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$$- \frac{x \cos{\left(2 x \right)}}{4} - \frac{\sin{\left(2 x \right)}}{8} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = -10.2345837013705$$
$$x_{2} = -40.0615464074251$$
$$x_{3} = 16.5085005166786$$
$$x_{4} = -25.927780364576$$
$$x_{5} = -71.4747305517771$$
$$x_{6} = 3.98933285620662$$
$$x_{7} = -3.98933285620662$$
$$x_{8} = -84.0405782018796$$
$$x_{9} = 24.3576053587789$$
$$x_{10} = 18.0779832097684$$
$$x_{11} = -63.6211806632638$$
$$x_{12} = -62.0504837986507$$
$$x_{13} = -41.6321073520443$$
$$x_{14} = 11.8021423864902$$
$$x_{15} = 41.6321073520443$$
$$x_{16} = -47.9145054045097$$
$$x_{17} = 52.6264272696834$$
$$x_{18} = -77.757633250469$$
$$x_{19} = 99.7480730445654$$
$$x_{20} = 55.7677523585655$$
$$x_{21} = 82.469838530885$$
$$x_{22} = -99.7480730445654$$
$$x_{23} = 46.3438858860085$$
$$x_{24} = -1.01437891905522$$
$$x_{25} = 77.757633250469$$
$$x_{26} = 2.45659021971744$$
$$x_{27} = 0$$
$$x_{28} = -79.3283659192419$$
$$x_{29} = 98.1773168157084$$
$$x_{30} = 85.6113199516972$$
$$x_{31} = 38.4910046652094$$
$$x_{32} = -33.7795214194042$$
$$x_{33} = -19.6476754907365$$
$$x_{34} = -90.3235565896713$$
$$x_{35} = -2.45659021971744$$
$$x_{36} = 47.9145054045097$$
$$x_{37} = 19.6476754907365$$
$$x_{38} = -11.8021423864902$$
$$x_{39} = 71.4747305517771$$
$$x_{40} = 32.2090858609196$$
$$x_{41} = -60.479792099527$$
$$x_{42} = 66.7625884309285$$
$$x_{43} = -93.4650562152248$$
$$x_{44} = -57.3384258953415$$
$$x_{45} = 25.927780364576$$
$$x_{46} = -98.1773168157084$$
$$x_{47} = -46.3438858860085$$
$$x_{48} = 62.0504837986507$$
$$x_{49} = 91.8943056074308$$
$$x_{50} = 27.4980262787482$$
$$x_{51} = -18.0779832097684$$
$$x_{52} = 74.6161759525405$$
$$x_{53} = 33.7795214194042$$
$$x_{54} = 90.3235565896713$$
$$x_{55} = 63.6211806632638$$
$$x_{56} = 7.10371836259559$$
$$x_{57} = 54.1970859376957$$
$$x_{58} = 30.6386872667848$$
$$x_{59} = 8.66818896199168$$
$$x_{60} = 68.3332986887281$$
$$x_{61} = -49.4851361441979$$
$$x_{62} = 96.6065618907118$$
$$x_{63} = -16.5085005166786$$
$$x_{64} = -27.4980262787482$$
$$x_{65} = -55.7677523585655$$
$$x_{66} = -38.4910046652094$$
$$x_{67} = 5.54276920324851$$
$$x_{68} = 69.9040128139871$$
$$x_{69} = -35.349989019305$$
$$x_{70} = -68.3332986887281$$
$$x_{71} = -85.6113199516972$$
$$x_{72} = -5.54276920324851$$
$$x_{73} = 84.0405782018796$$
$$x_{74} = -82.469838530885$$
$$x_{75} = -32.2090858609196$$
$$x_{76} = -54.1970859376957$$
$$x_{77} = 88.752809246359$$
$$x_{78} = 40.0615464074251$$
$$x_{79} = -13.3704580073937$$
$$x_{80} = -76.186903206326$$
$$x_{81} = 49.4851361441979$$
$$x_{82} = 76.186903206326$$
$$x_{83} = -69.9040128139871$$
$$x_{84} = -91.8943056074308$$
$$x_{85} = 60.479792099527$$
$$x_{86} = -24.3576053587789$$
$$x_{87} = 10.2345837013705$$
Signos de extremos en los puntos:
(-10.234583701370475, -1.15279900364077)
(-40.061546407425126, 5.1323033216037)
(16.508500516678623, -1.93761673317783)
(-25.927780364575984, -3.11537007813759)
(-71.47473055177714, 9.05912271826634)
(3.9893328562066204, -0.369795460724236)
(-3.9893328562066204, -0.369795460724236)
(-84.04057820187961, 10.6298863580799)
(24.357605358778862, 3.1690593890855)
(18.07798320976836, 2.38388408576513)
(-63.62118066326382, -7.82740199997356)
(-62.050483798650674, 7.88105867600272)
(-41.63210735204432, -5.07863814830958)
(11.802142386490203, 1.59894566560208)
(41.63210735204432, -5.07863814830958)
(-47.91450540450974, -5.8639871005291)
(52.6264272696834, 6.70300652474802)
(-77.75763325046901, 9.84450321763782)
(99.74807304456543, 12.5933524888925)
(55.7677523585655, 7.09568888187)
(82.46983853088497, -10.1835403583777)
(-99.74807304456543, 12.5933524888925)
(46.3438858860085, 5.9176486117537)
(-1.014378919055217, 0.0112683911775217)
(77.75763325046901, 9.84450321763782)
(2.456590219717442, 0.425904368107017)
(0, 1/8)
(-79.32836591924193, -9.79084877965922)
(98.17731681570837, -12.1470054542442)
(85.61131995169717, -10.57623248772)
(38.49100466520936, -4.68596969551648)
(-33.7795214194042, 4.34697769506078)
(-19.647675490736493, -2.33016456293074)
(-90.32355658967134, 11.4152715884823)
(-2.456590219717442, 0.425904368107017)
(47.91450540450974, -5.8639871005291)
(19.647675490736493, -2.33016456293074)
(-11.802142386490203, 1.59894566560208)
(71.47473055177714, 9.05912271826634)
(32.20908586091958, -3.90065070871096)
(-60.47979209952698, -7.43471567492743)
(66.76258843092853, -8.22008952545119)
(-93.46505621522485, 11.8079648557134)
(-57.338425895341494, -7.04203074759302)
(25.927780364575984, -3.11537007813759)
(-98.17731681570837, -12.1470054542442)
(-46.3438858860085, 5.9176486117537)
(62.050483798650674, 7.88105867600272)
(91.89430560743084, -11.3616181724056)
(27.498026278748195, 3.5616852031061)
(-18.07798320976836, 2.38388408576513)
(74.61617595254046, 9.45181259612382)
(33.7795214194042, 4.34697769506078)
(90.32355658967134, 11.4152715884823)
(63.62118066326382, -7.82740199997356)
(7.103718362595594, -0.760773382110893)
(54.197085937695654, -6.64934746098059)
(30.638687266784828, 3.95432603402979)
(8.66818896199168, 1.20672553799116)
(68.33329868872808, 8.66643368661981)
(-49.48513614419785, 6.31032629081992)
(96.6065618907118, 12.2006585010925)
(-16.508500516678623, -1.93761673317783)
(-27.498026278748195, 3.5616852031061)
(-55.7677523585655, 7.09568888187)
(-38.49100466520936, -4.68596969551648)
(5.542769203248511, 0.815044250995625)
(69.90401281398711, -8.61277808953703)
(-35.349989019305, -4.29330668509957)
(-68.33329868872808, 8.66643368661981)
(-85.61131995169717, -10.57623248772)
(-5.542769203248511, 0.815044250995625)
(84.04057820187961, 10.6298863580799)
(-82.46983853088497, -10.1835403583777)
(-32.20908586091958, -3.90065070871096)
(-54.197085937695654, -6.64934746098059)
(88.75280924635904, -10.9689251092199)
(40.061546407425126, 5.1323033216037)
(-13.370458007393655, -1.54513985404359)
(-76.186903206326, -9.39815781967275)
(49.48513614419785, 6.31032629081992)
(76.186903206326, -9.39815781967275)
(-69.90401281398711, -8.61277808953703)
(-91.89430560743084, -11.3616181724056)
(60.47979209952698, -7.43471567492743)
(-24.357605358778862, 3.1690593890855)
(10.234583701370475, -1.15279900364077)
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:
$$x_{1} = -10.2345837013705$$
$$x_{2} = 16.5085005166786$$
$$x_{3} = -25.927780364576$$
$$x_{4} = 3.98933285620662$$
$$x_{5} = -3.98933285620662$$
$$x_{6} = -63.6211806632638$$
$$x_{7} = -41.6321073520443$$
$$x_{8} = 41.6321073520443$$
$$x_{9} = -47.9145054045097$$
$$x_{10} = 82.469838530885$$
$$x_{11} = -1.01437891905522$$
$$x_{12} = -79.3283659192419$$
$$x_{13} = 98.1773168157084$$
$$x_{14} = 85.6113199516972$$
$$x_{15} = 38.4910046652094$$
$$x_{16} = -19.6476754907365$$
$$x_{17} = 47.9145054045097$$
$$x_{18} = 19.6476754907365$$
$$x_{19} = 32.2090858609196$$
$$x_{20} = -60.479792099527$$
$$x_{21} = 66.7625884309285$$
$$x_{22} = -57.3384258953415$$
$$x_{23} = 25.927780364576$$
$$x_{24} = -98.1773168157084$$
$$x_{25} = 91.8943056074308$$
$$x_{26} = 63.6211806632638$$
$$x_{27} = 7.10371836259559$$
$$x_{28} = 54.1970859376957$$
$$x_{29} = -16.5085005166786$$
$$x_{30} = -38.4910046652094$$
$$x_{31} = 69.9040128139871$$
$$x_{32} = -35.349989019305$$
$$x_{33} = -85.6113199516972$$
$$x_{34} = -82.469838530885$$
$$x_{35} = -32.2090858609196$$
$$x_{36} = -54.1970859376957$$
$$x_{37} = 88.752809246359$$
$$x_{38} = -13.3704580073937$$
$$x_{39} = -76.186903206326$$
$$x_{40} = 76.186903206326$$
$$x_{41} = -69.9040128139871$$
$$x_{42} = -91.8943056074308$$
$$x_{43} = 60.479792099527$$
$$x_{44} = 10.2345837013705$$
Puntos máximos de la función:
$$x_{44} = -40.0615464074251$$
$$x_{44} = -71.4747305517771$$
$$x_{44} = -84.0405782018796$$
$$x_{44} = 24.3576053587789$$
$$x_{44} = 18.0779832097684$$
$$x_{44} = -62.0504837986507$$
$$x_{44} = 11.8021423864902$$
$$x_{44} = 52.6264272696834$$
$$x_{44} = -77.757633250469$$
$$x_{44} = 99.7480730445654$$
$$x_{44} = 55.7677523585655$$
$$x_{44} = -99.7480730445654$$
$$x_{44} = 46.3438858860085$$
$$x_{44} = 77.757633250469$$
$$x_{44} = 2.45659021971744$$
$$x_{44} = 0$$
$$x_{44} = -33.7795214194042$$
$$x_{44} = -90.3235565896713$$
$$x_{44} = -2.45659021971744$$
$$x_{44} = -11.8021423864902$$
$$x_{44} = 71.4747305517771$$
$$x_{44} = -93.4650562152248$$
$$x_{44} = -46.3438858860085$$
$$x_{44} = 62.0504837986507$$
$$x_{44} = 27.4980262787482$$
$$x_{44} = -18.0779832097684$$
$$x_{44} = 74.6161759525405$$
$$x_{44} = 33.7795214194042$$
$$x_{44} = 90.3235565896713$$
$$x_{44} = 30.6386872667848$$
$$x_{44} = 8.66818896199168$$
$$x_{44} = 68.3332986887281$$
$$x_{44} = -49.4851361441979$$
$$x_{44} = 96.6065618907118$$
$$x_{44} = -27.4980262787482$$
$$x_{44} = -55.7677523585655$$
$$x_{44} = 5.54276920324851$$
$$x_{44} = -68.3332986887281$$
$$x_{44} = -5.54276920324851$$
$$x_{44} = 84.0405782018796$$
$$x_{44} = 40.0615464074251$$
$$x_{44} = 49.4851361441979$$
$$x_{44} = -24.3576053587789$$
Decrece en los intervalos
$$\left[98.1773168157084, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -98.1773168157084\right]$$