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$$- \frac{\sin{\left(\frac{1}{x} \right)}}{x^{2}} - \frac{\cos{\left(\frac{1}{x} \right)}}{x^{3}} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = -22820.7728941576$$
$$x_{2} = -30448.9416552006$$
$$x_{3} = 36513.2705382744$$
$$x_{4} = -35534.4517543759$$
$$x_{5} = 21256.8765949775$$
$$x_{6} = -21973.2101350335$$
$$x_{7} = -25363.4778078947$$
$$x_{8} = 11934.1321841624$$
$$x_{9} = -23668.3386246561$$
$$x_{10} = -37229.6287891721$$
$$x_{11} = -33839.277869644$$
$$x_{12} = -16040.3936060323$$
$$x_{13} = 17866.6787268417$$
$$x_{14} = 20409.3203021652$$
$$x_{15} = 33122.922370868$$
$$x_{16} = -27906.2023693417$$
$$x_{17} = -38077.2183563533$$
$$x_{18} = -42315.1750077004$$
$$x_{19} = -11802.9219858377$$
$$x_{20} = 40751.2218421629$$
$$x_{21} = -40619.9907312464$$
$$x_{22} = 42446.4062709612$$
$$x_{23} = -13497.8691947406$$
$$x_{24} = 14476.5835259143$$
$$x_{25} = -18582.9969472475$$
$$x_{26} = -24515.9070182535$$
$$x_{27} = 28037.4313121062$$
$$x_{28} = 25494.7058871718$$
$$x_{29} = -27058.6256744435$$
$$x_{30} = 27189.8543562071$$
$$x_{31} = -36382.0399055397$$
$$x_{32} = 22951.9998063081$$
$$x_{33} = 35665.6822704316$$
$$x_{34} = -12650.3868310499$$
$$x_{35} = -26211.0507610785$$
$$x_{36} = 24647.1347484752$$
$$x_{37} = 18714.2207495256$$
$$x_{38} = -31296.5240688142$$
$$x_{39} = -16887.9210447625$$
$$x_{40} = 16171.6142548589$$
$$x_{41} = -38924.808562425$$
$$x_{42} = 26342.2791561729$$
$$x_{43} = -20278.0950218298$$
$$x_{44} = 39903.6303938846$$
$$x_{45} = -17735.4558269403$$
$$x_{46} = 32275.3375870094$$
$$x_{47} = 38208.4491995411$$
$$x_{48} = 30580.1712546872$$
$$x_{49} = 28885.0098692674$$
$$x_{50} = -32991.6922603157$$
$$x_{51} = 29732.5898907212$$
$$x_{52} = -28753.7806881948$$
$$x_{53} = -39772.3993665363$$
$$x_{54} = -32144.1076334276$$
$$x_{55} = 37360.8595307203$$
$$x_{56} = 23799.5659677507$$
$$x_{57} = -15192.8747405778$$
$$x_{58} = 31427.7538525251$$
$$x_{59} = 12781.5999770302$$
$$x_{60} = 39056.0395006955$$
$$x_{61} = 41598.8138115962$$
$$x_{62} = 19561.7681653678$$
$$x_{63} = 22104.4365655516$$
$$x_{64} = 33970.5081255404$$
$$x_{65} = -21125.6507050534$$
$$x_{66} = -34686.8643893828$$
$$x_{67} = 17019.1429035105$$
$$x_{68} = 13629.0847527609$$
$$x_{69} = -14345.3659691272$$
$$x_{70} = -19430.5435758771$$
$$x_{71} = 15324.0939721606$$
$$x_{72} = -41467.5826221759$$
$$x_{73} = 34818.0947801175$$
$$x_{74} = -29601.3604914767$$
The values of the extrema at the points:
(-22820.772894157573, 1.92016832722739e-9)
(-30448.941655200615, 1.07858802910931e-9)
(36513.27053827435, 7.50064359890521e-10)
(-35534.45175437587, 7.91955467372039e-10)
(21256.876594977497, 2.21310030110204e-9)
(-21973.210135033478, 2.07115681341263e-9)
(-25363.47780789469, 1.55447017378804e-9)
(11934.132184162385, 7.02131264624273e-9)
(-23668.338624656142, 1.78510781631491e-9)
(-37229.62878917209, 7.21477161603512e-10)
(-33839.277869644044, 8.73288672042809e-10)
(-16040.393606032281, 3.88660099711682e-9)
(17866.678726841652, 3.13265337670453e-9)
(20409.320302165244, 2.4007277737314e-9)
(33122.922370867964, 9.11470687336713e-10)
(-27906.202369341747, 1.28409903998383e-9)
(-38077.21835635332, 6.89714840869954e-10)
(-42315.17500770036, 5.58480118576188e-10)
(-11802.92198583774, 7.17828878815265e-9)
(40751.22184216292, 6.02169466601296e-10)
(-40619.99073124638, 6.06066612668086e-10)
(42446.4062709612, 5.5503215853839e-10)
(-13497.869194740559, 5.48870095304181e-9)
(14476.583525914342, 4.77164184811957e-9)
(-18582.99694724754, 2.89579949932944e-9)
(-24515.90701825348, 1.66381129723624e-9)
(28037.43131210623, 1.27210674293587e-9)
(25494.705887171793, 1.53850880987833e-9)
(-27058.625674443527, 1.36580447228137e-9)
(27189.854356207117, 1.35265249323197e-9)
(-36382.03990553967, 7.55485108297283e-10)
(22951.999806308108, 1.89827416311931e-9)
(35665.68227043159, 7.8613824940945e-10)
(-12650.386831049884, 6.2487390379762e-9)
(-26211.05076107848, 1.45556343397359e-9)
(24647.13474847516, 1.64614133781089e-9)
(18714.220749525568, 2.85533128463339e-9)
(-31296.524068814153, 1.02095779385533e-9)
(-16887.92104476253, 3.50628827966937e-9)
(16171.614254858863, 3.82378312981834e-9)
(-38924.80856242503, 6.60004701622595e-10)
(26342.27915617286, 1.44109729949528e-9)
(-20278.09502182977, 2.43189988518305e-9)
(39903.63039388465, 6.28022468513138e-10)
(-17735.455826940255, 3.17918126065533e-9)
(32275.33758700941, 9.59971661769784e-10)
(38208.44919954114, 6.84985184164745e-10)
(30580.171254687153, 1.06935073800839e-9)
(28885.0098692674, 1.19854669695631e-9)
(-32991.69226031572, 9.1873617026553e-10)
(29732.58989072119, 1.13118729530794e-9)
(-28753.78068819479, 1.20951173961037e-9)
(-39772.39936653633, 6.32173689106629e-10)
(-32144.107633427648, 9.67825928054685e-10)
(37360.85953072034, 7.16417658076677e-10)
(23799.565967750717, 1.76547643881391e-9)
(-15192.874740577829, 4.33231558626961e-9)
(31427.75385252513, 1.01244936863584e-9)
(12781.599977030211, 6.12110115785768e-9)
(39056.03950069551, 6.555768321039e-10)
(41598.813811596214, 5.77880588151573e-10)
(19561.768165367816, 2.61326700203962e-9)
(22104.43656555161, 2.04663831900024e-9)
(33970.50812554042, 8.66554563844005e-10)
(-21125.650705053413, 2.24067985801215e-9)
(-34686.864389382834, 8.31131830416913e-10)
(17019.142903510474, 3.45242799797342e-9)
(13629.084752760904, 5.38352352507529e-9)
(-14345.365969127153, 4.85933382770265e-9)
(-19430.54357587707, 2.6486837018706e-9)
(15324.09397216062, 4.25843856872904e-9)
(-41467.58262217587, 5.81543977719108e-10)
(34818.09478011748, 8.24878518050218e-10)
(-29601.360491476687, 1.14123913560002e-9)
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
Increasing at the entire real axis