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$$- \frac{\sin{\left(x - 1 \right)}}{x - 1} - \frac{\cos{\left(x - 1 \right)}}{\left(x - 1\right)^{2}} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = -27.2389365752603$$
$$x_{2} = 73.2427897046973$$
$$x_{3} = 82.6691650818489$$
$$x_{4} = -55.5309801938186$$
$$x_{5} = 7.12125046689807$$
$$x_{6} = -30.3840740178899$$
$$x_{7} = 44.9595528888955$$
$$x_{8} = 85.811211299318$$
$$x_{9} = 35.5285657554621$$
$$x_{10} = -46.1026627703624$$
$$x_{11} = -99.5210170746866$$
$$x_{12} = 95.2371684817036$$
$$x_{13} = 70.100567727981$$
$$x_{14} = 22.945612879981$$
$$x_{15} = -93.2371684817036$$
$$x_{16} = -49.2455828375744$$
$$x_{17} = 32.3840740178899$$
$$x_{18} = -39.8162093266346$$
$$x_{19} = -83.811211299318$$
$$x_{20} = -52.3883466217256$$
$$x_{21} = -5.12125046689807$$
$$x_{22} = 334.005818339011$$
$$x_{23} = 98.3791034786112$$
$$x_{24} = -1.79838604578389$$
$$x_{25} = 3.79838604578389$$
$$x_{26} = 10.3178664617911$$
$$x_{27} = 26.0929104121121$$
$$x_{28} = 48.1026627703624$$
$$x_{29} = -33.5285657554621$$
$$x_{30} = -77.5270825679419$$
$$x_{31} = 60.6735041304405$$
$$x_{32} = 16.644128370333$$
$$x_{33} = -96.3791034786112$$
$$x_{34} = 19.7964043662102$$
$$x_{35} = -71.2427897046973$$
$$x_{36} = -130.939312353727$$
$$x_{37} = 63.8159348889734$$
$$x_{38} = 38.672573565113$$
$$x_{39} = -14.644128370333$$
$$x_{40} = -42.9595528888955$$
$$x_{41} = -8.31786646179107$$
$$x_{42} = -80.6691650818489$$
$$x_{43} = -61.8159348889734$$
$$x_{44} = -90.0952098694071$$
$$x_{45} = 57.5309801938186$$
$$x_{46} = 76.3849592185347$$
$$x_{47} = 13.4864543952238$$
$$x_{48} = 88.9532251106725$$
$$x_{49} = -11.4864543952238$$
$$x_{50} = 29.2389365752603$$
$$x_{51} = -68.100567727981$$
$$x_{52} = -24.0929104121121$$
$$x_{53} = 79.5270825679419$$
$$x_{54} = -58.6735041304405$$
$$x_{55} = 54.3883466217256$$
$$x_{56} = -86.9532251106725$$
$$x_{57} = 51.2455828375744$$
$$x_{58} = -64.9582857893902$$
$$x_{59} = 41.8162093266346$$
$$x_{60} = -17.7964043662102$$
$$x_{61} = 92.0952098694071$$
$$x_{62} = 66.9582857893902$$
$$x_{63} = -74.3849592185347$$
$$x_{64} = -20.945612879981$$
$$x_{65} = -36.672573565113$$
The values of the extrema at the points:
(-27.238936575260272, 0.0353899155541688)
(73.24278970469729, -0.0138408859131547)
(82.66916508184887, 0.0122436055670467)
(-55.53098019381864, -0.0176866485521696)
(7.1212504668980685, 0.161228034325064)
(-30.38407401788986, -0.0318471321112693)
(44.959552888895495, 0.0227423004725314)
(85.81121129931802, -0.0117900744410766)
(35.52856575546206, -0.0289493889114503)
(-46.10266277036235, 0.0212254394164143)
(-99.52101707468658, -0.00994767611536293)
(95.23716848170359, 0.01061092686295)
(70.10056772798097, 0.0144701459746764)
(22.945612879981045, -0.0455199604051285)
(-93.23716848170359, -0.01061092686295)
(-49.24558283757444, -0.0198983065303553)
(32.38407401788986, 0.0318471321112693)
(-39.81620932663458, 0.0244927205346957)
(-83.81121129931802, 0.0117900744410766)
(-52.38834662172563, 0.0187273944640866)
(-5.1212504668980685, -0.161228034325064)
(334.00581833901066, 0.00300293699420144)
(98.3791034786112, -0.0102686022030809)
(-1.7983860457838872, 0.336508416918395)
(3.798386045783887, -0.336508416918395)
(10.317866461791066, -0.106707947715237)
(26.092910412112097, 0.0398202855500511)
(48.10266277036235, -0.0212254394164143)
(-33.52856575546206, 0.0289493889114503)
(-77.52708256794193, 0.0127334276777468)
(60.67350413044053, -0.0167555036571887)
(16.64412837033303, -0.0637915530395936)
(-96.3791034786112, 0.0102686022030809)
(19.796404366210158, 0.0531265325613881)
(-71.24278970469729, 0.0138408859131547)
(-130.9393123537265, -0.00757902448438246)
(63.81593488897342, 0.015917510583426)
(38.67257356511297, 0.0265351630103045)
(-14.644128370333028, 0.0637915530395936)
(-42.959552888895495, -0.0227423004725314)
(-8.317866461791066, 0.106707947715237)
(-80.66916508184887, -0.0122436055670467)
(-61.81593488897342, -0.015917510583426)
(-90.09520986940714, 0.0109768642483425)
(57.53098019381864, 0.0176866485521696)
(76.38495921853475, 0.0132640786518247)
(13.486454395223781, 0.0798311807800032)
(88.95322511067255, 0.0113689449158811)
(-11.486454395223781, -0.0798311807800032)
(29.238936575260272, -0.0353899155541688)
(-68.10056772798097, -0.0144701459746764)
(-24.092910412112097, -0.0398202855500511)
(79.52708256794193, -0.0127334276777468)
(-58.67350413044053, 0.0167555036571887)
(54.38834662172563, -0.0187273944640866)
(-86.95322511067255, -0.0113689449158811)
(51.24558283757444, 0.0198983065303553)
(-64.95828578939016, 0.0151593553168405)
(41.81620932663458, -0.0244927205346957)
(-17.796404366210158, -0.0531265325613881)
(92.09520986940714, -0.0109768642483425)
(66.95828578939016, -0.0151593553168405)
(-74.38495921853475, -0.0132640786518247)
(-20.945612879981045, 0.0455199604051285)
(-36.67257356511297, -0.0265351630103045)
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:
$$x_{1} = 73.2427897046973$$
$$x_{2} = -55.5309801938186$$
$$x_{3} = -30.3840740178899$$
$$x_{4} = 85.811211299318$$
$$x_{5} = 35.5285657554621$$
$$x_{6} = -99.5210170746866$$
$$x_{7} = 22.945612879981$$
$$x_{8} = -93.2371684817036$$
$$x_{9} = -49.2455828375744$$
$$x_{10} = -5.12125046689807$$
$$x_{11} = 98.3791034786112$$
$$x_{12} = 3.79838604578389$$
$$x_{13} = 10.3178664617911$$
$$x_{14} = 48.1026627703624$$
$$x_{15} = 60.6735041304405$$
$$x_{16} = 16.644128370333$$
$$x_{17} = -130.939312353727$$
$$x_{18} = -42.9595528888955$$
$$x_{19} = -80.6691650818489$$
$$x_{20} = -61.8159348889734$$
$$x_{21} = -11.4864543952238$$
$$x_{22} = 29.2389365752603$$
$$x_{23} = -68.100567727981$$
$$x_{24} = -24.0929104121121$$
$$x_{25} = 79.5270825679419$$
$$x_{26} = 54.3883466217256$$
$$x_{27} = -86.9532251106725$$
$$x_{28} = 41.8162093266346$$
$$x_{29} = -17.7964043662102$$
$$x_{30} = 92.0952098694071$$
$$x_{31} = 66.9582857893902$$
$$x_{32} = -74.3849592185347$$
$$x_{33} = -36.672573565113$$
Maxima of the function at points:
$$x_{33} = -27.2389365752603$$
$$x_{33} = 82.6691650818489$$
$$x_{33} = 7.12125046689807$$
$$x_{33} = 44.9595528888955$$
$$x_{33} = -46.1026627703624$$
$$x_{33} = 95.2371684817036$$
$$x_{33} = 70.100567727981$$
$$x_{33} = 32.3840740178899$$
$$x_{33} = -39.8162093266346$$
$$x_{33} = -83.811211299318$$
$$x_{33} = -52.3883466217256$$
$$x_{33} = 334.005818339011$$
$$x_{33} = -1.79838604578389$$
$$x_{33} = 26.0929104121121$$
$$x_{33} = -33.5285657554621$$
$$x_{33} = -77.5270825679419$$
$$x_{33} = -96.3791034786112$$
$$x_{33} = 19.7964043662102$$
$$x_{33} = -71.2427897046973$$
$$x_{33} = 63.8159348889734$$
$$x_{33} = 38.672573565113$$
$$x_{33} = -14.644128370333$$
$$x_{33} = -8.31786646179107$$
$$x_{33} = -90.0952098694071$$
$$x_{33} = 57.5309801938186$$
$$x_{33} = 76.3849592185347$$
$$x_{33} = 13.4864543952238$$
$$x_{33} = 88.9532251106725$$
$$x_{33} = -58.6735041304405$$
$$x_{33} = 51.2455828375744$$
$$x_{33} = -64.9582857893902$$
$$x_{33} = -20.945612879981$$
Decreasing at intervals
$$\left[98.3791034786112, \infty\right)$$
Increasing at intervals
$$\left(-\infty, -130.939312353727\right]$$