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$$- \frac{\left(x - 1\right) \cos{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{1}{\sin{\left(x \right)}} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = -89.524344324422$$
$$x_{2} = 23.5175642862085$$
$$x_{3} = -61.2449925776205$$
$$x_{4} = -48.6745578196793$$
$$x_{5} = 64.3868745704895$$
$$x_{6} = -98.9501639357993$$
$$x_{7} = 89.5240947482585$$
$$x_{8} = -7.74006134563749$$
$$x_{9} = -67.5296508640044$$
$$x_{10} = 29.8104344913478$$
$$x_{11} = 14.0607507547713$$
$$x_{12} = -73.8140616841155$$
$$x_{13} = -36.1013688569472$$
$$x_{14} = -51.8173478727486$$
$$x_{15} = -80.0982825740805$$
$$x_{16} = -64.3873571144002$$
$$x_{17} = 54.9593410866482$$
$$x_{18} = 98.9499596481$$
$$x_{19} = -29.8126874326337$$
$$x_{20} = -54.9600034336304$$
$$x_{21} = -26.667409674396$$
$$x_{22} = 61.2444592323791$$
$$x_{23} = 4.42859686541094$$
$$x_{24} = -76.9561929992537$$
$$x_{25} = 42.3873435477815$$
$$x_{26} = 32.9554394950363$$
$$x_{27} = 92.6660745522577$$
$$x_{28} = -17.2239416992227$$
$$x_{29} = 95.808028695694$$
$$x_{30} = 83.2400463931404$$
$$x_{31} = 48.6737132861388$$
$$x_{32} = -70.6718831382477$$
$$x_{33} = -95.8082466033645$$
$$x_{34} = -42.3884572986485$$
$$x_{35} = -14.0709110733623$$
$$x_{36} = -58.1025459608263$$
$$x_{37} = -86.3823545152058$$
$$x_{38} = -10.9118204503674$$
$$x_{39} = -32.9572826072601$$
$$x_{40} = 58.1019533456208$$
$$x_{41} = 7.705951184346$$
$$x_{42} = 86.3820864505975$$
$$x_{43} = 17.2171745487221$$
$$x_{44} = 10.8948536303641$$
$$x_{45} = 76.9558552315675$$
$$x_{46} = 70.6714826164982$$
$$x_{47} = -4.53360450162482$$
$$x_{48} = 51.8166027158994$$
$$x_{49} = 45.5306408032197$$
$$x_{50} = 73.8136945424488$$
$$x_{51} = -39.2450655150134$$
$$x_{52} = -83.2403350796909$$
$$x_{53} = 20.3687685412497$$
$$x_{54} = -92.6663074889768$$
$$x_{55} = 36.0998330569238$$
$$x_{56} = -23.5211864259599$$
$$x_{57} = -45.5316060155777$$
$$x_{58} = -20.3735996512825$$
$$x_{59} = 26.6645930692902$$
$$x_{60} = 39.2437660743313$$
$$x_{61} = 67.5292121928599$$
$$x_{62} = -1.13226772527289$$
$$x_{63} = 80.0979707906851$$
Signos de extremos en los puntos:
(-89.52434432442197, 90.5298675320278)
(23.517564286208504, -22.5397582370249)
(-61.24499257762051, -62.2530248340435)
(-48.674557819679315, -49.6846223149645)
(64.38687457048954, 63.3947621481063)
(-98.95016393579934, -99.9551663036642)
(89.52409474825846, 88.5297427478395)
(-7.740061345637487, 8.79708317145556)
(-67.52965086400441, -68.5369465875329)
(29.810434491347824, -28.827784090704)
(14.06075075477126, 13.098977451628)
(-73.81406168411554, -74.8207446212255)
(-36.101368856947204, -37.114843001948)
(-51.81734787274865, 52.8268136111856)
(-80.09828257408046, -81.1044476984179)
(-64.38735711440016, 65.3950034055057)
(54.959341086648216, -53.9686065273622)
(98.94995964810002, -97.9550641624231)
(-29.812687432633666, -30.8289102438147)
(-54.96000343363045, -55.9689376734268)
(-26.667409674395977, 27.6854755800015)
(61.24445923237913, -60.2527581792053)
(4.428596865410944, -3.57145299080161)
(-76.95619299925373, 77.962606594039)
(42.38734354778151, -41.3994227730544)
(32.95543949503625, 31.9710824546327)
(92.66607455225765, -91.6715289706682)
(-17.22394169922269, -18.2513575127076)
(95.80802869569398, 94.8133023639802)
(83.24004639314036, 82.2461259315348)
(48.67371328613878, -47.6842000927872)
(-70.6718831382477, 71.6788590351621)
(-95.8082466033645, 96.8134113148475)
(-42.38845729864853, -43.3999795709243)
(-14.07091107336234, 15.1040511314414)
(-58.102545960826276, 59.1110052278895)
(-86.38235451520583, -87.3880763069031)
(-10.911820450367392, -11.9537218656697)
(-32.957282607260126, 33.972003798265)
(58.101953345620764, 57.1107089422417)
(7.705951184345997, 6.78010186404537)
(86.38208645059747, -85.3879422790906)
(17.217174548722074, -16.247976807705)
(10.894853630364137, -9.94525657619402)
(76.95585523156745, 75.9624377173273)
(70.6714826164982, 69.6786587843151)
(-4.53360450162482, -5.62323561487534)
(51.81660271589941, 50.8264410674754)
(45.5306408032197, 44.5418676117804)
(73.8136945424488, -72.820561058818)
(-39.24506551501339, 40.2574874813086)
(-83.24033507969094, 84.2462702696007)
(20.368768541249715, 19.394566115397)
(-92.66630748897677, -93.6716454356362)
(36.09983305692385, -35.1140752494484)
(-23.521186425959854, -24.5415684856669)
(-45.53160601557773, 46.5423501597088)
(-20.373599651282518, 21.3969802087422)
(26.664593069290245, 25.6840677738605)
(39.2437660743313, 38.2568378665589)
(67.52921219285986, -66.5367272639899)
(-1.1322677252728852, 2.35511478536406)
(80.09797079068508, -79.1042918127965)
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:
$$x_{1} = -89.524344324422$$
$$x_{2} = 64.3868745704895$$
$$x_{3} = 89.5240947482585$$
$$x_{4} = -7.74006134563749$$
$$x_{5} = 14.0607507547713$$
$$x_{6} = -51.8173478727486$$
$$x_{7} = -64.3873571144002$$
$$x_{8} = -26.667409674396$$
$$x_{9} = -76.9561929992537$$
$$x_{10} = 32.9554394950363$$
$$x_{11} = 95.808028695694$$
$$x_{12} = 83.2400463931404$$
$$x_{13} = -70.6718831382477$$
$$x_{14} = -95.8082466033645$$
$$x_{15} = -14.0709110733623$$
$$x_{16} = -58.1025459608263$$
$$x_{17} = -32.9572826072601$$
$$x_{18} = 58.1019533456208$$
$$x_{19} = 7.705951184346$$
$$x_{20} = 76.9558552315675$$
$$x_{21} = 70.6714826164982$$
$$x_{22} = 51.8166027158994$$
$$x_{23} = 45.5306408032197$$
$$x_{24} = -39.2450655150134$$
$$x_{25} = -83.2403350796909$$
$$x_{26} = 20.3687685412497$$
$$x_{27} = -45.5316060155777$$
$$x_{28} = -20.3735996512825$$
$$x_{29} = 26.6645930692902$$
$$x_{30} = 39.2437660743313$$
$$x_{31} = -1.13226772527289$$
Puntos máximos de la función:
$$x_{31} = 23.5175642862085$$
$$x_{31} = -61.2449925776205$$
$$x_{31} = -48.6745578196793$$
$$x_{31} = -98.9501639357993$$
$$x_{31} = -67.5296508640044$$
$$x_{31} = 29.8104344913478$$
$$x_{31} = -73.8140616841155$$
$$x_{31} = -36.1013688569472$$
$$x_{31} = -80.0982825740805$$
$$x_{31} = 54.9593410866482$$
$$x_{31} = 98.9499596481$$
$$x_{31} = -29.8126874326337$$
$$x_{31} = -54.9600034336304$$
$$x_{31} = 61.2444592323791$$
$$x_{31} = 4.42859686541094$$
$$x_{31} = 42.3873435477815$$
$$x_{31} = 92.6660745522577$$
$$x_{31} = -17.2239416992227$$
$$x_{31} = 48.6737132861388$$
$$x_{31} = -42.3884572986485$$
$$x_{31} = -86.3823545152058$$
$$x_{31} = -10.9118204503674$$
$$x_{31} = 86.3820864505975$$
$$x_{31} = 17.2171745487221$$
$$x_{31} = 10.8948536303641$$
$$x_{31} = -4.53360450162482$$
$$x_{31} = 73.8136945424488$$
$$x_{31} = -92.6663074889768$$
$$x_{31} = 36.0998330569238$$
$$x_{31} = -23.5211864259599$$
$$x_{31} = 67.5292121928599$$
$$x_{31} = 80.0979707906851$$
Decrece en los intervalos
$$\left[95.808028695694, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -95.8082466033645\right]$$