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$$- x \cos{\left(x \right)} - \sin{\left(x \right)} + 3 = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = -3.07425456038503$$
$$x_{2} = -23.4764939315032$$
$$x_{3} = 48.7767154978288$$
$$x_{4} = 61.326293199787$$
$$x_{5} = -42.3642472717267$$
$$x_{6} = -58.1882200818766$$
$$x_{7} = -83.3002290124857$$
$$x_{8} = -61.2283776163412$$
$$x_{9} = -20.6146964477378$$
$$x_{10} = 36.2387508361043$$
$$x_{11} = 17.5077348540241$$
$$x_{12} = 23.7307041587325$$
$$x_{13} = -80.0856329048869$$
$$x_{14} = -54.9414489247552$$
$$x_{15} = 152.35411544443$$
$$x_{16} = 58.0850147745936$$
$$x_{17} = -8.33861725086343$$
$$x_{18} = -4.17351860404203$$
$$x_{19} = -14.4155561688775$$
$$x_{20} = 95.797694812346$$
$$x_{21} = -26.8526373613253$$
$$x_{22} = 29.9786579713163$$
$$x_{23} = 51.7976429698733$$
$$x_{24} = -39.3715482074989$$
$$x_{25} = 89.5130428650002$$
$$x_{26} = -64.4647088184572$$
$$x_{27} = 13.9929980191756$$
$$x_{28} = 73.8815746752966$$
$$x_{29} = 7.58201102989815$$
$$x_{30} = 32.9258866785534$$
$$x_{31} = 55.050547940963$$
$$x_{32} = 67.6034192970942$$
$$x_{33} = 86.4400769367249$$
$$x_{34} = 76.9430194295277$$
$$x_{35} = -33.1076147952702$$
$$x_{36} = 70.6575196987241$$
$$x_{37} = -73.8003189069402$$
$$x_{38} = 26.6282518824098$$
$$x_{39} = -86.3706367799173$$
$$x_{40} = -77.0209597693217$$
$$x_{41} = -10.8077945444872$$
$$x_{42} = -95.8603063492954$$
$$x_{43} = 92.7201272074518$$
$$x_{44} = 99.0005751433479$$
$$x_{45} = -17.1615520588194$$
$$x_{46} = -89.5800471399892$$
$$x_{47} = 5.46394020948928$$
$$x_{48} = -48.6535501731887$$
$$x_{49} = -92.6553937288938$$
$$x_{50} = -51.9133493509289$$
$$x_{51} = -29.7778393958327$$
$$x_{52} = 64.3715672727068$$
$$x_{53} = 20.321533659266$$
$$x_{54} = 42.5056407968813$$
$$x_{55} = 39.2188569010297$$
$$x_{56} = -36.0728008640266$$
$$x_{57} = 83.2281692096228$$
$$x_{58} = 80.1605177248327$$
$$x_{59} = -45.6407625449963$$
$$x_{60} = -70.7423854333491$$
$$x_{61} = 45.5091108034916$$
$$x_{62} = -98.9399508638295$$
$$x_{63} = -67.5146079931799$$
$$x_{64} = 11.3498948460387$$
Signos de extremos en los puntos:
(-3.074254560385031, -9.42962170838235)
(-23.47649393150319, -47.0386468553682)
(48.77671549782876, 194.942849191742)
(61.32629319978696, 245.174723005962)
(-42.36424727172675, -84.7757832679427)
(-58.18822008187658, -232.615395407305)
(-83.30022901248567, -333.104877890626)
(-61.228377616341156, -122.48944599461)
(-20.614696447737813, -82.0707044576556)
(36.238750836104316, 144.7342446696)
(17.507734854024072, 69.5739788104759)
(23.73070415873247, 94.5856963550296)
(-80.08563290488694, -160.196250767202)
(-54.94144892475524, -109.91933646914)
(152.35411544443033, 304.721359897761)
(58.085014774593574, 116.204492494047)
(-8.338617250863432, -32.3942336062574)
(-4.17351860404203, -8.93847106115194)
(-14.415556168877535, -57.1072162107503)
(95.79769481234598, 191.616273781739)
(-26.852637361325307, -107.112624821251)
(29.978657971316284, 119.647774049135)
(51.79764296987326, 103.633940979053)
(-39.37154820749894, -157.283000074919)
(89.5130428650002, 179.048437212681)
(-64.46470881845724, -257.734736336575)
(13.992998019175591, 28.1311643166625)
(73.88157467529658, 295.418017291241)
(7.5820110298981485, 15.4427109809883)
(32.925886678553425, 65.9126846006064)
(55.050547940963035, 220.056870712781)
(67.60341929709422, 270.29533995608)
(86.44007693672488, 345.667758077152)
(76.94301942952774, 153.912045301638)
(-33.10761479527021, -132.188822120202)
(70.65751969872413, 141.343361971734)
(-73.80031890694019, -147.627752906802)
(26.62825188240977, 53.3319316961434)
(-86.37063677991728, -172.764438895229)
(-77.02095976932169, -307.979971234898)
(-10.807794544487214, -21.8055779082812)
(-95.86030634929537, -383.357770621333)
(92.72012720745177, 370.794227666546)
(99.00057514334786, 395.921492958671)
(-17.161552058819417, -34.4408485554685)
(-89.58004713998925, -358.23088295282)
(5.463940209489285, 20.3839426026258)
(-48.65355017318866, -97.3482595077823)
(-92.65539372889377, -185.332380364191)
(-51.91334935092886, -207.4992943741)
(-29.777839395832704, -59.6230711819135)
(64.3715672727068, 128.774226681005)
(20.32153365926604, 40.7422076445466)
(42.505640796881266, 169.834352642409)
(39.21885690102972, 78.4888094242779)
(-36.07280086402657, -72.2011734144758)
(83.22816920962278, 166.48037915865)
(80.16051772483273, 320.542271135026)
(-45.6407625449963, -182.387768108812)
(-70.74238543334906, -282.856455339296)
(45.50911080349164, 91.0622326430517)
(-98.93995086382952, -197.900122206644)
(-67.51460799317994, -135.058858720205)
(11.34989484603866, 44.6945516066719)
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:
$$x_{1} = -3.07425456038503$$
$$x_{2} = -58.1882200818766$$
$$x_{3} = -83.3002290124857$$
$$x_{4} = -20.6146964477378$$
$$x_{5} = 152.35411544443$$
$$x_{6} = 58.0850147745936$$
$$x_{7} = -8.33861725086343$$
$$x_{8} = -14.4155561688775$$
$$x_{9} = 95.797694812346$$
$$x_{10} = -26.8526373613253$$
$$x_{11} = 51.7976429698733$$
$$x_{12} = -39.3715482074989$$
$$x_{13} = 89.5130428650002$$
$$x_{14} = -64.4647088184572$$
$$x_{15} = 13.9929980191756$$
$$x_{16} = 7.58201102989815$$
$$x_{17} = 32.9258866785534$$
$$x_{18} = 76.9430194295277$$
$$x_{19} = -33.1076147952702$$
$$x_{20} = 70.6575196987241$$
$$x_{21} = 26.6282518824098$$
$$x_{22} = -77.0209597693217$$
$$x_{23} = -95.8603063492954$$
$$x_{24} = -89.5800471399892$$
$$x_{25} = -51.9133493509289$$
$$x_{26} = 64.3715672727068$$
$$x_{27} = 20.321533659266$$
$$x_{28} = 39.2188569010297$$
$$x_{29} = 83.2281692096228$$
$$x_{30} = -45.6407625449963$$
$$x_{31} = -70.7423854333491$$
$$x_{32} = 45.5091108034916$$
Puntos máximos de la función:
$$x_{32} = -23.4764939315032$$
$$x_{32} = 48.7767154978288$$
$$x_{32} = 61.326293199787$$
$$x_{32} = -42.3642472717267$$
$$x_{32} = -61.2283776163412$$
$$x_{32} = 36.2387508361043$$
$$x_{32} = 17.5077348540241$$
$$x_{32} = 23.7307041587325$$
$$x_{32} = -80.0856329048869$$
$$x_{32} = -54.9414489247552$$
$$x_{32} = -4.17351860404203$$
$$x_{32} = 29.9786579713163$$
$$x_{32} = 73.8815746752966$$
$$x_{32} = 55.050547940963$$
$$x_{32} = 67.6034192970942$$
$$x_{32} = 86.4400769367249$$
$$x_{32} = -73.8003189069402$$
$$x_{32} = -86.3706367799173$$
$$x_{32} = -10.8077945444872$$
$$x_{32} = 92.7201272074518$$
$$x_{32} = 99.0005751433479$$
$$x_{32} = -17.1615520588194$$
$$x_{32} = 5.46394020948928$$
$$x_{32} = -48.6535501731887$$
$$x_{32} = -92.6553937288938$$
$$x_{32} = -29.7778393958327$$
$$x_{32} = 42.5056407968813$$
$$x_{32} = -36.0728008640266$$
$$x_{32} = 80.1605177248327$$
$$x_{32} = -98.9399508638295$$
$$x_{32} = -67.5146079931799$$
$$x_{32} = 11.3498948460387$$
Decrece en los intervalos
$$\left[152.35411544443, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -95.8603063492954\right]$$