left(2 \sin{\left(2 x \right)} - 1 \right)} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = -41.6261026600648$$
$$x_{2} = 18.0641577581413$$
$$x_{3} = 25.9181393921158$$
$$x_{4} = 41.6261026600648$$
$$x_{5} = 40.0553063332699$$
$$x_{6} = -73.0420291959627$$
$$x_{7} = -47.9092879672443$$
$$x_{8} = 88.7499924639117$$
$$x_{9} = 96.6039740978861$$
$$x_{10} = 82.4668071567321$$
$$x_{11} = 65.1880475619882$$
$$x_{12} = -13.3517687777566$$
$$x_{13} = -62.0464549083984$$
$$x_{14} = 44.7676953136546$$
$$x_{15} = -24.3473430653209$$
$$x_{16} = -16.4933614313464$$
$$x_{17} = 19.6349540849362$$
$$x_{18} = -3.92699081698724$$
$$x_{19} = 52.621676947629$$
$$x_{20} = -84.037603483527$$
$$x_{21} = -49.4800842940392$$
$$x_{22} = -63.6172512351933$$
$$x_{23} = 55.7632696012188$$
$$x_{24} = 91.8915851175014$$
$$x_{25} = -117.02432634622$$
$$x_{26} = 60.4756585816035$$
$$x_{27} = -10.2101761241668$$
$$x_{28} = 32.2013246992954$$
$$x_{29} = -85.6083998103219$$
$$x_{30} = 33.7721210260903$$
$$x_{31} = -57.3340659280137$$
$$x_{32} = -32.2013246992954$$
$$x_{33} = -58.9048622548086$$
$$x_{34} = 30.6305283725005$$
$$x_{35} = -82.4668071567321$$
$$x_{36} = 3.92699081698724$$
$$x_{37} = 2.35619449019234$$
$$x_{38} = 1265.27644123329$$
$$x_{39} = 16.4933614313464$$
$$x_{40} = 76.1836218495525$$
$$x_{41} = -2.35619449019234$$
$$x_{42} = -90.3207887907066$$
$$x_{43} = -46.3384916404494$$
$$x_{44} = -98.174770424681$$
$$x_{45} = 46.3384916404494$$
$$x_{46} = 5.49778714378214$$
$$x_{47} = 47.9092879672443$$
$$x_{48} = 99.7455667514759$$
$$x_{49} = -91.8915851175014$$
$$x_{50} = -33.7721210260903$$
$$x_{51} = -19.6349540849362$$
$$x_{52} = -60.4756585816035$$
$$x_{53} = 27.4889357189107$$
$$x_{54} = -71.4712328691678$$
$$x_{55} = 84.037603483527$$
$$x_{56} = -76.1836218495525$$
$$x_{57} = -38.484510006475$$
$$x_{58} = 68.329640215578$$
$$x_{59} = 66.7588438887831$$
$$x_{60} = -11.7809724509617$$
$$x_{61} = -270.96236637212$$
$$x_{62} = -27.4889357189107$$
$$x_{63} = 62.0464549083984$$
$$x_{64} = -25.9181393921158$$
$$x_{65} = -40.0553063332699$$
$$x_{66} = -5.49778714378214$$
$$x_{67} = 54.1924732744239$$
$$x_{68} = -35.3429173528852$$
$$x_{69} = -52.621676947629$$
$$x_{70} = 90.3207887907066$$
$$x_{71} = -55.7632696012188$$
$$x_{72} = 11.7809724509617$$
$$x_{73} = 8.63937979737193$$
$$x_{74} = 74.6128255227576$$
$$x_{75} = -54.1924732744239$$
$$x_{76} = -99.7455667514759$$
$$x_{77} = -18.0641577581413$$
$$x_{78} = 38.484510006475$$
$$x_{79} = -77.7544181763474$$
$$x_{80} = -79.3252145031423$$
$$x_{81} = 10.2101761241668$$
$$x_{82} = 98.174770424681$$
$$x_{83} = -68.329640215578$$
$$x_{84} = -69.9004365423729$$
$$x_{85} = 69.9004365423729$$
$$x_{86} = 24.3473430653209$$
$$x_{87} = 219.126087587888$$
$$x_{88} = 14.9225651045515$$
$$x_{89} = -93.4623814442964$$
$$x_{90} = 77.7544181763474$$
Signos de extremos en los puntos:
(-41.62610266006476, 3)
(18.06415775814131, 3)
(25.918139392115794, 1)
(41.62610266006476, 1)
(40.05530633326986, 3)
(-73.0420291959627, 3)
(-47.909287967244346, 3)
(88.74999246391165, 1)
(96.60397409788614, 3)
(82.46680715673207, 1)
(65.18804756198821, 3)
(-13.351768777756622, 3)
(-62.04645490839842, 1)
(44.767695313654556, 1)
(-24.3473430653209, 1)
(-16.493361431346415, 3)
(19.634954084936208, 1)
(-3.9269908169872414, 3)
(52.621676947629034, 3)
(-84.03760348352696, 1)
(-49.480084294039244, 1)
(-63.617251235193315, 3)
(55.76326960121883, 3)
(91.89158511750145, 1)
(-117.0243263462198, 3)
(60.47565858160352, 1)
(-10.210176124166829, 3)
(32.201324699295384, 1)
(-85.60839981032187, 3)
(33.772121026090275, 3)
(-57.33406592801373, 3)
(-32.201324699295384, 3)
(-58.90486225480862, 1)
(30.630528372500486, 3)
(-82.46680715673207, 3)
(3.9269908169872414, 1)
(2.356194490192345, 3)
(1265.2764412332892, 3)
(16.493361431346415, 1)
(76.18362184955248, 1)
(-2.356194490192345, 1)
(-90.32078879070656, 1)
(-46.33849164044945, 1)
(-98.17477042468104, 3)
(46.33849164044945, 3)
(5.497787143782138, 3)
(47.909287967244346, 1)
(99.74556675147593, 3)
(-91.89158511750145, 3)
(-33.772121026090275, 1)
(-19.634954084936208, 3)
(-60.47565858160352, 3)
(27.488935718910692, 3)
(-71.47123286916779, 1)
(84.03760348352696, 3)
(-76.18362184955248, 3)
(-38.48451000647497, 3)
(68.329640215578, 3)
(66.7588438887831, 1)
(-11.780972450961725, 1)
(-270.9623663721197, 3)
(-27.488935718910692, 1)
(62.04645490839842, 3)
(-25.918139392115794, 3)
(-40.05530633326986, 1)
(-5.497787143782138, 1)
(54.19247327442393, 1)
(-35.34291735288517, 3)
(-52.621676947629034, 1)
(90.32078879070656, 3)
(-55.76326960121883, 1)
(11.780972450961725, 3)
(8.639379797371932, 3)
(74.61282552275759, 3)
(-54.19247327442393, 3)
(-99.74556675147593, 1)
(-18.06415775814131, 1)
(38.48451000647497, 1)
(-77.75441817634739, 1)
(-79.32521450314228, 3)
(10.210176124166829, 1)
(98.17477042468104, 1)
(-68.329640215578, 1)
(-69.9004365423729, 3)
(69.9004365423729, 1)
(24.3473430653209, 3)
(219.12608758788807, 3)
(14.922565104551518, 3)
(-93.46238144429635, 1)
(77.75441817634739, 3)
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:
$$x_{90} = -41.6261026600648$$
$$x_{90} = 18.0641577581413$$
$$x_{90} = 25.9181393921158$$
$$x_{90} = 41.6261026600648$$
$$x_{90} = 40.0553063332699$$
$$x_{90} = -73.0420291959627$$
$$x_{90} = -47.9092879672443$$
$$x_{90} = 88.7499924639117$$
$$x_{90} = 96.6039740978861$$
$$x_{90} = 82.4668071567321$$
$$x_{90} = 65.1880475619882$$
$$x_{90} = -13.3517687777566$$
$$x_{90} = -62.0464549083984$$
$$x_{90} = 44.7676953136546$$
$$x_{90} = -24.3473430653209$$
$$x_{90} = -16.4933614313464$$
$$x_{90} = 19.6349540849362$$
$$x_{90} = -3.92699081698724$$
$$x_{90} = 52.621676947629$$
$$x_{90} = -84.037603483527$$
$$x_{90} = -49.4800842940392$$
$$x_{90} = -63.6172512351933$$
$$x_{90} = 55.7632696012188$$
$$x_{90} = 91.8915851175014$$
$$x_{90} = -117.02432634622$$
$$x_{90} = 60.4756585816035$$
$$x_{90} = -10.2101761241668$$
$$x_{90} = 32.2013246992954$$
$$x_{90} = -85.6083998103219$$
$$x_{90} = 33.7721210260903$$
$$x_{90} = -57.3340659280137$$
$$x_{90} = -32.2013246992954$$
$$x_{90} = -58.9048622548086$$
$$x_{90} = 30.6305283725005$$
$$x_{90} = -82.4668071567321$$
$$x_{90} = 3.92699081698724$$
$$x_{90} = 2.35619449019234$$
$$x_{90} = 1265.27644123329$$
$$x_{90} = 16.4933614313464$$
$$x_{90} = 76.1836218495525$$
$$x_{90} = -2.35619449019234$$
$$x_{90} = -90.3207887907066$$
$$x_{90} = -46.3384916404494$$
$$x_{90} = -98.174770424681$$
$$x_{90} = 46.3384916404494$$
$$x_{90} = 5.49778714378214$$
$$x_{90} = 47.9092879672443$$
$$x_{90} = 99.7455667514759$$
$$x_{90} = -91.8915851175014$$
$$x_{90} = -33.7721210260903$$
$$x_{90} = -19.6349540849362$$
$$x_{90} = -60.4756585816035$$
$$x_{90} = 27.4889357189107$$
$$x_{90} = -71.4712328691678$$
$$x_{90} = 84.037603483527$$
$$x_{90} = -76.1836218495525$$
$$x_{90} = -38.484510006475$$
$$x_{90} = 68.329640215578$$
$$x_{90} = 66.7588438887831$$
$$x_{90} = -11.7809724509617$$
$$x_{90} = -270.96236637212$$
$$x_{90} = -27.4889357189107$$
$$x_{90} = 62.0464549083984$$
$$x_{90} = -25.9181393921158$$
$$x_{90} = -40.0553063332699$$
$$x_{90} = -5.49778714378214$$
$$x_{90} = 54.1924732744239$$
$$x_{90} = -35.3429173528852$$
$$x_{90} = -52.621676947629$$
$$x_{90} = 90.3207887907066$$
$$x_{90} = -55.7632696012188$$
$$x_{90} = 11.7809724509617$$
$$x_{90} = 8.63937979737193$$
$$x_{90} = 74.6128255227576$$
$$x_{90} = -54.1924732744239$$
$$x_{90} = -99.7455667514759$$
$$x_{90} = -18.0641577581413$$
$$x_{90} = 38.484510006475$$
$$x_{90} = -77.7544181763474$$
$$x_{90} = -79.3252145031423$$
$$x_{90} = 10.2101761241668$$
$$x_{90} = 98.174770424681$$
$$x_{90} = -68.329640215578$$
$$x_{90} = -69.9004365423729$$
$$x_{90} = 69.9004365423729$$
$$x_{90} = 24.3473430653209$$
$$x_{90} = 219.126087587888$$
$$x_{90} = 14.9225651045515$$
$$x_{90} = -93.4623814442964$$
$$x_{90} = 77.7544181763474$$
Decrece en los intervalos
$$\left(-\infty, -270.96236637212\right]$$
Crece en los intervalos
$$\left[1265.27644123329, \infty\right)$$