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$$- \sin{\left(x \right)} - \frac{2}{x^{2}} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = 3.32364732189927$$
$$x_{2} = -116.238780160263$$
$$x_{3} = -31.4179526995949$$
$$x_{4} = -84.82272367203$$
$$x_{5} = -15.6998491035096$$
$$x_{6} = -50.2662740043369$$
$$x_{7} = 18.8439235716796$$
$$x_{8} = -6.33307161321987$$
$$x_{9} = -72.2562479616373$$
$$x_{10} = -18.8551815474534$$
$$x_{11} = -9.40215171277304$$
$$x_{12} = 31.4138998493574$$
$$x_{13} = -25.1359067235501$$
$$x_{14} = -69.1154570564923$$
$$x_{15} = 78.5401405648612$$
$$x_{16} = -40.8395053587023$$
$$x_{17} = 97.389583126519$$
$$x_{18} = 62.8313464576863$$
$$x_{19} = 53.4077762773349$$
$$x_{20} = 47.1247904022886$$
$$x_{21} = -56.5492931902278$$
$$x_{22} = 84.8232796181751$$
$$x_{23} = -97.3891613942219$$
$$x_{24} = 100.530767021156$$
$$x_{25} = -94.2480047648058$$
$$x_{26} = 75.3978718731992$$
$$x_{27} = -2.90178022717344$$
$$x_{28} = -100.531162807032$$
$$x_{29} = 12.5536795051862$$
$$x_{30} = -21.987011448352$$
$$x_{31} = 138.229972087148$$
$$x_{32} = -59.689699071421$$
$$x_{33} = -87.9648527714054$$
$$x_{34} = 94.2475544484302$$
$$x_{35} = 40.8419034938155$$
$$x_{36} = -157.079713736353$$
$$x_{37} = -91.1059459987638$$
$$x_{38} = 69.1146196913135$$
$$x_{39} = 951.902576244924$$
$$x_{40} = 56.5480423113348$$
$$x_{41} = 59.6908217438756$$
$$x_{42} = 9.44718895618355$$
$$x_{43} = 65.9739052255028$$
$$x_{44} = -81.6817087579544$$
$$x_{45} = -65.9729862124666$$
$$x_{46} = -75.3985754925447$$
$$x_{47} = -78.5394921092745$$
$$x_{48} = -47.1229891365535$$
$$x_{49} = -62.8323596695666$$
$$x_{50} = 72.2570140953699$$
$$x_{51} = 21.9952825910084$$
$$x_{52} = 91.1064279068952$$
$$x_{53} = 87.9643358265851$$
$$x_{54} = 81.6811092243143$$
$$x_{55} = 50.2646908606744$$
$$x_{56} = 37.6977044988782$$
$$x_{57} = 28.2768351999549$$
$$x_{58} = 6.23166071387218$$
$$x_{59} = -53.4063739078945$$
$$x_{60} = 43.9812632118327$$
$$x_{61} = -34.555844296047$$
$$x_{62} = -12.5790106584862$$
$$x_{63} = -28.2718316792198$$
$$x_{64} = -37.700518977158$$
$$x_{65} = -43.9833309914671$$
$$x_{66} = 15.7160607006388$$
$$x_{67} = -1.42208339294766$$
$$x_{68} = 34.559193758284$$
$$x_{69} = 25.1295741382878$$
Signos de extremos en los puntos:
(3.3236473218992666, -0.38172520689039)
(-116.23878016026272, -1.01720595074903)
(-31.41795269959486, 0.936340075695571)
(-84.82272367202995, -1.02357854872246)
(-15.699849103509576, -1.12735683975392)
(-50.266274004336864, 0.960211577510091)
(18.84392357167961, 1.1061191474722)
(-6.3330716132198654, 0.682953412289677)
(-72.25624796163729, -1.02767919390861)
(-18.855181547453377, 0.893912537758828)
(-9.402151712773035, -1.21246130264159)
(31.41389984935736, 1.06366403069884)
(-25.135906723550132, 0.920427539882339)
(-69.11545705649226, 0.971062825266145)
(78.54014056486125, -0.974535261666692)
(-40.83950535870234, -1.04897147064403)
(97.38958312651904, -0.979463900542893)
(62.831346457686344, 1.03183111694523)
(53.407776277334875, -0.962552023919545)
(47.1247904022886, -0.957559087396356)
(-56.54929319022782, 0.964632430451372)
(84.8232796181751, -0.976421528546572)
(-97.38916139422187, -1.02053614392164)
(100.5307670211562, 1.01989438746737)
(-94.24800476480577, 0.978779366269064)
(75.3978718731992, 1.02652588573425)
(-2.901780227173435, -1.66061461702709)
(-100.53116280703239, 0.980105651694243)
(12.553679505186174, 1.15923530931656)
(-21.987011448351957, -1.09095423641922)
(138.22997208714793, 1.01446863666815)
(-59.689699071420996, -1.03350646136095)
(-87.96485277140538, 0.977263612962097)
(94.24755444843021, 1.02122068442714)
(40.84190349381554, -0.951029967118444)
(-157.079713736353, 0.987267607837759)
(-91.10594599876377, -1.02195243497329)
(69.11461969131352, 1.02893735002896)
(951.9025762449241, -0.997898944648087)
(56.5480423113348, 1.03536796072309)
(59.69082174387558, -0.966493853737384)
(9.447188956183549, -0.78804569823414)
(65.97390522550282, -0.969684878316045)
(-81.6817087579544, 0.975514669069451)
(-65.97298621246661, -1.03031533283018)
(-75.39857549254467, 0.973474238035794)
(-78.5394921092745, -1.02546484345697)
(-47.12298913655351, -1.04244172374304)
(-62.832359669566564, 0.968169139704333)
(72.25701409536993, -0.97232095283161)
(21.99528259100837, -0.90906286645302)
(91.10642790689518, -0.978047623085567)
(87.96433582658508, 1.02273645384589)
(81.68110922431434, 1.02448542079069)
(50.26469086067438, 1.03978904907577)
(37.69770449887822, 1.05305263793172)
(28.2768351999549, -0.929267598580522)
(6.231660713872177, 1.31961463654648)
(-53.4063739078945, -1.03744846774041)
(43.98126321183269, 1.04547337537246)
(-34.555844296046956, -1.05787592725776)
(-12.579010658486215, 0.840925099778197)
(-28.27183167921975, -1.07073866021423)
(-37.70051897715799, 0.946949342389277)
(-43.983330991467085, 0.954527693555406)
(15.716060700638788, -0.872708862997222)
(-1.4220833929476613, -1.25822188829491)
(34.55919375828402, -0.942126877464538)
(25.1295741382878, 1.07958248550029)
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:
$$x_{1} = 3.32364732189927$$
$$x_{2} = -116.238780160263$$
$$x_{3} = -84.82272367203$$
$$x_{4} = -15.6998491035096$$
$$x_{5} = -72.2562479616373$$
$$x_{6} = -9.40215171277304$$
$$x_{7} = 78.5401405648612$$
$$x_{8} = -40.8395053587023$$
$$x_{9} = 97.389583126519$$
$$x_{10} = 53.4077762773349$$
$$x_{11} = 47.1247904022886$$
$$x_{12} = 84.8232796181751$$
$$x_{13} = -97.3891613942219$$
$$x_{14} = -2.90178022717344$$
$$x_{15} = -21.987011448352$$
$$x_{16} = -59.689699071421$$
$$x_{17} = 40.8419034938155$$
$$x_{18} = -91.1059459987638$$
$$x_{19} = 951.902576244924$$
$$x_{20} = 59.6908217438756$$
$$x_{21} = 9.44718895618355$$
$$x_{22} = 65.9739052255028$$
$$x_{23} = -65.9729862124666$$
$$x_{24} = -78.5394921092745$$
$$x_{25} = -47.1229891365535$$
$$x_{26} = 72.2570140953699$$
$$x_{27} = 21.9952825910084$$
$$x_{28} = 91.1064279068952$$
$$x_{29} = 28.2768351999549$$
$$x_{30} = -53.4063739078945$$
$$x_{31} = -34.555844296047$$
$$x_{32} = -28.2718316792198$$
$$x_{33} = 15.7160607006388$$
$$x_{34} = 34.559193758284$$
Puntos máximos de la función:
$$x_{34} = -31.4179526995949$$
$$x_{34} = -50.2662740043369$$
$$x_{34} = 18.8439235716796$$
$$x_{34} = -6.33307161321987$$
$$x_{34} = -18.8551815474534$$
$$x_{34} = 31.4138998493574$$
$$x_{34} = -25.1359067235501$$
$$x_{34} = -69.1154570564923$$
$$x_{34} = 62.8313464576863$$
$$x_{34} = -56.5492931902278$$
$$x_{34} = 100.530767021156$$
$$x_{34} = -94.2480047648058$$
$$x_{34} = 75.3978718731992$$
$$x_{34} = -100.531162807032$$
$$x_{34} = 12.5536795051862$$
$$x_{34} = 138.229972087148$$
$$x_{34} = -87.9648527714054$$
$$x_{34} = 94.2475544484302$$
$$x_{34} = -157.079713736353$$
$$x_{34} = 69.1146196913135$$
$$x_{34} = 56.5480423113348$$
$$x_{34} = -81.6817087579544$$
$$x_{34} = -75.3985754925447$$
$$x_{34} = -62.8323596695666$$
$$x_{34} = 87.9643358265851$$
$$x_{34} = 81.6811092243143$$
$$x_{34} = 50.2646908606744$$
$$x_{34} = 37.6977044988782$$
$$x_{34} = 6.23166071387218$$
$$x_{34} = 43.9812632118327$$
$$x_{34} = -12.5790106584862$$
$$x_{34} = -37.700518977158$$
$$x_{34} = -43.9833309914671$$
$$x_{34} = -1.42208339294766$$
$$x_{34} = 25.1295741382878$$
Decrece en los intervalos
$$\left[951.902576244924, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -116.238780160263\right]$$