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$$2 x \cos{\left(2 x \right)} + \sin{\left(2 x \right)} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = 63.6211806632638$$
$$x_{2} = 85.6113199516972$$
$$x_{3} = -19.6476754907365$$
$$x_{4} = 54.1970859376957$$
$$x_{5} = -98.1773168157084$$
$$x_{6} = -16.5085005166786$$
$$x_{7} = -82.469838530885$$
$$x_{8} = -5.54276920324851$$
$$x_{9} = 3.98933285620662$$
$$x_{10} = -79.3283659192419$$
$$x_{11} = -10.2345837013705$$
$$x_{12} = 7.10371836259559$$
$$x_{13} = -77.757633250469$$
$$x_{14} = 11.8021423864902$$
$$x_{15} = -40.0615464074251$$
$$x_{16} = 52.6264272696834$$
$$x_{17} = -76.186903206326$$
$$x_{18} = -32.2090858609196$$
$$x_{19} = 24.3576053587789$$
$$x_{20} = 2.45659021971744$$
$$x_{21} = 16.5085005166786$$
$$x_{22} = 25.927780364576$$
$$x_{23} = -71.4747305517771$$
$$x_{24} = 62.0504837986507$$
$$x_{25} = 41.6321073520443$$
$$x_{26} = 18.0779832097684$$
$$x_{27} = 82.469838530885$$
$$x_{28} = 74.6161759525405$$
$$x_{29} = 71.4747305517771$$
$$x_{30} = 91.8943056074308$$
$$x_{31} = -1.01437891905522$$
$$x_{32} = 69.9040128139871$$
$$x_{33} = 0$$
$$x_{34} = 99.7480730445654$$
$$x_{35} = -18.0779832097684$$
$$x_{36} = 8.66818896199168$$
$$x_{37} = 84.0405782018796$$
$$x_{38} = -35.349989019305$$
$$x_{39} = -41.6321073520443$$
$$x_{40} = -99.7480730445654$$
$$x_{41} = -11.8021423864902$$
$$x_{42} = -85.6113199516972$$
$$x_{43} = -68.3332986887281$$
$$x_{44} = 49.4851361441979$$
$$x_{45} = -25.927780364576$$
$$x_{46} = -24.3576053587789$$
$$x_{47} = -27.4980262787482$$
$$x_{48} = -54.1970859376957$$
$$x_{49} = -90.3235565896713$$
$$x_{50} = -13.3704580073937$$
$$x_{51} = 98.1773168157084$$
$$x_{52} = -62.0504837986507$$
$$x_{53} = -60.479792099527$$
$$x_{54} = -3.98933285620662$$
$$x_{55} = -49.4851361441979$$
$$x_{56} = 46.3438858860085$$
$$x_{57} = -93.4650562152248$$
$$x_{58} = -47.9145054045097$$
$$x_{59} = -84.0405782018796$$
$$x_{60} = 77.757633250469$$
$$x_{61} = 38.4910046652094$$
$$x_{62} = 19.6476754907365$$
$$x_{63} = -33.7795214194042$$
$$x_{64} = -2.45659021971744$$
$$x_{65} = 47.9145054045097$$
$$x_{66} = -38.4910046652094$$
$$x_{67} = 55.7677523585655$$
$$x_{68} = 30.6386872667848$$
$$x_{69} = -69.9040128139871$$
$$x_{70} = -63.6211806632638$$
$$x_{71} = -46.3438858860085$$
$$x_{72} = 10.2345837013705$$
$$x_{73} = 96.6065618907118$$
$$x_{74} = 88.752809246359$$
$$x_{75} = -57.3384258953415$$
$$x_{76} = -91.8943056074308$$
$$x_{77} = 76.186903206326$$
$$x_{78} = 60.479792099527$$
$$x_{79} = -55.7677523585655$$
$$x_{80} = 27.4980262787482$$
$$x_{81} = 66.7625884309285$$
$$x_{82} = 40.0615464074251$$
$$x_{83} = 33.7795214194042$$
$$x_{84} = 68.3332986887281$$
$$x_{85} = 5.54276920324851$$
$$x_{86} = 32.2090858609196$$
$$x_{87} = 90.3235565896713$$
Signos de extremos en los puntos:
(63.62118066326382, 63.6192159997885)
(85.61131995169717, 85.6098599017601)
(-19.647675490736493, 19.6413165034459)
(54.197085937695654, 54.1947796878447)
(-98.17731681570837, 98.1760436339539)
(-16.508500516678623, 16.5009338654227)
(-82.46983853088497, 82.4683228670218)
(-5.542769203248511, -5.520354007965)
(3.9893328562066204, 3.95836368579389)
(-79.32836591924193, 79.3267902372738)
(-10.234583701370475, 10.2223920291261)
(7.103718362595594, 7.08618705688714)
(-77.75763325046901, -77.7560257411026)
(11.802142386490203, -11.7915653248167)
(-40.061546407425126, -40.0584265728296)
(52.6264272696834, -52.6240521979841)
(-76.186903206326, 76.185262557382)
(-32.20908586091958, 32.2052056696877)
(24.357605358778862, -24.352475112684)
(2.456590219717442, -2.40723494485613)
(16.508500516678623, 16.5009338654227)
(25.927780364575984, 25.9229606251007)
(-71.47473055177714, -71.4729817461307)
(62.050483798650674, -62.0484694080218)
(41.63210735204432, 41.6291051864766)
(18.07798320976836, -18.071072686121)
(82.46983853088497, 82.4683228670218)
(74.61617595254046, -74.6145007689905)
(71.47473055177714, -71.4729817461307)
(91.89430560743084, 91.8929453792449)
(-1.014378919055217, 0.909852870579826)
(69.90401281398711, 69.9022247162962)
(0, 0)
(99.74807304456543, -99.7468199111401)
(-18.07798320976836, -18.071072686121)
(8.66818896199168, -8.65380430392926)
(84.04057820187961, -84.0390908646392)
(-35.349989019305, 35.3464534807966)
(-41.63210735204432, 41.6291051864766)
(-99.74807304456543, -99.7468199111401)
(-11.802142386490203, -11.7915653248167)
(-85.61131995169717, 85.6098599017601)
(-68.33329868872808, -68.3314694929585)
(49.48513614419785, -49.4826103265593)
(-25.927780364575984, 25.9229606251007)
(-24.357605358778862, -24.352475112684)
(-27.498026278748195, -27.4934816248488)
(-54.197085937695654, 54.1947796878447)
(-90.32355658967134, -90.3221727078584)
(-13.370458007393655, 13.3611188323487)
(98.17731681570837, 98.1760436339539)
(-62.050483798650674, -62.0484694080218)
(-60.47979209952698, 60.4777253994195)
(-3.9893328562066204, 3.95836368579389)
(-49.48513614419785, -49.4826103265593)
(46.3438858860085, -46.3411888940296)
(-93.46505621522485, -93.4637188457076)
(-47.91450540450974, 47.9118968042328)
(-84.04057820187961, -84.0390908646392)
(77.75763325046901, -77.7560257411026)
(38.49100466520936, 38.4877575641319)
(19.647675490736493, 19.6413165034459)
(-33.7795214194042, -33.7758215604863)
(-2.456590219717442, -2.40723494485613)
(47.91450540450974, 47.9118968042328)
(-38.49100466520936, 38.4877575641319)
(55.7677523585655, -55.76551105496)
(30.638687266784828, -30.6346082722383)
(-69.90401281398711, 69.9022247162962)
(-63.62118066326382, 63.6192159997885)
(-46.3438858860085, -46.3411888940296)
(10.234583701370475, 10.2223920291261)
(96.6065618907118, -96.6052680087403)
(88.75280924635904, 88.7514008737596)
(-57.338425895341494, 57.3362459807442)
(-91.89430560743084, 91.8929453792449)
(76.186903206326, 76.185262557382)
(60.47979209952698, 60.4777253994195)
(-55.7677523585655, -55.76551105496)
(27.498026278748195, -27.4934816248488)
(66.76258843092853, 66.7607162036095)
(40.061546407425126, -40.0584265728296)
(33.7795214194042, -33.7758215604863)
(68.33329868872808, -68.3314694929585)
(5.542769203248511, -5.520354007965)
(32.20908586091958, 32.2052056696877)
(90.32355658967134, -90.3221727078584)
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:
$$x_{1} = -5.54276920324851$$
$$x_{2} = -77.757633250469$$
$$x_{3} = 11.8021423864902$$
$$x_{4} = -40.0615464074251$$
$$x_{5} = 52.6264272696834$$
$$x_{6} = 24.3576053587789$$
$$x_{7} = 2.45659021971744$$
$$x_{8} = -71.4747305517771$$
$$x_{9} = 62.0504837986507$$
$$x_{10} = 18.0779832097684$$
$$x_{11} = 74.6161759525405$$
$$x_{12} = 71.4747305517771$$
$$x_{13} = 0$$
$$x_{14} = 99.7480730445654$$
$$x_{15} = -18.0779832097684$$
$$x_{16} = 8.66818896199168$$
$$x_{17} = 84.0405782018796$$
$$x_{18} = -99.7480730445654$$
$$x_{19} = -11.8021423864902$$
$$x_{20} = -68.3332986887281$$
$$x_{21} = 49.4851361441979$$
$$x_{22} = -24.3576053587789$$
$$x_{23} = -27.4980262787482$$
$$x_{24} = -90.3235565896713$$
$$x_{25} = -62.0504837986507$$
$$x_{26} = -49.4851361441979$$
$$x_{27} = 46.3438858860085$$
$$x_{28} = -93.4650562152248$$
$$x_{29} = -84.0405782018796$$
$$x_{30} = 77.757633250469$$
$$x_{31} = -33.7795214194042$$
$$x_{32} = -2.45659021971744$$
$$x_{33} = 55.7677523585655$$
$$x_{34} = 30.6386872667848$$
$$x_{35} = -46.3438858860085$$
$$x_{36} = 96.6065618907118$$
$$x_{37} = -55.7677523585655$$
$$x_{38} = 27.4980262787482$$
$$x_{39} = 40.0615464074251$$
$$x_{40} = 33.7795214194042$$
$$x_{41} = 68.3332986887281$$
$$x_{42} = 5.54276920324851$$
$$x_{43} = 90.3235565896713$$
Puntos máximos de la función:
$$x_{43} = 63.6211806632638$$
$$x_{43} = 85.6113199516972$$
$$x_{43} = -19.6476754907365$$
$$x_{43} = 54.1970859376957$$
$$x_{43} = -98.1773168157084$$
$$x_{43} = -16.5085005166786$$
$$x_{43} = -82.469838530885$$
$$x_{43} = 3.98933285620662$$
$$x_{43} = -79.3283659192419$$
$$x_{43} = -10.2345837013705$$
$$x_{43} = 7.10371836259559$$
$$x_{43} = -76.186903206326$$
$$x_{43} = -32.2090858609196$$
$$x_{43} = 16.5085005166786$$
$$x_{43} = 25.927780364576$$
$$x_{43} = 41.6321073520443$$
$$x_{43} = 82.469838530885$$
$$x_{43} = 91.8943056074308$$
$$x_{43} = -1.01437891905522$$
$$x_{43} = 69.9040128139871$$
$$x_{43} = -35.349989019305$$
$$x_{43} = -41.6321073520443$$
$$x_{43} = -85.6113199516972$$
$$x_{43} = -25.927780364576$$
$$x_{43} = -54.1970859376957$$
$$x_{43} = -13.3704580073937$$
$$x_{43} = 98.1773168157084$$
$$x_{43} = -60.479792099527$$
$$x_{43} = -3.98933285620662$$
$$x_{43} = -47.9145054045097$$
$$x_{43} = 38.4910046652094$$
$$x_{43} = 19.6476754907365$$
$$x_{43} = 47.9145054045097$$
$$x_{43} = -38.4910046652094$$
$$x_{43} = -69.9040128139871$$
$$x_{43} = -63.6211806632638$$
$$x_{43} = 10.2345837013705$$
$$x_{43} = 88.752809246359$$
$$x_{43} = -57.3384258953415$$
$$x_{43} = -91.8943056074308$$
$$x_{43} = 76.186903206326$$
$$x_{43} = 60.479792099527$$
$$x_{43} = 66.7625884309285$$
$$x_{43} = 32.2090858609196$$
Decrece en los intervalos
$$\left[99.7480730445654, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -99.7480730445654\right]$$