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$$\frac{2 x \cos{\left(2 x \right)}}{3} + \frac{\sin{\left(2 x \right)}}{3} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = -98.1773168157084$$
$$x_{2} = 99.7480730445654$$
$$x_{3} = -46.3438858860085$$
$$x_{4} = 11.8021423864902$$
$$x_{5} = -18.0779832097684$$
$$x_{6} = -63.6211806632638$$
$$x_{7} = -35.349989019305$$
$$x_{8} = -2.45659021971744$$
$$x_{9} = -79.3283659192419$$
$$x_{10} = 88.752809246359$$
$$x_{11} = 52.6264272696834$$
$$x_{12} = -90.3235565896713$$
$$x_{13} = -55.7677523585655$$
$$x_{14} = -60.479792099527$$
$$x_{15} = -69.9040128139871$$
$$x_{16} = 33.7795214194042$$
$$x_{17} = 76.186903206326$$
$$x_{18} = 40.0615464074251$$
$$x_{19} = -49.4851361441979$$
$$x_{20} = -47.9145054045097$$
$$x_{21} = 8.66818896199168$$
$$x_{22} = 46.3438858860085$$
$$x_{23} = -16.5085005166786$$
$$x_{24} = -99.7480730445654$$
$$x_{25} = 74.6161759525405$$
$$x_{26} = -19.6476754907365$$
$$x_{27} = 24.3576053587789$$
$$x_{28} = -13.3704580073937$$
$$x_{29} = 3.98933285620662$$
$$x_{30} = 18.0779832097684$$
$$x_{31} = 63.6211806632638$$
$$x_{32} = 19.6476754907365$$
$$x_{33} = 66.7625884309285$$
$$x_{34} = 54.1970859376957$$
$$x_{35} = -5.54276920324851$$
$$x_{36} = 98.1773168157084$$
$$x_{37} = -1.01437891905522$$
$$x_{38} = 2.45659021971744$$
$$x_{39} = 68.3332986887281$$
$$x_{40} = -76.186903206326$$
$$x_{41} = 16.5085005166786$$
$$x_{42} = 0$$
$$x_{43} = -71.4747305517771$$
$$x_{44} = -33.7795214194042$$
$$x_{45} = 96.6065618907118$$
$$x_{46} = 38.4910046652094$$
$$x_{47} = 41.6321073520443$$
$$x_{48} = -32.2090858609196$$
$$x_{49} = -82.469838530885$$
$$x_{50} = -25.927780364576$$
$$x_{51} = 30.6386872667848$$
$$x_{52} = -85.6113199516972$$
$$x_{53} = -84.0405782018796$$
$$x_{54} = -10.2345837013705$$
$$x_{55} = 47.9145054045097$$
$$x_{56} = 69.9040128139871$$
$$x_{57} = -24.3576053587789$$
$$x_{58} = -41.6321073520443$$
$$x_{59} = -38.4910046652094$$
$$x_{60} = -77.757633250469$$
$$x_{61} = 62.0504837986507$$
$$x_{62} = 77.757633250469$$
$$x_{63} = -62.0504837986507$$
$$x_{64} = 85.6113199516972$$
$$x_{65} = -40.0615464074251$$
$$x_{66} = 84.0405782018796$$
$$x_{67} = 71.4747305517771$$
$$x_{68} = 49.4851361441979$$
$$x_{69} = -54.1970859376957$$
$$x_{70} = -57.3384258953415$$
$$x_{71} = -27.4980262787482$$
$$x_{72} = -93.4650562152248$$
$$x_{73} = 27.4980262787482$$
$$x_{74} = 5.54276920324851$$
$$x_{75} = -43.2026854058443$$
$$x_{76} = 25.927780364576$$
$$x_{77} = 82.469838530885$$
$$x_{78} = -91.8943056074308$$
$$x_{79} = 91.8943056074308$$
$$x_{80} = 55.7677523585655$$
$$x_{81} = 32.2090858609196$$
$$x_{82} = 60.479792099527$$
$$x_{83} = 90.3235565896713$$
$$x_{84} = -3.98933285620662$$
$$x_{85} = 7.10371836259559$$
$$x_{86} = -11.8021423864902$$
$$x_{87} = 10.2345837013705$$
$$x_{88} = -68.3332986887281$$
Signos de extremos en los puntos:
(-98.17731681570837, 32.7253478779846)
(99.74807304456543, -33.24893997038)
(-46.3438858860085, -15.4470629646765)
(11.802142386490203, -3.93052177493889)
(-18.07798320976836, -6.02369089537368)
(-63.62118066326382, 21.2064053332628)
(-35.349989019305, 11.7821511602655)
(-2.456590219717442, -0.802411648285378)
(-79.32836591924193, 26.4422634124246)
(88.75280924635904, 29.5838002912532)
(52.6264272696834, -17.5413507326614)
(-90.32355658967134, -30.1073909026194)
(-55.7677523585655, -18.5885036849867)
(-60.47979209952698, 20.1592417998065)
(-69.90401281398711, 23.3007415720987)
(33.7795214194042, -11.2586071868288)
(76.186903206326, 25.3950875191273)
(40.061546407425126, -13.3528088576099)
(-49.48513614419785, -16.4942034421864)
(-47.91450540450974, 15.9706322680776)
(8.66818896199168, -2.88460143464309)
(46.3438858860085, -15.4470629646765)
(-16.508500516678623, 5.50031128847423)
(-99.74807304456543, -33.24893997038)
(74.61617595254046, -24.8715002563302)
(-19.647675490736493, 6.54710550114863)
(24.357605358778862, -8.11749170422799)
(-13.370458007393655, 4.45370627744957)
(3.9893328562066204, 1.3194545619313)
(18.07798320976836, -6.02369089537368)
(63.62118066326382, 21.2064053332628)
(19.647675490736493, 6.54710550114863)
(66.76258843092853, 22.2535720678698)
(54.197085937695654, 18.0649265626149)
(-5.542769203248511, -1.840118002655)
(98.17731681570837, 32.7253478779846)
(-1.014378919055217, 0.303284290193275)
(2.456590219717442, -0.802411648285378)
(68.33329868872808, -22.7771564976528)
(-76.186903206326, 25.3950875191273)
(16.508500516678623, 5.50031128847423)
(0, 0)
(-71.47473055177714, -23.8243272487102)
(-33.7795214194042, -11.2586071868288)
(96.6065618907118, -32.2017560029134)
(38.49100466520936, 12.8292525213773)
(41.63210735204432, 13.8763683954922)
(-32.20908586091958, 10.7350685565625)
(-82.46983853088497, 27.4894409556739)
(-25.927780364575984, 8.64098687503358)
(30.638687266784828, -10.2115360907461)
(-85.61131995169717, 28.5366199672534)
(-84.04057820187961, -28.0130302882131)
(-10.234583701370475, 3.40746400970871)
(47.91450540450974, 15.9706322680776)
(69.90401281398711, 23.3007415720987)
(-24.357605358778862, -8.11749170422799)
(-41.63210735204432, 13.8763683954922)
(-38.49100466520936, 12.8292525213773)
(-77.75763325046901, -25.9186752470342)
(62.050483798650674, -20.6828231360073)
(77.75763325046901, -25.9186752470342)
(-62.050483798650674, -20.6828231360073)
(85.61131995169717, 28.5366199672534)
(-40.061546407425126, -13.3528088576099)
(84.04057820187961, -28.0130302882131)
(71.47473055177714, -23.8243272487102)
(49.48513614419785, -16.4942034421864)
(-54.197085937695654, 18.0649265626149)
(-57.338425895341494, 19.1120819935814)
(-27.498026278748195, -9.1644938749496)
(-93.46505621522485, -31.1545729485692)
(27.498026278748195, -9.1644938749496)
(5.542769203248511, -1.840118002655)
(-43.20268540584427, -14.3999307859351)
(25.927780364575984, 8.64098687503358)
(82.46983853088497, 27.4894409556739)
(-91.89430560743084, 30.6309817930816)
(91.89430560743084, 30.6309817930816)
(55.7677523585655, -18.5885036849867)
(32.20908586091958, 10.7350685565625)
(60.47979209952698, 20.1592417998065)
(90.32355658967134, -30.1073909026194)
(-3.9893328562066204, 1.3194545619313)
(7.103718362595594, 2.36206235229571)
(-11.802142386490203, -3.93052177493889)
(10.234583701370475, 3.40746400970871)
(-68.33329868872808, -22.7771564976528)
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:
$$x_{1} = 99.7480730445654$$
$$x_{2} = -46.3438858860085$$
$$x_{3} = 11.8021423864902$$
$$x_{4} = -18.0779832097684$$
$$x_{5} = -2.45659021971744$$
$$x_{6} = 52.6264272696834$$
$$x_{7} = -90.3235565896713$$
$$x_{8} = -55.7677523585655$$
$$x_{9} = 33.7795214194042$$
$$x_{10} = 40.0615464074251$$
$$x_{11} = -49.4851361441979$$
$$x_{12} = 8.66818896199168$$
$$x_{13} = 46.3438858860085$$
$$x_{14} = -99.7480730445654$$
$$x_{15} = 74.6161759525405$$
$$x_{16} = 24.3576053587789$$
$$x_{17} = 18.0779832097684$$
$$x_{18} = -5.54276920324851$$
$$x_{19} = 2.45659021971744$$
$$x_{20} = 68.3332986887281$$
$$x_{21} = 0$$
$$x_{22} = -71.4747305517771$$
$$x_{23} = -33.7795214194042$$
$$x_{24} = 96.6065618907118$$
$$x_{25} = 30.6386872667848$$
$$x_{26} = -84.0405782018796$$
$$x_{27} = -24.3576053587789$$
$$x_{28} = -77.757633250469$$
$$x_{29} = 62.0504837986507$$
$$x_{30} = 77.757633250469$$
$$x_{31} = -62.0504837986507$$
$$x_{32} = -40.0615464074251$$
$$x_{33} = 84.0405782018796$$
$$x_{34} = 71.4747305517771$$
$$x_{35} = 49.4851361441979$$
$$x_{36} = -27.4980262787482$$
$$x_{37} = -93.4650562152248$$
$$x_{38} = 27.4980262787482$$
$$x_{39} = 5.54276920324851$$
$$x_{40} = -43.2026854058443$$
$$x_{41} = 55.7677523585655$$
$$x_{42} = 90.3235565896713$$
$$x_{43} = -11.8021423864902$$
$$x_{44} = -68.3332986887281$$
Puntos máximos de la función:
$$x_{44} = -98.1773168157084$$
$$x_{44} = -63.6211806632638$$
$$x_{44} = -35.349989019305$$
$$x_{44} = -79.3283659192419$$
$$x_{44} = 88.752809246359$$
$$x_{44} = -60.479792099527$$
$$x_{44} = -69.9040128139871$$
$$x_{44} = 76.186903206326$$
$$x_{44} = -47.9145054045097$$
$$x_{44} = -16.5085005166786$$
$$x_{44} = -19.6476754907365$$
$$x_{44} = -13.3704580073937$$
$$x_{44} = 3.98933285620662$$
$$x_{44} = 63.6211806632638$$
$$x_{44} = 19.6476754907365$$
$$x_{44} = 66.7625884309285$$
$$x_{44} = 54.1970859376957$$
$$x_{44} = 98.1773168157084$$
$$x_{44} = -1.01437891905522$$
$$x_{44} = -76.186903206326$$
$$x_{44} = 16.5085005166786$$
$$x_{44} = 38.4910046652094$$
$$x_{44} = 41.6321073520443$$
$$x_{44} = -32.2090858609196$$
$$x_{44} = -82.469838530885$$
$$x_{44} = -25.927780364576$$
$$x_{44} = -85.6113199516972$$
$$x_{44} = -10.2345837013705$$
$$x_{44} = 47.9145054045097$$
$$x_{44} = 69.9040128139871$$
$$x_{44} = -41.6321073520443$$
$$x_{44} = -38.4910046652094$$
$$x_{44} = 85.6113199516972$$
$$x_{44} = -54.1970859376957$$
$$x_{44} = -57.3384258953415$$
$$x_{44} = 25.927780364576$$
$$x_{44} = 82.469838530885$$
$$x_{44} = -91.8943056074308$$
$$x_{44} = 91.8943056074308$$
$$x_{44} = 32.2090858609196$$
$$x_{44} = 60.479792099527$$
$$x_{44} = -3.98933285620662$$
$$x_{44} = 7.10371836259559$$
$$x_{44} = 10.2345837013705$$
Decrece en los intervalos
$$\left[99.7480730445654, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -99.7480730445654\right]$$