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$$x \cos{\left(x \right)} - \cos{\left(x \right)} + \frac{1}{x - 1} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = 61.2607813662104$$
$$x_{2} = 54.977528216687$$
$$x_{3} = 45.5535972495666$$
$$x_{4} = 26.7050509886075$$
$$x_{5} = -45.5535548948216$$
$$x_{6} = -67.5440292085082$$
$$x_{7} = 10.9855451469272$$
$$x_{8} = -14.1415287027946$$
$$x_{9} = 161.791982981232$$
$$x_{10} = 80.1104528829182$$
$$x_{11} = -10.9886165929728$$
$$x_{12} = 42.4109176868425$$
$$x_{13} = -86.3936670434456$$
$$x_{14} = 73.8272388155349$$
$$x_{15} = 4.63670598778003$$
$$x_{16} = 95.8186871619757$$
$$x_{17} = 64.4028981592989$$
$$x_{18} = 14.1429561112565$$
$$x_{19} = 29.8439282455701$$
$$x_{20} = 2.25659975598781$$
$$x_{21} = -58.119750202589$$
$$x_{22} = 51.8366657262513$$
$$x_{23} = -48.6942811938986$$
$$x_{24} = -73.8272487597743$$
$$x_{25} = -64.4028831779781$$
$$x_{26} = 17.2749842216247$$
$$x_{27} = -20.4225312560761$$
$$x_{28} = -26.7048403887156$$
$$x_{29} = -29.8440790790236$$
$$x_{30} = 92.6768642991083$$
$$x_{31} = -61.2607987741546$$
$$x_{32} = -70.6860293004454$$
$$x_{33} = -54.9775523044659$$
$$x_{34} = -80.1104606656025$$
$$x_{35} = -7.8667016459742$$
$$x_{36} = -1.70762553238458$$
$$x_{37} = 86.3936608385848$$
$$x_{38} = 36.1275051050789$$
$$x_{39} = 20.4230029921347$$
$$x_{40} = 83.2523531299668$$
$$x_{41} = -23.560287097399$$
$$x_{42} = 70.6860406304616$$
$$x_{43} = 23.5599800815366$$
$$x_{44} = 7.87513937945483$$
$$x_{45} = -98.9600685081675$$
$$x_{46} = -42.4109701821765$$
$$x_{47} = -17.2757656106429$$
$$x_{48} = -51.8366369876076$$
$$x_{49} = 67.5440162219978$$
$$x_{50} = -89.5355126279385$$
$$x_{51} = -92.6768693254059$$
$$x_{52} = -83.2523461957994$$
$$x_{53} = 76.9691932835873$$
$$x_{54} = 48.6942465201693$$
$$x_{55} = -36.1275900678989$$
$$x_{56} = -4.68140350043458$$
$$x_{57} = 58.119770589025$$
$$x_{58} = -32.9875885466119$$
$$x_{59} = 98.9600643798826$$
$$x_{60} = -95.8186826141585$$
$$x_{61} = 32.9877001765031$$
$$x_{62} = 89.5355182021202$$
$$x_{63} = 39.2705909324313$$
$$x_{64} = -39.2705248010067$$
$$x_{65} = -76.9691845084486$$
Signos de extremos en los puntos:
(61.26078136621038, -56.1623729599054)
(54.97752821668704, -49.9893004427013)
(45.55359724956661, 48.3497807205169)
(26.705050988607464, 28.9501956267596)
(-45.55355489482161, 50.3936918893375 + pi*I)
(-67.5440292085082, -64.3167641986204 + pi*I)
(10.985545146927166, -7.69393337033291)
(-14.141528702794568, 17.8544641357235 + pi*I)
(161.79198298123237, -155.711910041179)
(80.11045288291817, -74.7397666429213)
(-10.988616592972788, -9.51132646873339 + pi*I)
(42.410917686842495, -37.687949224265)
(-86.39366704344562, -82.9233744040566 + pi*I)
(73.82723881553494, -68.539336020123)
(4.636705987780029, -2.41082802510856)
(95.81868716197569, 99.3705418597584)
(64.4028981592989, 67.5521570095379)
(14.142956111256495, 15.7128326925369)
(29.843928245570083, -25.4832098617902)
(2.2565997559878066, 0.567610200608466)
(-58.11975020258898, 63.1990267230257 + pi*I)
(51.83666572625129, 54.7648928390048)
(-48.694281193898554, -45.7887921963908 + pi*I)
(-73.82724875977429, -70.5122440590706 + pi*I)
(-64.4028831779781, 69.5832139541491 + pi*I)
(17.274984221624663, -13.4890143800309)
(-20.42253125607606, 24.4847446227787 + pi*I)
(-26.704840388715592, 31.0251211843448 + pi*I)
(-29.844079079023626, -27.4161683636881 + pi*I)
(92.67686429910835, -87.1587125822689)
(-61.26079877415462, -58.129722678719 + pi*I)
(-70.68602930044538, 75.9581292277548 + pi*I)
(-54.97755230446593, -51.9529178286017 + pi*I)
(-80.11046066560252, -76.7147997919567 + pi*I)
(-7.866701645974201, 11.0355675483802 + pi*I)
(-1.7076255323845784, 3.54198807320733 + pi*I)
(86.39366083858478, -80.9465253017773)
(36.127505105078896, -31.5693195360851)
(20.4230029921347, 22.3867420990701)
(83.25235312996678, 86.6619964203348)
(-23.560287097399023, -21.3607803586146 + pi*I)
(70.68604063046158, 73.9298332480638)
(23.55998008153659, -19.4457238103368)
(7.875139379454832, 8.78035635787239)
(-98.96006850816748, -95.3553972961565 + pi*I)
(-42.41097018217647, -39.640782533263 + pi*I)
(-17.275765610642857, -15.373101780254 + pi*I)
(-51.83663698760759, 56.8034802269421 + pi*I)
(67.54401622199784, -62.3463767285514)
(-89.53551262793849, 95.0411321320011 + pi*I)
(-92.67686932540586, -89.1371313727456 + pi*I)
(-83.2523461957994, 88.686020900962 + pi*I)
(76.96919328358727, 80.2993467791276)
(48.694246520169315, -43.8298707494019)
(-36.12759006789888, -33.5139453857235 + pi*I)
(-4.6814035004345795, -3.97245858561825 + pi*I)
(58.119770589025016, 61.1646117102026)
(-32.9875885466119, 37.5127055427305 + pi*I)
(98.96006437988264, -93.3756081668975)
(-95.8186826141585, 101.391415361436 + pi*I)
(32.98770017650311, 35.4520590458737)
(89.53551820212023, 93.0187937139544)
(39.2705909324313, 41.9145809901848)
(-39.2705248010067, 43.9655203206385 + pi*I)
(-76.96918450844859, 82.3253326362998 + pi*I)
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:
$$x_{1} = 61.2607813662104$$
$$x_{2} = 54.977528216687$$
$$x_{3} = 10.9855451469272$$
$$x_{4} = 161.791982981232$$
$$x_{5} = 80.1104528829182$$
$$x_{6} = 42.4109176868425$$
$$x_{7} = 73.8272388155349$$
$$x_{8} = 4.63670598778003$$
$$x_{9} = 29.8439282455701$$
$$x_{10} = 17.2749842216247$$
$$x_{11} = 92.6768642991083$$
$$x_{12} = 86.3936608385848$$
$$x_{13} = 36.1275051050789$$
$$x_{14} = 23.5599800815366$$
$$x_{15} = 67.5440162219978$$
$$x_{16} = 48.6942465201693$$
$$x_{17} = 98.9600643798826$$
Puntos máximos de la función:
$$x_{17} = 45.5535972495666$$
$$x_{17} = 26.7050509886075$$
$$x_{17} = 95.8186871619757$$
$$x_{17} = 64.4028981592989$$
$$x_{17} = 14.1429561112565$$
$$x_{17} = 2.25659975598781$$
$$x_{17} = 51.8366657262513$$
$$x_{17} = 20.4230029921347$$
$$x_{17} = 83.2523531299668$$
$$x_{17} = 70.6860406304616$$
$$x_{17} = 7.87513937945483$$
$$x_{17} = 76.9691932835873$$
$$x_{17} = 58.119770589025$$
$$x_{17} = 32.9877001765031$$
$$x_{17} = 89.5355182021202$$
$$x_{17} = 39.2705909324313$$
Decrece en los intervalos
$$\left[161.791982981232, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, 4.63670598778003\right]$$