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$$\frac{\cos{\left(x \right)}}{x} - \frac{\sin{\left(x \right)}}{x^{2}} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = -10.9041216594289$$
$$x_{2} = -48.6741442319544$$
$$x_{3} = 45.5311340139913$$
$$x_{4} = -26.6660542588127$$
$$x_{5} = -73.8138806006806$$
$$x_{6} = -17.2207552719308$$
$$x_{7} = -64.3871195905574$$
$$x_{8} = 14.0661939128315$$
$$x_{9} = 39.2444323611642$$
$$x_{10} = 76.9560263103312$$
$$x_{11} = 23.519452498689$$
$$x_{12} = -23.519452498689$$
$$x_{13} = 51.8169824872797$$
$$x_{14} = -394.267341680887$$
$$x_{15} = 58.1022547544956$$
$$x_{16} = 54.9596782878889$$
$$x_{17} = 83.2401924707234$$
$$x_{18} = 36.1006222443756$$
$$x_{19} = -67.5294347771441$$
$$x_{20} = 86.3822220347287$$
$$x_{21} = -70.6716857116195$$
$$x_{22} = -76.9560263103312$$
$$x_{23} = -32.9563890398225$$
$$x_{24} = 26.6660542588127$$
$$x_{25} = -4.49340945790906$$
$$x_{26} = 42.3879135681319$$
$$x_{27} = -95.8081387868617$$
$$x_{28} = -51.8169824872797$$
$$x_{29} = 67.5294347771441$$
$$x_{30} = -20.3713029592876$$
$$x_{31} = -92.6661922776228$$
$$x_{32} = -86.3822220347287$$
$$x_{33} = -89.5242209304172$$
$$x_{34} = 10.9041216594289$$
$$x_{35} = -98.9500628243319$$
$$x_{36} = 17.2207552719308$$
$$x_{37} = -4355.81798462425$$
$$x_{38} = 32.9563890398225$$
$$x_{39} = -45.5311340139913$$
$$x_{40} = -54.9596782878889$$
$$x_{41} = -80.0981286289451$$
$$x_{42} = 80.0981286289451$$
$$x_{43} = -36.1006222443756$$
$$x_{44} = -14.0661939128315$$
$$x_{45} = -83.2401924707234$$
$$x_{46} = 95.8081387868617$$
$$x_{47} = 73.8138806006806$$
$$x_{48} = -39.2444323611642$$
$$x_{49} = -58.1022547544956$$
$$x_{50} = -42.3879135681319$$
$$x_{51} = -61.2447302603744$$
$$x_{52} = 108.375719651675$$
$$x_{53} = 92.6661922776228$$
$$x_{54} = 4.49340945790906$$
$$x_{55} = 61.2447302603744$$
$$x_{56} = 48.6741442319544$$
$$x_{57} = 98.9500628243319$$
$$x_{58} = -29.811598790893$$
$$x_{59} = 20.3713029592876$$
$$x_{60} = 89.5242209304172$$
$$x_{61} = 29.811598790893$$
$$x_{62} = 70.6716857116195$$
$$x_{63} = -7.72525183693771$$
$$x_{64} = 7.72525183693771$$
$$x_{65} = 64.3871195905574$$
The values of the extrema at the points:
(-10.904121659428899, -0.0913252028230577)
(-48.674144231954386, -0.0205404540417537)
(45.53113401399128, 0.0219576982284824)
(-26.666054258812675, 0.0374745199939312)
(-73.81388060068065, -0.01354634434514)
(-17.22075527193077, -0.0579718023461539)
(-64.38711959055742, 0.0155291838074613)
(14.066193912831473, 0.0709134594504622)
(39.24443236116419, 0.0254730530928808)
(76.95602631033118, 0.0129933369870427)
(23.519452498689006, -0.0424796169776126)
(-23.519452498689006, -0.0424796169776126)
(51.81698248727967, 0.019295099487588)
(-394.26734168088706, -0.00253634191261283)
(58.10225475449559, 0.0172084874716279)
(54.959678287888934, -0.0181921463218031)
(83.2401924707234, 0.0120125604820527)
(36.10062224437561, -0.0276897323011492)
(-67.52943477714412, -0.0148067339465492)
(86.38222203472871, -0.0115756804584678)
(-70.6716857116195, 0.0141485220648664)
(-76.95602631033118, 0.0129933369870427)
(-32.956389039822476, 0.0303291711863103)
(26.666054258812675, 0.0374745199939312)
(-4.493409457909064, -0.217233628211222)
(42.38791356813192, -0.0235850682290164)
(-95.8081387868617, 0.0104369581345658)
(-51.81698248727967, 0.019295099487588)
(67.52943477714412, -0.0148067339465492)
(-20.37130295928756, 0.0490296240140742)
(-92.66619227762284, -0.0107907938495342)
(-86.38222203472871, -0.0115756804584678)
(-89.52422093041719, 0.0111694646341736)
(10.904121659428899, -0.0913252028230577)
(-98.95006282433188, -0.010105591736504)
(17.22075527193077, -0.0579718023461539)
(-4355.817984624248, 0.000229577998248987)
(32.956389039822476, 0.0303291711863103)
(-45.53113401399128, 0.0219576982284824)
(-54.959678287888934, -0.0181921463218031)
(-80.09812862894512, -0.012483713321779)
(80.09812862894512, -0.012483713321779)
(-36.10062224437561, -0.0276897323011492)
(-14.066193912831473, 0.0709134594504622)
(-83.2401924707234, 0.0120125604820527)
(95.8081387868617, 0.0104369581345658)
(73.81388060068065, -0.01354634434514)
(-39.24443236116419, 0.0254730530928808)
(-58.10225475449559, 0.0172084874716279)
(-42.38791356813192, -0.0235850682290164)
(-61.2447302603744, -0.0163257593209978)
(108.37571965167469, 0.00922676625078197)
(92.66619227762284, -0.0107907938495342)
(4.493409457909064, -0.217233628211222)
(61.2447302603744, -0.0163257593209978)
(48.674144231954386, -0.0205404540417537)
(98.95006282433188, -0.010105591736504)
(-29.81159879089296, -0.0335251350213988)
(20.37130295928756, 0.0490296240140742)
(89.52422093041719, 0.0111694646341736)
(29.81159879089296, -0.0335251350213988)
(70.6716857116195, 0.0141485220648664)
(-7.725251836937707, 0.128374553525899)
(7.725251836937707, 0.128374553525899)
(64.38711959055742, 0.0155291838074613)
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:
$$x_{1} = -10.9041216594289$$
$$x_{2} = -48.6741442319544$$
$$x_{3} = -73.8138806006806$$
$$x_{4} = -17.2207552719308$$
$$x_{5} = 23.519452498689$$
$$x_{6} = -23.519452498689$$
$$x_{7} = -394.267341680887$$
$$x_{8} = 54.9596782878889$$
$$x_{9} = 36.1006222443756$$
$$x_{10} = -67.5294347771441$$
$$x_{11} = 86.3822220347287$$
$$x_{12} = -4.49340945790906$$
$$x_{13} = 42.3879135681319$$
$$x_{14} = 67.5294347771441$$
$$x_{15} = -92.6661922776228$$
$$x_{16} = -86.3822220347287$$
$$x_{17} = 10.9041216594289$$
$$x_{18} = -98.9500628243319$$
$$x_{19} = 17.2207552719308$$
$$x_{20} = -54.9596782878889$$
$$x_{21} = -80.0981286289451$$
$$x_{22} = 80.0981286289451$$
$$x_{23} = -36.1006222443756$$
$$x_{24} = 73.8138806006806$$
$$x_{25} = -42.3879135681319$$
$$x_{26} = -61.2447302603744$$
$$x_{27} = 92.6661922776228$$
$$x_{28} = 4.49340945790906$$
$$x_{29} = 61.2447302603744$$
$$x_{30} = 48.6741442319544$$
$$x_{31} = 98.9500628243319$$
$$x_{32} = -29.811598790893$$
$$x_{33} = 29.811598790893$$
Maxima of the function at points:
$$x_{33} = 45.5311340139913$$
$$x_{33} = -26.6660542588127$$
$$x_{33} = -64.3871195905574$$
$$x_{33} = 14.0661939128315$$
$$x_{33} = 39.2444323611642$$
$$x_{33} = 76.9560263103312$$
$$x_{33} = 51.8169824872797$$
$$x_{33} = 58.1022547544956$$
$$x_{33} = 83.2401924707234$$
$$x_{33} = -70.6716857116195$$
$$x_{33} = -76.9560263103312$$
$$x_{33} = -32.9563890398225$$
$$x_{33} = 26.6660542588127$$
$$x_{33} = -95.8081387868617$$
$$x_{33} = -51.8169824872797$$
$$x_{33} = -20.3713029592876$$
$$x_{33} = -89.5242209304172$$
$$x_{33} = -4355.81798462425$$
$$x_{33} = 32.9563890398225$$
$$x_{33} = -45.5311340139913$$
$$x_{33} = -14.0661939128315$$
$$x_{33} = -83.2401924707234$$
$$x_{33} = 95.8081387868617$$
$$x_{33} = -39.2444323611642$$
$$x_{33} = -58.1022547544956$$
$$x_{33} = 108.375719651675$$
$$x_{33} = 20.3713029592876$$
$$x_{33} = 89.5242209304172$$
$$x_{33} = 70.6716857116195$$
$$x_{33} = -7.72525183693771$$
$$x_{33} = 7.72525183693771$$
$$x_{33} = 64.3871195905574$$
Decreasing at intervals
$$\left[98.9500628243319, \infty\right)$$
Increasing at intervals
$$\left(-\infty, -394.267341680887\right]$$