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$$\cos{\left(\frac{1}{x} \right)} \cos{\left(x \right)} + \frac{\sin{\left(\frac{1}{x} \right)} \sin{\left(x \right)}}{x^{2}} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = -14.1375214322216$$
$$x_{2} = -98.9601696199687$$
$$x_{3} = -42.4115139342614$$
$$x_{4} = 70.6858375373694$$
$$x_{5} = -54.9778774562622$$
$$x_{6} = -70.6858375373694$$
$$x_{7} = -95.818577071243$$
$$x_{8} = 48.6946947926006$$
$$x_{9} = -1.77032184622602$$
$$x_{10} = 54.9778774562622$$
$$x_{11} = 51.8362859646913$$
$$x_{12} = 26.7035900960034$$
$$x_{13} = -89.5353920205745$$
$$x_{14} = 73.8274298446286$$
$$x_{15} = -45.5531040577872$$
$$x_{16} = -92.6769845372215$$
$$x_{17} = 92.6769845372215$$
$$x_{18} = -10.9963284351197$$
$$x_{19} = -29.8451678396463$$
$$x_{20} = -83.2522070532694$$
$$x_{21} = 202.632726276733$$
$$x_{22} = -17.278953653359$$
$$x_{23} = 45.5531040577872$$
$$x_{24} = 95.818577071243$$
$$x_{25} = -67.5442452975689$$
$$x_{26} = -36.1283367275698$$
$$x_{27} = 23.5620213951938$$
$$x_{28} = 29.8451678396463$$
$$x_{29} = 42.4115139342614$$
$$x_{30} = 67.5442452975689$$
$$x_{31} = 1.77032184622602$$
$$x_{32} = 32.986750731248$$
$$x_{33} = -23.5620213951938$$
$$x_{34} = 39.2699246862074$$
$$x_{35} = 20.4204697787209$$
$$x_{36} = 80.1106146116865$$
$$x_{37} = 36.1283367275698$$
$$x_{38} = 98.9601696199687$$
$$x_{39} = 7.85605530705307$$
$$x_{40} = 64.4026531424855$$
$$x_{41} = 86.393799524576$$
$$x_{42} = -32.986750731248$$
$$x_{43} = -64.4026531424855$$
$$x_{44} = 10.9963284351197$$
$$x_{45} = 17.278953653359$$
$$x_{46} = -20.4204697787209$$
$$x_{47} = 14.1375214322216$$
$$x_{48} = 1983.91576074208$$
$$x_{49} = -86.393799524576$$
$$x_{50} = -39.2699246862074$$
$$x_{51} = -51.8362859646913$$
$$x_{52} = -48.6946947926006$$
$$x_{53} = 76.9690222061413$$
$$x_{54} = -61.2610610949586$$
$$x_{55} = 4.72203085083256$$
$$x_{56} = 58.1194691856341$$
$$x_{57} = -58.1194691856341$$
$$x_{58} = -4.72203085083256$$
$$x_{59} = 83.2522070532694$$
$$x_{60} = -80.1106146116865$$
$$x_{61} = -7.85605530705307$$
$$x_{62} = -26.7035900960034$$
$$x_{63} = 61.2610610949586$$
$$x_{64} = 89.5353920205745$$
$$x_{65} = -73.8274298446286$$
$$x_{66} = -76.9690222061413$$
The values of the extrema at the points:
(-14.13752143222161, -0.997499348016665)
(-98.9601696199687, 0.999948944157013)
(-42.41151393426139, 0.999722039894878)
(70.68583753736935, 0.999899931368176)
(-54.97787745626216, 0.999834582238175)
(-70.68583753736935, -0.999899931368176)
(-95.81857707124303, -0.999945541379844)
(48.69469479260059, -0.99978914130114)
(-1.7703218462260202, -0.827901301432779)
(54.97787745626216, -0.999834582238175)
(51.83628596469129, 0.999813924651387)
(26.703590096003428, 0.999298898656899)
(-89.53539202057446, -0.999937629960711)
(73.82742984462863, -0.999908266517767)
(-45.55310405778716, -0.999759055668925)
(-92.6769845372215, 0.999941786728398)
(92.6769845372215, -0.999941786728398)
(-10.996328435119693, 0.995867574424807)
(-29.845167839646347, 0.999438717024592)
(-83.25220705326942, -0.999927860474679)
(202.63272627673345, 0.99998782272965)
(-17.278953653359007, 0.998325755891758)
(45.55310405778716, 0.999759055668925)
(95.81857707124303, 0.999945541379844)
(-67.5442452975689, 0.999890406351476)
(-36.128336727569845, 0.999616957824081)
(23.56202139519381, -0.99909950536129)
(29.845167839646347, -0.999438717024592)
(42.41151393426139, -0.999722039894878)
(67.5442452975689, -0.999890406351476)
(1.7703218462260202, 0.827901301432779)
(32.98675073124796, 0.999540529076331)
(-23.56202139519381, 0.99909950536129)
(39.269924686207396, 0.999675789868926)
(20.42046977872091, 0.998801179244391)
(80.11061461168646, -0.999922091606656)
(36.128336727569845, -0.999616957824081)
(98.9601696199687, -0.999948944157013)
(7.856055307053065, 0.991907384132263)
(64.40265314248548, 0.999879453727449)
(86.39379952457602, -0.999933011536799)
(-32.98675073124796, -0.999540529076331)
(-64.40265314248548, -0.999879453727449)
(10.996328435119693, -0.995867574424807)
(17.278953653359007, -0.998325755891758)
(-20.42046977872091, -0.998801179244391)
(14.13752143222161, 0.997499348016665)
(1983.9157607420825, -0.999999872964957)
(-86.39379952457602, 0.999933011536799)
(-39.269924686207396, -0.999675789868926)
(-51.83628596469129, -0.999813924651387)
(-48.69469479260059, 0.99978914130114)
(76.9690222061413, 0.999915602036213)
(-61.26106109495865, 0.99986677327201)
(4.722030850832559, -0.977614273674147)
(58.11946918563406, 0.999851981482606)
(-58.11946918563406, -0.999851981482606)
(-4.722030850832559, 0.977614273674147)
(83.25220705326942, 0.999927860474679)
(-80.11061461168646, 0.999922091606656)
(-7.856055307053065, -0.991907384132263)
(-26.703590096003428, -0.999298898656899)
(61.26106109495865, -0.99986677327201)
(89.53539202057446, 0.999937629960711)
(-73.82742984462863, 0.999908266517767)
(-76.9690222061413, -0.999915602036213)
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:
$$x_{1} = -14.1375214322216$$
$$x_{2} = -70.6858375373694$$
$$x_{3} = -95.818577071243$$
$$x_{4} = 48.6946947926006$$
$$x_{5} = -1.77032184622602$$
$$x_{6} = 54.9778774562622$$
$$x_{7} = -89.5353920205745$$
$$x_{8} = 73.8274298446286$$
$$x_{9} = -45.5531040577872$$
$$x_{10} = 92.6769845372215$$
$$x_{11} = -83.2522070532694$$
$$x_{12} = 23.5620213951938$$
$$x_{13} = 29.8451678396463$$
$$x_{14} = 42.4115139342614$$
$$x_{15} = 67.5442452975689$$
$$x_{16} = 80.1106146116865$$
$$x_{17} = 36.1283367275698$$
$$x_{18} = 98.9601696199687$$
$$x_{19} = 86.393799524576$$
$$x_{20} = -32.986750731248$$
$$x_{21} = -64.4026531424855$$
$$x_{22} = 10.9963284351197$$
$$x_{23} = 17.278953653359$$
$$x_{24} = -20.4204697787209$$
$$x_{25} = 1983.91576074208$$
$$x_{26} = -39.2699246862074$$
$$x_{27} = -51.8362859646913$$
$$x_{28} = 4.72203085083256$$
$$x_{29} = -58.1194691856341$$
$$x_{30} = -7.85605530705307$$
$$x_{31} = -26.7035900960034$$
$$x_{32} = 61.2610610949586$$
$$x_{33} = -76.9690222061413$$
Maxima of the function at points:
$$x_{33} = -98.9601696199687$$
$$x_{33} = -42.4115139342614$$
$$x_{33} = 70.6858375373694$$
$$x_{33} = -54.9778774562622$$
$$x_{33} = 51.8362859646913$$
$$x_{33} = 26.7035900960034$$
$$x_{33} = -92.6769845372215$$
$$x_{33} = -10.9963284351197$$
$$x_{33} = -29.8451678396463$$
$$x_{33} = 202.632726276733$$
$$x_{33} = -17.278953653359$$
$$x_{33} = 45.5531040577872$$
$$x_{33} = 95.818577071243$$
$$x_{33} = -67.5442452975689$$
$$x_{33} = -36.1283367275698$$
$$x_{33} = 1.77032184622602$$
$$x_{33} = 32.986750731248$$
$$x_{33} = -23.5620213951938$$
$$x_{33} = 39.2699246862074$$
$$x_{33} = 20.4204697787209$$
$$x_{33} = 7.85605530705307$$
$$x_{33} = 64.4026531424855$$
$$x_{33} = 14.1375214322216$$
$$x_{33} = -86.393799524576$$
$$x_{33} = -48.6946947926006$$
$$x_{33} = 76.9690222061413$$
$$x_{33} = -61.2610610949586$$
$$x_{33} = 58.1194691856341$$
$$x_{33} = -4.72203085083256$$
$$x_{33} = 83.2522070532694$$
$$x_{33} = -80.1106146116865$$
$$x_{33} = 89.5353920205745$$
$$x_{33} = -73.8274298446286$$
Decreasing at intervals
$$\left[1983.91576074208, \infty\right)$$
Increasing at intervals
$$\left(-\infty, -95.818577071243\right]$$