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$$- \frac{2 x \cos{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{2}{\sin{\left(x \right)}} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = 32.9563890398225$$
$$x_{2} = 80.0981286289451$$
$$x_{3} = -76.9560263103312$$
$$x_{4} = -86.3822220347287$$
$$x_{5} = 36.1006222443756$$
$$x_{6} = -98.9500628243319$$
$$x_{7} = -39.2444323611642$$
$$x_{8} = 70.6716857116195$$
$$x_{9} = 86.3822220347287$$
$$x_{10} = -89.5242209304172$$
$$x_{11} = 73.8138806006806$$
$$x_{12} = 23.519452498689$$
$$x_{13} = -48.6741442319544$$
$$x_{14} = -36.1006222443756$$
$$x_{15} = -45.5311340139913$$
$$x_{16} = 26.6660542588127$$
$$x_{17} = 83.2401924707234$$
$$x_{18} = 10.9041216594289$$
$$x_{19} = -7.72525183693771$$
$$x_{20} = 42.3879135681319$$
$$x_{21} = -70.6716857116195$$
$$x_{22} = 4.93829501990806 \cdot 10^{-17}$$
$$x_{23} = -23.519452498689$$
$$x_{24} = 45.5311340139913$$
$$x_{25} = -26.6660542588127$$
$$x_{26} = 64.3871195905574$$
$$x_{27} = -80.0981286289451$$
$$x_{28} = 98.9500628243319$$
$$x_{29} = -32.9563890398225$$
$$x_{30} = 4.49340945790906$$
$$x_{31} = 17.2207552719308$$
$$x_{32} = 51.8169824872797$$
$$x_{33} = 76.9560263103312$$
$$x_{34} = 7.72525183693771$$
$$x_{35} = 89.5242209304172$$
$$x_{36} = 92.6661922776228$$
$$x_{37} = -92.6661922776228$$
$$x_{38} = -42.3879135681319$$
$$x_{39} = 20.3713029592876$$
$$x_{40} = 14.0661939128315$$
$$x_{41} = -83.2401924707234$$
$$x_{42} = -51.8169824872797$$
$$x_{43} = -58.1022547544956$$
$$x_{44} = 95.8081387868617$$
$$x_{45} = 2.01537897347346 \cdot 10^{-17}$$
$$x_{46} = 39.2444323611642$$
$$x_{47} = -17.2207552719308$$
$$x_{48} = -29.811598790893$$
$$x_{49} = -10.9041216594289$$
$$x_{50} = -73.8138806006806$$
$$x_{51} = -54.9596782878889$$
$$x_{52} = 54.9596782878889$$
$$x_{53} = -14.0661939128315$$
$$x_{54} = 29.811598790893$$
$$x_{55} = -61.2447302603744$$
$$x_{56} = -95.8081387868617$$
$$x_{57} = -4.49340945790906$$
$$x_{58} = 67.5294347771441$$
$$x_{59} = 58.1022547544956$$
$$x_{60} = -67.5294347771441$$
$$x_{61} = 48.6741442319544$$
$$x_{62} = -64.3871195905574$$
$$x_{63} = -20.3713029592876$$
$$x_{64} = 61.2447302603744$$
Signos de extremos en los puntos:
(32.956389039822476, 65.9431142286784)
(80.09812862894512, -160.208741457625)
(-76.95602631033118, 153.925046506102)
(-86.38222203472871, -172.776020137717)
(36.10062224437561, -72.2289395306648)
(-98.95006282433188, -197.910231498417)
(-39.24443236116419, 78.5143419089784)
(70.6716857116195, 141.35752065344)
(86.38222203472871, -172.776020137717)
(-89.52422093041719, 179.059611673857)
(73.81388060068065, -147.641308167214)
(23.519452498689006, -47.0814037954727)
(-48.674144231954386, -97.3688310849647)
(-36.10062224437561, -72.2289395306648)
(-45.53113401399128, 91.0842283735232)
(26.666054258812675, 53.3695962036042)
(83.2401924707234, 166.492397935318)
(10.904121659428899, -21.8997597396525)
(-7.725251836937707, 15.5794115349854)
(42.38791356813192, -84.799415485236)
(-70.6716857116195, 141.35752065344)
(4.938295019908061e-17, 2)
(-23.519452498689006, -47.0814037954727)
(45.53113401399128, 91.0842283735232)
(-26.666054258812675, 53.3695962036042)
(64.38711959055742, 128.789769301272)
(-80.09812862894512, -160.208741457625)
(98.95006282433188, -197.910231498417)
(-32.956389039822476, 65.9431142286784)
(4.493409457909064, -9.2066776975034)
(17.22075527193077, -34.4995311351173)
(51.81698248727967, 103.653261870277)
(76.95602631033118, 153.925046506102)
(7.725251836937707, 15.5794115349854)
(89.52422093041719, 179.059611673857)
(92.66619227762284, -185.343175663237)
(-92.66619227762284, -185.343175663237)
(-42.38791356813192, -84.799415485236)
(20.37130295928756, 40.7916650436865)
(14.066193912831473, 28.2033906609384)
(-83.2401924707234, 166.492397935318)
(-51.81698248727967, 103.653261870277)
(-58.10225475449559, 116.221719270648)
(95.8081387868617, 191.626714816098)
(2.0153789734734588e-17, 2)
(39.24443236116419, 78.5143419089784)
(-17.22075527193077, -34.4995311351173)
(-29.81159879089296, -59.6567321421202)
(-10.904121659428899, -21.8997597396525)
(-73.81388060068065, -147.641308167214)
(-54.959678287888934, -109.937550227541)
(54.959678287888934, -109.937550227541)
(-14.066193912831473, 28.2033906609384)
(29.81159879089296, -59.6567321421202)
(-61.2447302603744, -122.505787368043)
(-95.8081387868617, 191.626714816098)
(-4.493409457909064, -9.2066776975034)
(67.52943477714412, -135.073677099879)
(58.10225475449559, 116.221719270648)
(-67.52943477714412, -135.073677099879)
(48.674144231954386, -97.3688310849647)
(-64.38711959055742, 128.789769301272)
(-20.37130295928756, 40.7916650436865)
(61.2447302603744, -122.505787368043)
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:
$$x_{1} = 32.9563890398225$$
$$x_{2} = -76.9560263103312$$
$$x_{3} = -39.2444323611642$$
$$x_{4} = 70.6716857116195$$
$$x_{5} = -89.5242209304172$$
$$x_{6} = -45.5311340139913$$
$$x_{7} = 26.6660542588127$$
$$x_{8} = 83.2401924707234$$
$$x_{9} = -7.72525183693771$$
$$x_{10} = -70.6716857116195$$
$$x_{11} = 4.93829501990806 \cdot 10^{-17}$$
$$x_{12} = 45.5311340139913$$
$$x_{13} = -26.6660542588127$$
$$x_{14} = 64.3871195905574$$
$$x_{15} = -32.9563890398225$$
$$x_{16} = 51.8169824872797$$
$$x_{17} = 76.9560263103312$$
$$x_{18} = 7.72525183693771$$
$$x_{19} = 89.5242209304172$$
$$x_{20} = 20.3713029592876$$
$$x_{21} = 14.0661939128315$$
$$x_{22} = -83.2401924707234$$
$$x_{23} = -51.8169824872797$$
$$x_{24} = -58.1022547544956$$
$$x_{25} = 95.8081387868617$$
$$x_{26} = 2.01537897347346 \cdot 10^{-17}$$
$$x_{27} = 39.2444323611642$$
$$x_{28} = -14.0661939128315$$
$$x_{29} = -95.8081387868617$$
$$x_{30} = 58.1022547544956$$
$$x_{31} = -64.3871195905574$$
$$x_{32} = -20.3713029592876$$
Puntos máximos de la función:
$$x_{32} = 80.0981286289451$$
$$x_{32} = -86.3822220347287$$
$$x_{32} = 36.1006222443756$$
$$x_{32} = -98.9500628243319$$
$$x_{32} = 86.3822220347287$$
$$x_{32} = 73.8138806006806$$
$$x_{32} = 23.519452498689$$
$$x_{32} = -48.6741442319544$$
$$x_{32} = -36.1006222443756$$
$$x_{32} = 10.9041216594289$$
$$x_{32} = 42.3879135681319$$
$$x_{32} = -23.519452498689$$
$$x_{32} = -80.0981286289451$$
$$x_{32} = 98.9500628243319$$
$$x_{32} = 4.49340945790906$$
$$x_{32} = 17.2207552719308$$
$$x_{32} = 92.6661922776228$$
$$x_{32} = -92.6661922776228$$
$$x_{32} = -42.3879135681319$$
$$x_{32} = -17.2207552719308$$
$$x_{32} = -29.811598790893$$
$$x_{32} = -10.9041216594289$$
$$x_{32} = -73.8138806006806$$
$$x_{32} = -54.9596782878889$$
$$x_{32} = 54.9596782878889$$
$$x_{32} = 29.811598790893$$
$$x_{32} = -61.2447302603744$$
$$x_{32} = -4.49340945790906$$
$$x_{32} = 67.5294347771441$$
$$x_{32} = -67.5294347771441$$
$$x_{32} = 48.6741442319544$$
$$x_{32} = 61.2447302603744$$
Decrece en los intervalos
$$\left[95.8081387868617, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -95.8081387868617\right]$$