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$$- \frac{\sin{\left(\frac{1}{x} \right)}}{x^{2} x^{2}} + \frac{2 \cos{\left(\frac{1}{x} \right)}}{x^{3}} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = -21127.8065465298$$
$$x_{2} = 26344.0078318137$$
$$x_{3} = -39773.5440970563$$
$$x_{4} = 42447.4788863631$$
$$x_{5} = 39057.2052525216$$
$$x_{6} = 14479.7313150492$$
$$x_{7} = 33971.8484557919$$
$$x_{8} = -14348.5424650725$$
$$x_{9} = 11937.9523841008$$
$$x_{10} = -19432.887686874$$
$$x_{11} = 39904.771377312$$
$$x_{12} = 40752.3390879025$$
$$x_{13} = -16043.2338667424$$
$$x_{14} = 22953.9840195429$$
$$x_{15} = -15195.8737165349$$
$$x_{16} = -21975.2827488203$$
$$x_{17} = -35535.7330502043$$
$$x_{18} = 30581.6602478055$$
$$x_{19} = 25496.4920696083$$
$$x_{20} = 31429.2026711056$$
$$x_{21} = 27191.5291133465$$
$$x_{22} = -37230.8517263672$$
$$x_{23} = -18585.4480940284$$
$$x_{24} = 19564.09661214$$
$$x_{25} = 33124.297012115$$
$$x_{26} = 12785.166209279$$
$$x_{27} = 17021.8196886792$$
$$x_{28} = -29602.8987100326$$
$$x_{29} = -24517.764514177$$
$$x_{30} = 20411.5519538498$$
$$x_{31} = 41599.9082873259$$
$$x_{32} = -13501.2455714192$$
$$x_{33} = 23801.4794641224$$
$$x_{34} = 13632.4287111505$$
$$x_{35} = -40621.1115695252$$
$$x_{36} = 38209.640819107$$
$$x_{37} = 28886.5862895034$$
$$x_{38} = -12653.9899632038$$
$$x_{39} = -28755.3642709925$$
$$x_{40} = 32276.7483423665$$
$$x_{41} = -38078.4140636161$$
$$x_{42} = 24648.982396646$$
$$x_{43} = -34688.1770045647$$
$$x_{44} = -41468.6805452608$$
$$x_{45} = 36514.5174977468$$
$$x_{46} = -22822.7684696829$$
$$x_{47} = -16890.6185554691$$
$$x_{48} = -38925.9782260908$$
$$x_{49} = -31297.9789349291$$
$$x_{50} = 29734.1213504079$$
$$x_{51} = -30450.4370366936$$
$$x_{52} = -17738.0242579883$$
$$x_{53} = 21259.0191811792$$
$$x_{54} = 17869.2283625365$$
$$x_{55} = 37362.078192229$$
$$x_{56} = -25365.273191964$$
$$x_{57} = 15327.0673496263$$
$$x_{58} = -33840.6233739026$$
$$x_{59} = -27907.8340740483$$
$$x_{60} = -27060.3085181287$$
$$x_{61} = 34819.4024707177$$
$$x_{62} = -32993.0723444395$$
$$x_{63} = 18716.6547734102$$
$$x_{64} = 22106.496925216$$
$$x_{65} = -23670.2626854463$$
$$x_{66} = -36383.2913419629$$
$$x_{67} = -11806.7845665482$$
$$x_{68} = 28039.0554132717$$
$$x_{69} = -26212.7880538116$$
$$x_{70} = -42316.2509339008$$
$$x_{71} = 16174.4315462694$$
$$x_{72} = -32145.5241220507$$
$$x_{73} = -20280.3410578956$$
$$x_{74} = 35666.9588734903$$
The values of the extrema at the points:
(-21127.806546529755, -2.24022261018734e-9)
(26344.00783181366, -1.44090817738494e-9)
(-39773.54409705632, -6.32137300056214e-10)
(42447.47888636313, -5.55004108311864e-10)
(39057.20525252162, -6.55537698157871e-10)
(14479.731315049154, -4.76956742501154e-9)
(33971.84845579192, -8.66486186602981e-10)
(-14348.542465072484, -4.85718252877262e-9)
(11937.952384100769, -7.0168196438427e-9)
(-19432.88768687398, -2.64804473797296e-9)
(39904.77137731199, -6.27986555233478e-10)
(40752.339087902525, -6.0213644937939e-10)
(-16043.233866742383, -3.88522496240994e-9)
(22953.984019542928, -1.89794599074655e-9)
(-15195.873716534898, -4.33060574372337e-9)
(-21975.282748820264, -2.07076614527114e-9)
(-35535.733050204326, -7.91898357833571e-10)
(30581.660247805507, -1.06924660873998e-9)
(25496.492069608328, -1.5382932530713e-9)
(31429.202671105595, -1.01235602697324e-9)
(27191.529113346467, -1.35248587494898e-9)
(-37230.85172636723, -7.21429764890631e-10)
(-18585.448094028365, -2.89503572032555e-9)
(19564.096612140027, -2.61264499394054e-9)
(33124.29701211498, -9.11395037502451e-10)
(12785.16620927896, -6.11768684110306e-9)
(17021.819688679218, -3.45134224857833e-9)
(-29602.89871003259, -1.14112053667174e-9)
(-24517.764514177044, -1.66355920109288e-9)
(20411.551953849834, -2.40020284403386e-9)
(41599.908287325896, -5.77850180859856e-10)
(-13501.245571419151, -5.4859560696488e-9)
(23801.479464122403, -1.76519258172597e-9)
(13632.428711150475, -5.3808827430285e-9)
(-40621.111569525165, -6.06033167213377e-10)
(38209.64081910699, -6.84942460287209e-10)
(28886.586289503364, -1.19841588408651e-9)
(-12653.98996320384, -6.24518096509154e-9)
(-28755.364270992468, -1.20937852507445e-9)
(32276.748342366496, -9.59887746221052e-10)
(-38078.414063616125, -6.89671525667098e-10)
(24648.982396646003, -1.64589456193039e-9)
(-34688.17700456475, -8.31068930621954e-10)
(-41468.680545260824, -5.81513184145046e-10)
(36514.5174977468, -7.50013131646209e-10)
(-22822.768469682887, -1.91983254960971e-9)
(-16890.61855546906, -3.50516842376101e-9)
(-38925.97822609083, -6.59965037892555e-10)
(-31297.97893492907, -1.0208628786012e-9)
(29734.12135040786, -1.13107077399244e-9)
(-30450.43703669357, -1.07848209520273e-9)
(-17738.024257988258, -3.17826064554935e-9)
(21259.019181179232, -2.21265422816479e-9)
(17869.228362536483, -3.13175948424326e-9)
(37362.07819222895, -7.16370923011444e-10)
(-25365.273191964017, -1.55425012629189e-9)
(15327.067349626257, -4.25678648968596e-9)
(-33840.62337390262, -8.73219229198603e-10)
(-27907.834074048253, -1.28394888738824e-9)
(-27060.308518128684, -1.36563460190668e-9)
(34819.40247071765, -8.24816560090003e-10)
(-32993.07234443946, -9.18659311019976e-10)
(18716.6547734102, -2.85458868205407e-9)
(22106.496925216008, -2.04625683567672e-9)
(-23670.262685446316, -1.78481761852769e-9)
(-36383.29134196288, -7.55433137804917e-10)
(-11806.784566548204, -7.17359279569276e-9)
(28039.055413271715, -1.27195937864428e-9)
(-26212.7880538116, -1.45537050026306e-9)
(-42316.25093390082, -5.58451719181666e-10)
(16174.431546269394, -3.82245117419289e-9)
(-32145.524122050676, -9.6774063535741e-10)
(-20280.34105789558, -2.43136125002401e-9)
(35666.95887349029, -7.8608197479349e-10)
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:
The function has no minima
The function has no maxima
Decreasing at the entire real axis