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$$\frac{1 - \cos{\left(x \right)}}{x + \sin{\left(x \right)}} + \frac{\left(x - \sin{\left(x \right)}\right) \left(- \cos{\left(x \right)} - 1\right)}{\left(x + \sin{\left(x \right)}\right)^{2}} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = 17.2207552719308$$
$$x_{2} = 4.05735660895802 \cdot 10^{-15}$$
$$x_{3} = -80.0981286289451$$
$$x_{4} = -83.2401924707234$$
$$x_{5} = 98.9500628243319$$
$$x_{6} = -45.5311340139913$$
$$x_{7} = 161.785840727966$$
$$x_{8} = -86.3822220347287$$
$$x_{9} = 7.72525183693771$$
$$x_{10} = -4.49340945790906$$
$$x_{11} = 4.49340945790906$$
$$x_{12} = 39.2444323611642$$
$$x_{13} = -70.6716857116195$$
$$x_{14} = 10.9041216594289$$
$$x_{15} = -42.3879135681319$$
$$x_{16} = 80.0981286289451$$
$$x_{17} = 171.210958939446$$
$$x_{18} = -48.6741442319544$$
$$x_{19} = 89.5242209304172$$
$$x_{20} = 14.0661939128315$$
$$x_{21} = -36.1006222443756$$
$$x_{22} = 102.091966464908$$
$$x_{23} = -95.8081387868617$$
$$x_{24} = 64.3871195905574$$
$$x_{25} = 61.2447302603744$$
$$x_{26} = -54.9596782878889$$
$$x_{27} = 76.9560263103312$$
$$x_{28} = -76.9560263103312$$
$$x_{29} = -98.9500628243319$$
$$x_{30} = -7.72525183693771$$
$$x_{31} = -20.3713029592876$$
$$x_{32} = -2.79413003615298 \cdot 10^{-14}$$
$$x_{33} = -39.2444323611642$$
$$x_{34} = -14.0661939128315$$
$$x_{35} = -32.9563890398225$$
$$x_{36} = -227.761076847648$$
$$x_{37} = 73.8138806006806$$
$$x_{38} = 26.6660542588127$$
$$x_{39} = 54.9596782878889$$
$$x_{40} = -26.6660542588127$$
$$x_{41} = -61.2447302603744$$
$$x_{42} = -67.5294347771441$$
$$x_{43} = 29.811598790893$$
$$x_{44} = 51.8169824872797$$
$$x_{45} = 23.519452498689$$
$$x_{46} = -58.1022547544956$$
$$x_{47} = 67.5294347771441$$
$$x_{48} = -10.9041216594289$$
$$x_{49} = -89.5242209304172$$
$$x_{50} = -13215.1094216545$$
$$x_{51} = 86.3822220347287$$
$$x_{52} = -23.519452498689$$
$$x_{53} = -17.2207552719308$$
$$x_{54} = 58.1022547544956$$
$$x_{55} = -92.6661922776228$$
$$x_{56} = -29.811598790893$$
$$x_{57} = 92.6661922776228$$
$$x_{58} = -64.3871195905574$$
$$x_{59} = 127.226642643334$$
$$x_{60} = 32.9563890398225$$
$$x_{61} = 20.3713029592876$$
$$x_{62} = 48.6741442319544$$
$$x_{63} = 45.5311340139913$$
$$x_{64} = 36.1006222443756$$
$$x_{65} = 70.6716857116195$$
$$x_{66} = 83.2401924707234$$
$$x_{67} = 95.8081387868617$$
$$x_{68} = -73.8138806006806$$
$$x_{69} = 42.3879135681319$$
$$x_{70} = -51.8169824872797$$
Signos de extremos en los puntos:
(17.22075527193077, 1.12307869868553)
(4.057356608958019e-15, 0)
(-80.09812862894512, 1.02528305302948)
(-83.2401924707234, 0.976260056542518)
(98.95006282433188, 1.02041751453922)
(-45.53113401399128, 0.957028166104047)
(161.78584072796568, 1.01243866717085)
(-86.38222203472871, 1.02342249220221)
(7.725251836937707, 0.772461097913349)
(-4.493409457909064, 1.55504077855261)
(4.493409457909064, 1.55504077855261)
(39.24443236116419, 0.950319410117988)
(-70.6716857116195, 0.972097731728565)
(10.904121659428899, 1.20100745196584)
(-42.38791356813192, 1.04830951977811)
(80.09812862894512, 1.02528305302948)
(171.21095893944562, 0.988386534184906)
(-48.674144231954386, 1.0419424245269)
(89.52422093041719, 0.977907828460357)
(14.066193912831473, 0.86756453787246)
(-36.10062224437561, 1.05695657697143)
(102.09196646490764, 0.98060076781689)
(-95.8081387868617, 0.979341693609794)
(64.38711959055742, 0.969416568120202)
(61.2447302603744, 1.03319342653464)
(-54.959678287888934, 1.0370584656736)
(76.95602631033118, 0.974346648664662)
(-76.95602631033118, 0.974346648664662)
(-98.95006282433188, 1.02041751453922)
(-7.725251836937707, 0.772461097913349)
(-20.37130295928756, 0.906523852345629)
(-2.7941300361529808e-14, 0)
(-39.24443236116419, 0.950319410117988)
(-14.066193912831473, 0.86756453787246)
(-32.956389039822476, 0.941127220242848)
(-227.7610768476483, 0.99125733785207)
(73.81388060068065, 1.02746473545409)
(26.666054258812675, 0.927758187267769)
(54.959678287888934, 1.0370584656736)
(-26.666054258812675, 0.927758187267769)
(-61.2447302603744, 1.03319342653464)
(-67.52943477714412, 1.03005853664807)
(29.81159879089296, 1.06937611361914)
(51.81698248727967, 0.96214030755708)
(23.519452498689006, 1.08872838162155)
(-58.10225475449559, 0.966165269591092)
(67.52943477714412, 1.03005853664807)
(-10.904121659428899, 1.20100745196584)
(-89.52422093041719, 0.977907828460357)
(-13215.109421654506, 0.999848669534243)
(86.38222203472871, 1.02342249220221)
(-23.519452498689006, 1.08872838162155)
(-17.22075527193077, 1.12307869868553)
(58.10225475449559, 0.966165269591092)
(-92.66619227762284, 1.02181701056246)
(-29.81159879089296, 1.06937611361914)
(92.66619227762284, 1.02181701056246)
(-64.38711959055742, 0.969416568120202)
(127.22664264333433, 0.984403095456837)
(32.956389039822476, 0.941127220242848)
(20.37130295928756, 0.906523852345629)
(48.674144231954386, 1.0419424245269)
(45.53113401399128, 0.957028166104047)
(36.10062224437561, 1.05695657697143)
(70.6716857116195, 0.972097731728565)
(83.2401924707234, 0.976260056542518)
(95.8081387868617, 0.979341693609794)
(-73.81388060068065, 1.02746473545409)
(42.38791356813192, 1.04830951977811)
(-51.81698248727967, 0.96214030755708)
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:
$$x_{1} = 4.05735660895802 \cdot 10^{-15}$$
$$x_{2} = -83.2401924707234$$
$$x_{3} = -45.5311340139913$$
$$x_{4} = 7.72525183693771$$
$$x_{5} = 39.2444323611642$$
$$x_{6} = -70.6716857116195$$
$$x_{7} = 171.210958939446$$
$$x_{8} = 89.5242209304172$$
$$x_{9} = 14.0661939128315$$
$$x_{10} = 102.091966464908$$
$$x_{11} = -95.8081387868617$$
$$x_{12} = 64.3871195905574$$
$$x_{13} = 76.9560263103312$$
$$x_{14} = -76.9560263103312$$
$$x_{15} = -7.72525183693771$$
$$x_{16} = -20.3713029592876$$
$$x_{17} = -2.79413003615298 \cdot 10^{-14}$$
$$x_{18} = -39.2444323611642$$
$$x_{19} = -14.0661939128315$$
$$x_{20} = -32.9563890398225$$
$$x_{21} = -227.761076847648$$
$$x_{22} = 26.6660542588127$$
$$x_{23} = -26.6660542588127$$
$$x_{24} = 51.8169824872797$$
$$x_{25} = -58.1022547544956$$
$$x_{26} = -89.5242209304172$$
$$x_{27} = -13215.1094216545$$
$$x_{28} = 58.1022547544956$$
$$x_{29} = -64.3871195905574$$
$$x_{30} = 127.226642643334$$
$$x_{31} = 32.9563890398225$$
$$x_{32} = 20.3713029592876$$
$$x_{33} = 45.5311340139913$$
$$x_{34} = 70.6716857116195$$
$$x_{35} = 83.2401924707234$$
$$x_{36} = 95.8081387868617$$
$$x_{37} = -51.8169824872797$$
Puntos máximos de la función:
$$x_{37} = 17.2207552719308$$
$$x_{37} = -80.0981286289451$$
$$x_{37} = 98.9500628243319$$
$$x_{37} = 161.785840727966$$
$$x_{37} = -86.3822220347287$$
$$x_{37} = -4.49340945790906$$
$$x_{37} = 4.49340945790906$$
$$x_{37} = 10.9041216594289$$
$$x_{37} = -42.3879135681319$$
$$x_{37} = 80.0981286289451$$
$$x_{37} = -48.6741442319544$$
$$x_{37} = -36.1006222443756$$
$$x_{37} = 61.2447302603744$$
$$x_{37} = -54.9596782878889$$
$$x_{37} = -98.9500628243319$$
$$x_{37} = 73.8138806006806$$
$$x_{37} = 54.9596782878889$$
$$x_{37} = -61.2447302603744$$
$$x_{37} = -67.5294347771441$$
$$x_{37} = 29.811598790893$$
$$x_{37} = 23.519452498689$$
$$x_{37} = 67.5294347771441$$
$$x_{37} = -10.9041216594289$$
$$x_{37} = 86.3822220347287$$
$$x_{37} = -23.519452498689$$
$$x_{37} = -17.2207552719308$$
$$x_{37} = -92.6661922776228$$
$$x_{37} = -29.811598790893$$
$$x_{37} = 92.6661922776228$$
$$x_{37} = 48.6741442319544$$
$$x_{37} = 36.1006222443756$$
$$x_{37} = -73.8138806006806$$
$$x_{37} = 42.3879135681319$$
Decrece en los intervalos
$$\left[171.210958939446, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -13215.1094216545\right]$$