In order to find the extrema, we need to solve the equation
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative$$- \frac{\sin{\left(x \right)}}{x - 1} - \frac{\cos{\left(x \right)}}{\left(x - 1\right)^{2}} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = 100.520917114109$$
$$x_{2} = -25.0944376288815$$
$$x_{3} = 2.57625015820118$$
$$x_{4} = -37.673259943911$$
$$x_{5} = 191.631906205502$$
$$x_{6} = 15.6397620877646$$
$$x_{7} = 188.490225654589$$
$$x_{8} = 56.5306616416093$$
$$x_{9} = -69.1007741687956$$
$$x_{10} = 91.0950880256329$$
$$x_{11} = 87.9530943542027$$
$$x_{12} = -15.6479679638982$$
$$x_{13} = 50.2451786914948$$
$$x_{14} = -6.14411351301787$$
$$x_{15} = -12.492390025579$$
$$x_{16} = -53.388691007263$$
$$x_{17} = -94.2372799036618$$
$$x_{18} = 28.2376364595748$$
$$x_{19} = -2.88996969767843$$
$$x_{20} = -59.6737803264459$$
$$x_{21} = -1077.56535302402$$
$$x_{22} = 34.527701946778$$
$$x_{23} = 31.3830252979972$$
$$x_{24} = 9.30494468339504$$
$$x_{25} = -100.521115065812$$
$$x_{26} = 12.4794779911025$$
$$x_{27} = 21.9434371567881$$
$$x_{28} = 75.3847808857452$$
$$x_{29} = -78.5272426949571$$
$$x_{30} = 94.2370546693974$$
$$x_{31} = 81.6690132946536$$
$$x_{32} = -72.242978694986$$
$$x_{33} = -122.514017419839$$
$$x_{34} = 25.0912562079058$$
$$x_{35} = 69.1003552230555$$
$$x_{36} = 43.9590233567938$$
$$x_{37} = -50.2459712046114$$
$$x_{38} = 78.5269183093816$$
$$x_{39} = 40.8155939881502$$
$$x_{40} = 59.6732185170696$$
$$x_{41} = 65.9580523911179$$
$$x_{42} = 72.24259540785$$
$$x_{43} = 37.6718497263809$$
$$x_{44} = 18.7934144113698$$
$$x_{45} = -62.8161843480611$$
$$x_{46} = -43.9600588531378$$
$$x_{47} = -97.379207861883$$
$$x_{48} = -28.2401476526276$$
$$x_{49} = -87.9533529268738$$
$$x_{50} = -75.3851328811964$$
$$x_{51} = -546.635295693876$$
$$x_{52} = -18.7990914357831$$
$$x_{53} = 62.815677356778$$
$$x_{54} = -84.8113487041494$$
$$x_{55} = 84.8110706151124$$
$$x_{56} = 6.08916120309943$$
$$x_{57} = 53.3879890840753$$
$$x_{58} = -47.1031041186137$$
$$x_{59} = 47.1022022669651$$
$$x_{60} = -21.9475985837942$$
$$x_{61} = -65.9585122146304$$
$$x_{62} = -9.32825706323943$$
$$x_{63} = -34.5293808983144$$
$$x_{64} = -81.6693131963402$$
$$x_{65} = -40.8167952172419$$
$$x_{66} = 97.378996929011$$
$$x_{67} = -31.38505790634$$
$$x_{68} = -56.5312876685112$$
$$x_{69} = -91.0953290668266$$
The values of the extrema at the points:
(100.52091711410945, 0.0100476316966419)
(-25.094437628881476, -0.0382942342355763)
(2.5762501582011796, -0.535705052303484)
(-37.673259943911006, -0.0258490197028825)
(191.63190620550185, -0.00524563941809883)
(15.63976208776456, -0.0681483206400774)
(188.4902256545889, 0.00533353551155559)
(56.53066164160934, 0.0180051500304447)
(-69.10077416879557, -0.0142637264671467)
(91.09508802563293, -0.0110987005999837)
(87.95309435420273, 0.0114996928375307)
(-15.647967963898166, 0.0599593189797558)
(50.245178691494786, 0.0203023709567303)
(-6.1441135130178655, -0.138623930394573)
(-12.492390025578958, -0.0739131230459364)
(-53.388691007263, 0.0183830682189117)
(-94.23727990366179, -0.0104995111118831)
(28.237636459574798, -0.0366891865463047)
(-2.8899696976784344, 0.248976134877405)
(-59.67378032644585, 0.0164793457895915)
(-1077.5653530240243, 0.000927157142018311)
(34.52770194677802, -0.0298128246468963)
(31.38302529799723, 0.0328953023371544)
(9.304944683395044, -0.119546681963348)
(-100.52111506581193, -0.00984968979094353)
(12.479477991102517, 0.0867833198945747)
(21.94343715678808, -0.0476933188520339)
(75.38478088574516, 0.0134423955413013)
(-78.52724269495707, 0.0125733134820883)
(94.23705466939735, 0.010724732692878)
(81.66901329465364, 0.0123953812433342)
(-72.242978694986, 0.0136519134817116)
(-122.51401741983913, 0.00809598171844709)
(25.091256207905772, 0.0414731225016059)
(69.1003552230555, 0.0146826283229769)
(43.95902335679378, 0.0232716924030311)
(-50.24597120461141, -0.0195100148956696)
(78.5269183093816, -0.0128976727485698)
(40.81559398815024, -0.0251078697468112)
(59.673218517069586, -0.0170410762454831)
(65.9580523911179, -0.0153927263543733)
(72.24259540785, -0.0140351638863266)
(37.67184972638089, 0.0272587398500595)
(18.793414411369753, 0.0561120230339157)
(-62.81618434806106, -0.0156680826074814)
(-43.960058853137774, -0.022236464203186)
(-97.37920786188297, 0.0101642243790071)
(-28.240147652627645, 0.0341795711715136)
(-87.9533529268738, -0.0112411368826843)
(-75.38513288119637, -0.0130904310684593)
(-546.6352956938762, -0.00182602973305305)
(-18.79909143578314, -0.0504430691319447)
(62.815677356778, 0.0161750096209984)
(-84.81134870414938, 0.0116526790492257)
(84.81107061511238, -0.0119307487512748)
(6.089161203099427, 0.192809042427521)
(53.387989084075315, -0.0190848682073296)
(-47.10310411861372, 0.0207841885412821)
(47.10220226696507, -0.0216858368023364)
(-21.947598583794207, 0.0435362264748061)
(-65.95851221463039, 0.0149329557083856)
(-9.328257063239425, 0.0963710979823201)
(-34.5293808983144, 0.0281345781753277)
(-81.66931319634023, -0.0120955020439642)
(-40.81679521724192, 0.023907001519389)
(97.37899692901101, -0.0103751461271118)
(-31.385057906339963, -0.0308637274812354)
(-56.53128766851124, -0.0173792211238612)
(-91.09532906682657, 0.0108576739325778)
Intervals of increase and decrease of the function:Let's find intervals where the function increases and decreases, as well as minima and maxima of the function, for this let's look how the function behaves itself in the extremas and at the slightest deviation from:
Minima of the function at points:
$$x_{1} = -25.0944376288815$$
$$x_{2} = 2.57625015820118$$
$$x_{3} = -37.673259943911$$
$$x_{4} = 191.631906205502$$
$$x_{5} = 15.6397620877646$$
$$x_{6} = -69.1007741687956$$
$$x_{7} = 91.0950880256329$$
$$x_{8} = -6.14411351301787$$
$$x_{9} = -12.492390025579$$
$$x_{10} = -94.2372799036618$$
$$x_{11} = 28.2376364595748$$
$$x_{12} = 34.527701946778$$
$$x_{13} = 9.30494468339504$$
$$x_{14} = -100.521115065812$$
$$x_{15} = 21.9434371567881$$
$$x_{16} = -50.2459712046114$$
$$x_{17} = 78.5269183093816$$
$$x_{18} = 40.8155939881502$$
$$x_{19} = 59.6732185170696$$
$$x_{20} = 65.9580523911179$$
$$x_{21} = 72.24259540785$$
$$x_{22} = -62.8161843480611$$
$$x_{23} = -43.9600588531378$$
$$x_{24} = -87.9533529268738$$
$$x_{25} = -75.3851328811964$$
$$x_{26} = -546.635295693876$$
$$x_{27} = -18.7990914357831$$
$$x_{28} = 84.8110706151124$$
$$x_{29} = 53.3879890840753$$
$$x_{30} = 47.1022022669651$$
$$x_{31} = -81.6693131963402$$
$$x_{32} = 97.378996929011$$
$$x_{33} = -31.38505790634$$
$$x_{34} = -56.5312876685112$$
Maxima of the function at points:
$$x_{34} = 100.520917114109$$
$$x_{34} = 188.490225654589$$
$$x_{34} = 56.5306616416093$$
$$x_{34} = 87.9530943542027$$
$$x_{34} = -15.6479679638982$$
$$x_{34} = 50.2451786914948$$
$$x_{34} = -53.388691007263$$
$$x_{34} = -2.88996969767843$$
$$x_{34} = -59.6737803264459$$
$$x_{34} = -1077.56535302402$$
$$x_{34} = 31.3830252979972$$
$$x_{34} = 12.4794779911025$$
$$x_{34} = 75.3847808857452$$
$$x_{34} = -78.5272426949571$$
$$x_{34} = 94.2370546693974$$
$$x_{34} = 81.6690132946536$$
$$x_{34} = -72.242978694986$$
$$x_{34} = -122.514017419839$$
$$x_{34} = 25.0912562079058$$
$$x_{34} = 69.1003552230555$$
$$x_{34} = 43.9590233567938$$
$$x_{34} = 37.6718497263809$$
$$x_{34} = 18.7934144113698$$
$$x_{34} = -97.379207861883$$
$$x_{34} = -28.2401476526276$$
$$x_{34} = 62.815677356778$$
$$x_{34} = -84.8113487041494$$
$$x_{34} = 6.08916120309943$$
$$x_{34} = -47.1031041186137$$
$$x_{34} = -21.9475985837942$$
$$x_{34} = -65.9585122146304$$
$$x_{34} = -9.32825706323943$$
$$x_{34} = -34.5293808983144$$
$$x_{34} = -40.8167952172419$$
$$x_{34} = -91.0953290668266$$
Decreasing at intervals
$$\left[191.631906205502, \infty\right)$$
Increasing at intervals
$$\left(-\infty, -546.635295693876\right]$$