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$$\frac{x \cos{\left(2 x \right)}}{2} + \frac{\sin{\left(2 x \right)}}{4} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = -93.4650562152248$$
$$x_{2} = -82.469838530885$$
$$x_{3} = -33.7795214194042$$
$$x_{4} = 60.479792099527$$
$$x_{5} = -77.757633250469$$
$$x_{6} = -11.8021423864902$$
$$x_{7} = 25.927780364576$$
$$x_{8} = 46.3438858860085$$
$$x_{9} = 55.7677523585655$$
$$x_{10} = -16.5085005166786$$
$$x_{11} = -18.0779832097684$$
$$x_{12} = -62.0504837986507$$
$$x_{13} = -3.98933285620662$$
$$x_{14} = -71.4747305517771$$
$$x_{15} = 74.6161759525405$$
$$x_{16} = 98.1773168157084$$
$$x_{17} = 71.4747305517771$$
$$x_{18} = -27.4980262787482$$
$$x_{19} = -57.3384258953415$$
$$x_{20} = -46.3438858860085$$
$$x_{21} = 99.7480730445654$$
$$x_{22} = -38.4910046652094$$
$$x_{23} = 7.10371836259559$$
$$x_{24} = -25.927780364576$$
$$x_{25} = 41.6321073520443$$
$$x_{26} = 88.752809246359$$
$$x_{27} = -98.1773168157084$$
$$x_{28} = -63.6211806632638$$
$$x_{29} = 19.6476754907365$$
$$x_{30} = -41.6321073520443$$
$$x_{31} = -91.8943056074308$$
$$x_{32} = 62.0504837986507$$
$$x_{33} = 84.0405782018796$$
$$x_{34} = 54.1970859376957$$
$$x_{35} = -2.45659021971744$$
$$x_{36} = 47.9145054045097$$
$$x_{37} = 32.2090858609196$$
$$x_{38} = -1.01437891905522$$
$$x_{39} = 52.6264272696834$$
$$x_{40} = 77.757633250469$$
$$x_{41} = -69.9040128139871$$
$$x_{42} = 66.7625884309285$$
$$x_{43} = -99.7480730445654$$
$$x_{44} = 8.66818896199168$$
$$x_{45} = 16.5085005166786$$
$$x_{46} = -47.9145054045097$$
$$x_{47} = 69.9040128139871$$
$$x_{48} = -60.479792099527$$
$$x_{49} = 38.4910046652094$$
$$x_{50} = -79.3283659192419$$
$$x_{51} = 10.2345837013705$$
$$x_{52} = 76.186903206326$$
$$x_{53} = 33.7795214194042$$
$$x_{54} = 2.45659021971744$$
$$x_{55} = -85.6113199516972$$
$$x_{56} = 18.0779832097684$$
$$x_{57} = 49.4851361441979$$
$$x_{58} = -68.3332986887281$$
$$x_{59} = 63.6211806632638$$
$$x_{60} = -76.186903206326$$
$$x_{61} = -5.54276920324851$$
$$x_{62} = -35.349989019305$$
$$x_{63} = -54.1970859376957$$
$$x_{64} = 0$$
$$x_{65} = 3.98933285620662$$
$$x_{66} = -55.7677523585655$$
$$x_{67} = 24.3576053587789$$
$$x_{68} = -40.0615464074251$$
$$x_{69} = 30.6386872667848$$
$$x_{70} = 96.6065618907118$$
$$x_{71} = -13.3704580073937$$
$$x_{72} = -24.3576053587789$$
$$x_{73} = -84.0405782018796$$
$$x_{74} = 11.8021423864902$$
$$x_{75} = -32.2090858609196$$
$$x_{76} = 5.54276920324851$$
$$x_{77} = 91.8943056074308$$
$$x_{78} = -49.4851361441979$$
$$x_{79} = 90.3235565896713$$
$$x_{80} = -19.6476754907365$$
$$x_{81} = -10.2345837013705$$
$$x_{82} = 68.3332986887281$$
$$x_{83} = 27.4980262787482$$
$$x_{84} = 82.469838530885$$
$$x_{85} = -90.3235565896713$$
$$x_{86} = 85.6113199516972$$
$$x_{87} = 40.0615464074251$$
Signos de extremos en los puntos:
(-93.46505621522485, -23.3659297114269)
(-82.46983853088497, 20.6170807167555)
(-33.7795214194042, -8.44395539012157)
(60.47979209952698, 15.1194313498549)
(-77.75763325046901, -19.4390064352756)
(-11.802142386490203, -2.94789133120417)
(25.927780364575984, 6.48074015627518)
(46.3438858860085, -11.5852972235074)
(55.7677523585655, -13.94137776374)
(-16.508500516678623, 4.12523346635567)
(-18.07798320976836, -4.51776817153026)
(-62.050483798650674, -15.5121173520054)
(-3.9893328562066204, 0.989590921448473)
(-71.47473055177714, -17.8682454365327)
(74.61617595254046, -18.6536251922476)
(98.17731681570837, 24.5440109084885)
(71.47473055177714, -17.8682454365327)
(-27.498026278748195, -6.8733704062122)
(-57.338425895341494, 14.334061495186)
(-46.3438858860085, -11.5852972235074)
(99.74807304456543, -24.936704977785)
(-38.49100466520936, 9.62193939103296)
(7.103718362595594, 1.77154676422179)
(-25.927780364575984, 6.48074015627518)
(41.63210735204432, 10.4072762966192)
(88.75280924635904, 22.1878502184399)
(-98.17731681570837, 24.5440109084885)
(-63.62118066326382, 15.9048039999471)
(19.647675490736493, 4.91032912586147)
(-41.63210735204432, 10.4072762966192)
(-91.89430560743084, 22.9732363448112)
(62.050483798650674, -15.5121173520054)
(84.04057820187961, -21.0097727161598)
(54.197085937695654, 13.5486949219612)
(-2.456590219717442, -0.601808736214034)
(47.91450540450974, 11.9779742010582)
(32.20908586091958, 8.05130141742191)
(-1.014378919055217, 0.227463217644957)
(52.6264272696834, -13.156013049496)
(77.75763325046901, -19.4390064352756)
(-69.90401281398711, 17.4755561790741)
(66.76258843092853, 16.6901790509024)
(-99.74807304456543, -24.936704977785)
(8.66818896199168, -2.16345107598231)
(16.508500516678623, 4.12523346635567)
(-47.91450540450974, 11.9779742010582)
(69.90401281398711, 17.4755561790741)
(-60.47979209952698, 15.1194313498549)
(38.49100466520936, 9.62193939103296)
(-79.32836591924193, 19.8316975593184)
(10.234583701370475, 2.55559800728153)
(76.186903206326, 19.0463156393455)
(33.7795214194042, -8.44395539012157)
(2.456590219717442, -0.601808736214034)
(-85.61131995169717, 21.40246497544)
(18.07798320976836, -4.51776817153026)
(49.48513614419785, -12.3706525816398)
(-68.33329868872808, -17.0828673732396)
(63.62118066326382, 15.9048039999471)
(-76.186903206326, 19.0463156393455)
(-5.542769203248511, -1.38008850199125)
(-35.349989019305, 8.83661337019914)
(-54.197085937695654, 13.5486949219612)
(0, 0)
(3.9893328562066204, 0.989590921448473)
(-55.7677523585655, -13.94137776374)
(24.357605358778862, -6.08811877817099)
(-40.061546407425126, -10.0146066432074)
(30.638687266784828, -7.65865206805957)
(96.6065618907118, -24.1513170021851)
(-13.370458007393655, 3.34027970808717)
(-24.357605358778862, -6.08811877817099)
(-84.04057820187961, -21.0097727161598)
(11.802142386490203, -2.94789133120417)
(-32.20908586091958, 8.05130141742191)
(5.542769203248511, -1.38008850199125)
(91.89430560743084, 22.9732363448112)
(-49.48513614419785, -12.3706525816398)
(90.32355658967134, -22.5805431769646)
(-19.647675490736493, 4.91032912586147)
(-10.234583701370475, 2.55559800728153)
(68.33329868872808, -17.0828673732396)
(27.498026278748195, -6.8733704062122)
(82.46983853088497, 20.6170807167555)
(-90.32355658967134, -22.5805431769646)
(85.61131995169717, 21.40246497544)
(40.061546407425126, -10.0146066432074)
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:
$$x_{1} = -93.4650562152248$$
$$x_{2} = -33.7795214194042$$
$$x_{3} = -77.757633250469$$
$$x_{4} = -11.8021423864902$$
$$x_{5} = 46.3438858860085$$
$$x_{6} = 55.7677523585655$$
$$x_{7} = -18.0779832097684$$
$$x_{8} = -62.0504837986507$$
$$x_{9} = -71.4747305517771$$
$$x_{10} = 74.6161759525405$$
$$x_{11} = 71.4747305517771$$
$$x_{12} = -27.4980262787482$$
$$x_{13} = -46.3438858860085$$
$$x_{14} = 99.7480730445654$$
$$x_{15} = 62.0504837986507$$
$$x_{16} = 84.0405782018796$$
$$x_{17} = -2.45659021971744$$
$$x_{18} = 52.6264272696834$$
$$x_{19} = 77.757633250469$$
$$x_{20} = -99.7480730445654$$
$$x_{21} = 8.66818896199168$$
$$x_{22} = 33.7795214194042$$
$$x_{23} = 2.45659021971744$$
$$x_{24} = 18.0779832097684$$
$$x_{25} = 49.4851361441979$$
$$x_{26} = -68.3332986887281$$
$$x_{27} = -5.54276920324851$$
$$x_{28} = 0$$
$$x_{29} = -55.7677523585655$$
$$x_{30} = 24.3576053587789$$
$$x_{31} = -40.0615464074251$$
$$x_{32} = 30.6386872667848$$
$$x_{33} = 96.6065618907118$$
$$x_{34} = -24.3576053587789$$
$$x_{35} = -84.0405782018796$$
$$x_{36} = 11.8021423864902$$
$$x_{37} = 5.54276920324851$$
$$x_{38} = -49.4851361441979$$
$$x_{39} = 90.3235565896713$$
$$x_{40} = 68.3332986887281$$
$$x_{41} = 27.4980262787482$$
$$x_{42} = -90.3235565896713$$
$$x_{43} = 40.0615464074251$$
Puntos máximos de la función:
$$x_{43} = -82.469838530885$$
$$x_{43} = 60.479792099527$$
$$x_{43} = 25.927780364576$$
$$x_{43} = -16.5085005166786$$
$$x_{43} = -3.98933285620662$$
$$x_{43} = 98.1773168157084$$
$$x_{43} = -57.3384258953415$$
$$x_{43} = -38.4910046652094$$
$$x_{43} = 7.10371836259559$$
$$x_{43} = -25.927780364576$$
$$x_{43} = 41.6321073520443$$
$$x_{43} = 88.752809246359$$
$$x_{43} = -98.1773168157084$$
$$x_{43} = -63.6211806632638$$
$$x_{43} = 19.6476754907365$$
$$x_{43} = -41.6321073520443$$
$$x_{43} = -91.8943056074308$$
$$x_{43} = 54.1970859376957$$
$$x_{43} = 47.9145054045097$$
$$x_{43} = 32.2090858609196$$
$$x_{43} = -1.01437891905522$$
$$x_{43} = -69.9040128139871$$
$$x_{43} = 66.7625884309285$$
$$x_{43} = 16.5085005166786$$
$$x_{43} = -47.9145054045097$$
$$x_{43} = 69.9040128139871$$
$$x_{43} = -60.479792099527$$
$$x_{43} = 38.4910046652094$$
$$x_{43} = -79.3283659192419$$
$$x_{43} = 10.2345837013705$$
$$x_{43} = 76.186903206326$$
$$x_{43} = -85.6113199516972$$
$$x_{43} = 63.6211806632638$$
$$x_{43} = -76.186903206326$$
$$x_{43} = -35.349989019305$$
$$x_{43} = -54.1970859376957$$
$$x_{43} = 3.98933285620662$$
$$x_{43} = -13.3704580073937$$
$$x_{43} = -32.2090858609196$$
$$x_{43} = 91.8943056074308$$
$$x_{43} = -19.6476754907365$$
$$x_{43} = -10.2345837013705$$
$$x_{43} = 82.469838530885$$
$$x_{43} = 85.6113199516972$$
Decrece en los intervalos
$$\left[99.7480730445654, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -99.7480730445654\right]$$