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$$\frac{e^{x} - \cos{\left(x \right)} + 1}{- x + \left(e^{x} - \sin{\left(x \right)}\right)} + \frac{\left(x + \left(e^{x} - \sin{\left(x \right)}\right)\right) \left(- e^{x} + \cos{\left(x \right)} + 1\right)}{\left(- x + \left(e^{x} - \sin{\left(x \right)}\right)\right)^{2}} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = 68.4026026533356$$
$$x_{2} = 55.6879491257269$$
$$x_{3} = -633.02933999405$$
$$x_{4} = 80$$
$$x_{5} = -70.6716857116195$$
$$x_{6} = 66.5$$
$$x_{7} = -92.6661922776228$$
$$x_{8} = -86.3822220347287$$
$$x_{9} = 66.223996865801$$
$$x_{10} = -89.5242209304172$$
$$x_{11} = 74$$
$$x_{12} = 72$$
$$x_{13} = -45.5311340139913$$
$$x_{14} = 65.6834795893247$$
$$x_{15} = -80.0981286289451$$
$$x_{16} = 34.3069734414766$$
$$x_{17} = 49.7754698215909$$
$$x_{18} = -14.0661930771291$$
$$x_{19} = -58.1022547544956$$
$$x_{20} = -42.3879135681319$$
$$x_{21} = -23.5194524987527$$
$$x_{22} = 40.0202154992634$$
$$x_{23} = -29.8115987908931$$
$$x_{24} = 63.5975419173648$$
$$x_{25} = 47.8119589618645$$
$$x_{26} = 45.8533704858379$$
$$x_{27} = -20.3713029577941$$
$$x_{28} = 66.2441343042238$$
$$x_{29} = 86$$
$$x_{30} = -73.8138806006806$$
$$x_{31} = 35.9035726630501$$
$$x_{32} = -67.5294347771441$$
$$x_{33} = -95.8081387868617$$
$$x_{34} = 78$$
$$x_{35} = 41.9557498041686$$
$$x_{36} = 67.3055555555556$$
$$x_{37} = 38.0970665022932$$
$$x_{38} = -17.2207553071165$$
$$x_{39} = 66.3753305431907$$
$$x_{40} = 68$$
$$x_{41} = 90$$
$$x_{42} = 43.9008089821284$$
$$x_{43} = 70$$
$$x_{44} = -7.72474872891224$$
$$x_{45} = 67.0420892015282$$
$$x_{46} = 65.710495061406$$
$$x_{47} = -32.9563890398225$$
$$x_{48} = 59.6426199192496$$
$$x_{49} = 61.6276851753556$$
$$x_{50} = 57.6644222037334$$
$$x_{51} = -98.9500628243319$$
$$x_{52} = 53.7140644634518$$
$$x_{53} = 96$$
$$x_{54} = 36.190501914492$$
$$x_{55} = 65.5294651682181$$
$$x_{56} = -54.9596782878889$$
$$x_{57} = 98$$
$$x_{58} = -39.2444323611642$$
$$x_{59} = -83.2401924707234$$
$$x_{60} = 82$$
$$x_{61} = -61.2447302603744$$
$$x_{62} = 88$$
$$x_{63} = 94$$
$$x_{64} = -48.6741442319544$$
$$x_{65} = -4.50721566039406$$
$$x_{66} = 32.4565264847902$$
$$x_{67} = -64.3871195905574$$
$$x_{68} = 84$$
$$x_{69} = 51.7430578258417$$
$$x_{70} = -26.6660542588099$$
$$x_{71} = -76.9560263103312$$
$$x_{72} = 67.6734773926687$$
$$x_{73} = -51.8169824872797$$
$$x_{74} = -36.1006222443756$$
$$x_{75} = 92$$
$$x_{76} = 76$$
$$x_{77} = -10.904141811359$$
$$x_{78} = 65.687962582344$$
$$x_{79} = 153.051567263916$$
$$x_{80} = 100$$
Signos de extremos en los puntos:
(68.40260265333565, 1)
(55.68794912572692, 1)
(-633.0293399940497, -1.00316440610274)
(80, 1)
(-70.6716857116195, -0.972097731728565)
(66.5, 1)
(-92.66619227762284, -1.02181701056246)
(-86.38222203472871, -1.02342249220221)
(66.22399686580096, 1)
(-89.52422093041719, -0.977907828460357)
(74, 1)
(72, 1)
(-45.53113401399128, -0.957028166104047)
(65.68347958932469, 1)
(-80.09812862894512, -1.02528305302948)
(34.30697344147655, 1.00000000000009)
(49.77546982159091, 1)
(-14.066193077129078, -0.867564441384213)
(-58.10225475449559, -0.966165269591092)
(-42.38791356813192, -1.04830951977811)
(-23.519452498752702, -1.08872838161589)
(40.02021549926337, 1)
(-29.811598790893076, -1.06937611361913)
(63.59754191736475, 1)
(47.81195896186453, 1)
(45.85337048583789, 1)
(-20.371302957794114, -0.906523852218779)
(66.24413430422379, 1)
(86, 1)
(-73.81388060068065, -1.02746473545409)
(35.903572663050056, 1.00000000000002)
(-67.52943477714412, -1.03005853664807)
(-95.8081387868617, -0.979341693609794)
(78, 1)
(41.95574980416862, 1)
(67.30555555555556, 1)
(38.09706650229323, 1)
(-17.220755307116466, -1.12307869434072)
(66.3753305431907, 1)
(68, 1)
(90, 1)
(43.90080898212835, 1)
(70, 1)
(-7.724748728912244, -0.772371297588025)
(67.04208920152821, 1)
(65.71049506140604, 1)
(-32.95638903982247, -0.941127220242848)
(59.64261991924964, 1)
(61.62768517535564, 1)
(57.664422203733444, 1)
(-98.95006282433188, -1.02041751453922)
(53.71406446345178, 1)
(96, 1)
(36.190501914492025, 1.00000000000001)
(65.52946516821812, 1)
(-54.959678287888934, -1.0370584656736)
(98, 1)
(-39.24443236116419, -0.950319410117988)
(-83.2401924707234, -0.976260056542518)
(82, 1)
(-61.2447302603744, -1.03319342653464)
(88, 1)
(94, 1)
(-48.674144231954386, -1.0419424245269)
(-4.507215660394062, -1.5470113797114)
(32.456526484790174, 1.00000000000052)
(-64.38711959055742, -0.969416568120202)
(84, 1)
(51.74305782584166, 1)
(-26.666054258809947, -0.927758187267586)
(-76.95602631033118, -0.974346648664662)
(67.67347739266866, 1)
(-51.81698248727967, -0.96214030755708)
(-36.100622244375614, -1.05695657697143)
(92, 1)
(76, 1)
(-10.904141811359024, -1.20100336861316)
(65.68796258234401, 1)
(153.05156726391584, 1)
(100, 1)
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:
$$x_{1} = -633.02933999405$$
$$x_{2} = -92.6661922776228$$
$$x_{3} = -86.3822220347287$$
$$x_{4} = 72$$
$$x_{5} = -80.0981286289451$$
$$x_{6} = -42.3879135681319$$
$$x_{7} = -23.5194524987527$$
$$x_{8} = -29.8115987908931$$
$$x_{9} = 63.5975419173648$$
$$x_{10} = -73.8138806006806$$
$$x_{11} = -67.5294347771441$$
$$x_{12} = -17.2207553071165$$
$$x_{13} = 90$$
$$x_{14} = 65.710495061406$$
$$x_{15} = 61.6276851753556$$
$$x_{16} = -98.9500628243319$$
$$x_{17} = -54.9596782878889$$
$$x_{18} = -61.2447302603744$$
$$x_{19} = -48.6741442319544$$
$$x_{20} = -4.50721566039406$$
$$x_{21} = -36.1006222443756$$
$$x_{22} = -10.904141811359$$
Puntos máximos de la función:
$$x_{22} = -70.6716857116195$$
$$x_{22} = -89.5242209304172$$
$$x_{22} = -45.5311340139913$$
$$x_{22} = -14.0661930771291$$
$$x_{22} = -58.1022547544956$$
$$x_{22} = -20.3713029577941$$
$$x_{22} = -95.8081387868617$$
$$x_{22} = 38.0970665022932$$
$$x_{22} = -7.72474872891224$$
$$x_{22} = -32.9563890398225$$
$$x_{22} = -39.2444323611642$$
$$x_{22} = -83.2401924707234$$
$$x_{22} = -64.3871195905574$$
$$x_{22} = -26.6660542588099$$
$$x_{22} = -76.9560263103312$$
$$x_{22} = -51.8169824872797$$
Decrece en los intervalos
$$\left[90, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -633.02933999405\right]$$