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$$\left(x - 1\right) \cos{\left(x \right)} + 2 \sin{\left(x \right)} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = 67.5742746536542$$
$$x_{2} = 36.185096663082$$
$$x_{3} = 11.1893928335334$$
$$x_{4} = -95.83922578684$$
$$x_{5} = 42.4597030690427$$
$$x_{6} = -26.7754196064401$$
$$x_{7} = -17.3871055208222$$
$$x_{8} = 98.9805779624548$$
$$x_{9} = 89.5579708655131$$
$$x_{10} = -48.7348777037979$$
$$x_{11} = -70.7137161464592$$
$$x_{12} = -92.6983251388017$$
$$x_{13} = -64.4332054060865$$
$$x_{14} = -76.9946571767899$$
$$x_{15} = -20.5130526124428$$
$$x_{16} = -58.1532616938786$$
$$x_{17} = -8.07099262916911$$
$$x_{18} = -33.0454005126875$$
$$x_{19} = -80.1352578714972$$
$$x_{20} = 73.8548723024594$$
$$x_{21} = -42.4574903585174$$
$$x_{22} = 83.276508808943$$
$$x_{23} = -51.8740864721756$$
$$x_{24} = 20.5224422884564$$
$$x_{25} = 92.6987903619905$$
$$x_{26} = -89.5574724595818$$
$$x_{27} = 80.135880273854$$
$$x_{28} = 51.8755701510007$$
$$x_{29} = -5.03252136188435$$
$$x_{30} = 8.12754793711753$$
$$x_{31} = -55.0135619144274$$
$$x_{32} = 70.7145152693381$$
$$x_{33} = -14.2674231119067$$
$$x_{34} = 33.0490463971436$$
$$x_{35} = 29.914190397935$$
$$x_{36} = -29.9097446501775$$
$$x_{37} = -23.6429266008895$$
$$x_{38} = 26.7809591256787$$
$$x_{39} = 17.4001107835996$$
$$x_{40} = 23.6500166357864$$
$$x_{41} = 64.4341677074013$$
$$x_{42} = -11.1586068457488$$
$$x_{43} = 76.9953313457533$$
$$x_{44} = -73.8541396332586$$
$$x_{45} = -98.9801698872744$$
$$x_{46} = -2.1381977537389$$
$$x_{47} = 61.2942152651978$$
$$x_{48} = 45.5979086029841$$
$$x_{49} = 0.325342230148932$$
$$x_{50} = -86.4166729136966$$
$$x_{51} = 14.2865731358648$$
$$x_{52} = 2.49855432221968$$
$$x_{53} = 39.3220501388138$$
$$x_{54} = 95.839661033084$$
$$x_{55} = -61.2931519778376$$
$$x_{56} = 55.0148813637847$$
$$x_{57} = 5.160488137379$$
$$x_{58} = -45.5959892966016$$
$$x_{59} = -67.5733996147845$$
$$x_{60} = -36.1820531210707$$
$$x_{61} = 58.1544427241882$$
$$x_{62} = -83.2759324342754$$
$$x_{63} = 86.4172081819273$$
$$x_{64} = -39.3194713707251$$
$$x_{65} = 48.7365582531144$$
Signos de extremos en los puntos:
(67.57427465365424, -66.5742814256842)
(36.18509666308205, -35.1851424300885)
(11.189392833533422, -10.1912130731545)
(-95.83922578684005, 96.839227988199)
(42.45970306904274, -41.4597310679983)
(-26.77541960644013, 27.7755124606153)
(-17.387105520822153, -18.3874234848694)
(98.98057796245483, -97.9805800877981)
(89.55797086551311, 88.5579737437436)
(-48.73487770379787, -49.7348939347886)
(-70.71371614645922, 71.7137215650411)
(-92.69832513880175, -93.6983275689719)
(-64.43320540608647, 65.4332125383903)
(-76.99465717678993, 77.9946613893872)
(-20.513052612442788, 21.5132517643465)
(-58.1532616938786, 59.1532713454425)
(-8.070992629169108, 9.07354761945228)
(-33.04540051268751, 34.0454510204543)
(-80.13525787149717, -81.1352616137861)
(73.85487230245944, -72.8548774705131)
(-42.45749035851739, -43.4575146759137)
(83.27650880894296, 82.276512397715)
(-51.874086472175605, 52.874099982954)
(20.522442288456393, 19.5227082937804)
(92.69879036199049, -91.6987929545732)
(-89.55747245958176, 90.5574751513975)
(80.135880273854, -79.1358843068925)
(51.87557015100067, 50.8755853156168)
(-5.032521361884348, -6.04072389489312)
(8.127547937117532, 7.13266638174111)
(-55.01356191442744, -56.0135732801501)
(70.71451526933805, 69.7145211673148)
(-14.26742311190666, 15.267975615937)
(33.04904639714357, 32.049106916773)
(29.91419039793503, -28.9142727402996)
(-29.90974465017745, -30.9098120919251)
(-23.642926600889457, -24.6430593711288)
(26.780959125678717, 25.7810751438454)
(17.400110783599633, -16.4005575417742)
(23.650016635786415, -22.6501874207992)
(64.43416770740131, 63.4341755350032)
(-11.158606845748825, -12.1596901846029)
(76.99533134575333, 75.9953358994967)
(-73.85413963325861, -74.8541443983649)
(-98.9801698872744, -99.9801718876642)
(-2.1381977537389036, 3.18388592282072)
(61.29421526519784, -60.2942243795422)
(45.5979086029841, 44.5979311046089)
(0.3253422301489321, -1.16318433596694)
(-86.41667291369656, -87.4166759060983)
(14.286573135864806, 13.2874069067504)
(2.498554322219676, 1.69885528016381)
(39.32205013881378, 38.3220855795146)
(95.83966103308398, 94.8396633765946)
(-61.29315197783757, -62.2931602431943)
(55.01488136378469, -54.014894037229)
(5.160488137379001, -4.18298575484622)
(-45.595989296601644, 46.5960090292294)
(-67.57339961478453, -68.5734058119581)
(-36.18205312107074, -37.1820919159977)
(58.15444272418823, 57.1544534233191)
(-83.27593243427538, 84.275935773724)
(86.41720818192732, -85.4172113893473)
(-39.31947137072508, 40.3195018088614)
(48.73655825311442, -47.7365766064473)
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:
$$x_{1} = 67.5742746536542$$
$$x_{2} = 36.185096663082$$
$$x_{3} = 11.1893928335334$$
$$x_{4} = 42.4597030690427$$
$$x_{5} = -17.3871055208222$$
$$x_{6} = 98.9805779624548$$
$$x_{7} = -48.7348777037979$$
$$x_{8} = -92.6983251388017$$
$$x_{9} = -80.1352578714972$$
$$x_{10} = 73.8548723024594$$
$$x_{11} = -42.4574903585174$$
$$x_{12} = 92.6987903619905$$
$$x_{13} = 80.135880273854$$
$$x_{14} = -5.03252136188435$$
$$x_{15} = -55.0135619144274$$
$$x_{16} = 29.914190397935$$
$$x_{17} = -29.9097446501775$$
$$x_{18} = -23.6429266008895$$
$$x_{19} = 17.4001107835996$$
$$x_{20} = 23.6500166357864$$
$$x_{21} = -11.1586068457488$$
$$x_{22} = -73.8541396332586$$
$$x_{23} = -98.9801698872744$$
$$x_{24} = 61.2942152651978$$
$$x_{25} = 0.325342230148932$$
$$x_{26} = -86.4166729136966$$
$$x_{27} = -61.2931519778376$$
$$x_{28} = 55.0148813637847$$
$$x_{29} = 5.160488137379$$
$$x_{30} = -67.5733996147845$$
$$x_{31} = -36.1820531210707$$
$$x_{32} = 86.4172081819273$$
$$x_{33} = 48.7365582531144$$
Puntos máximos de la función:
$$x_{33} = -95.83922578684$$
$$x_{33} = -26.7754196064401$$
$$x_{33} = 89.5579708655131$$
$$x_{33} = -70.7137161464592$$
$$x_{33} = -64.4332054060865$$
$$x_{33} = -76.9946571767899$$
$$x_{33} = -20.5130526124428$$
$$x_{33} = -58.1532616938786$$
$$x_{33} = -8.07099262916911$$
$$x_{33} = -33.0454005126875$$
$$x_{33} = 83.276508808943$$
$$x_{33} = -51.8740864721756$$
$$x_{33} = 20.5224422884564$$
$$x_{33} = -89.5574724595818$$
$$x_{33} = 51.8755701510007$$
$$x_{33} = 8.12754793711753$$
$$x_{33} = 70.7145152693381$$
$$x_{33} = -14.2674231119067$$
$$x_{33} = 33.0490463971436$$
$$x_{33} = 26.7809591256787$$
$$x_{33} = 64.4341677074013$$
$$x_{33} = 76.9953313457533$$
$$x_{33} = -2.1381977537389$$
$$x_{33} = 45.5979086029841$$
$$x_{33} = 14.2865731358648$$
$$x_{33} = 2.49855432221968$$
$$x_{33} = 39.3220501388138$$
$$x_{33} = 95.839661033084$$
$$x_{33} = -45.5959892966016$$
$$x_{33} = 58.1544427241882$$
$$x_{33} = -83.2759324342754$$
$$x_{33} = -39.3194713707251$$
Decrece en los intervalos
$$\left[98.9805779624548, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -98.9801698872744\right]$$