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$$\frac{x}{x^{2} + 1} - \operatorname{acot}{\left(x \right)} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = -36383.6664480288$$
$$x_{2} = 30582.10639855$$
$$x_{3} = -16891.4245591221$$
$$x_{4} = 33972.2501557217$$
$$x_{5} = 19564.7930550097$$
$$x_{6} = 8551.45303022722$$
$$x_{7} = 42447.8004663952$$
$$x_{8} = 28887.0585662606$$
$$x_{9} = 39905.113426091$$
$$x_{10} = -32993.4859340089$$
$$x_{11} = 35667.3415095204$$
$$x_{12} = 40752.6740304247$$
$$x_{13} = -35536.1170902848$$
$$x_{14} = -34688.5704142599$$
$$x_{15} = -17738.7919818582$$
$$x_{16} = 31429.6368125843$$
$$x_{17} = -25365.810848258$$
$$x_{18} = 24649.5356458528$$
$$x_{19} = 16175.2730314267$$
$$x_{20} = -38078.7724924216$$
$$x_{21} = 39057.5547154358$$
$$x_{22} = -15196.7689456217$$
$$x_{23} = 34819.7944072015$$
$$x_{24} = -19433.5887869564$$
$$x_{25} = -39773.8872675524$$
$$x_{26} = -21975.9030580391$$
$$x_{27} = -10960.8778764425$$
$$x_{28} = -38926.3288599295$$
$$x_{29} = 20412.2196081348$$
$$x_{30} = 36514.8912641685$$
$$x_{31} = 26344.5255783821$$
$$x_{32} = 22954.5779890773$$
$$x_{33} = -22823.365825763$$
$$x_{34} = 14480.6704897809$$
$$x_{35} = 38209.9980244722$$
$$x_{36} = -27060.8125814987$$
$$x_{37} = -20281.012992025$$
$$x_{38} = 23802.0523520614$$
$$x_{39} = -14349.4901108771$$
$$x_{40} = -12655.0631680385$$
$$x_{41} = -31298.4148843908$$
$$x_{42} = -24518.3207010738$$
$$x_{43} = 11092.020896022$$
$$x_{44} = 25497.0269799384$$
$$x_{45} = -29603.3595724168$$
$$x_{46} = -37231.2183043527$$
$$x_{47} = 15327.9550113123$$
$$x_{48} = 11939.0892257274$$
$$x_{49} = -21128.451639271$$
$$x_{50} = 28039.5419336107$$
$$x_{51} = -30450.8850969459$$
$$x_{52} = 17869.99050832$$
$$x_{53} = 12786.2285767811$$
$$x_{54} = -27908.3228655288$$
$$x_{55} = 33124.7089739066$$
$$x_{56} = 37362.4434906472$$
$$x_{57} = -23670.8387235676$$
$$x_{58} = -9267.06673722176$$
$$x_{59} = -28755.8386877142$$
$$x_{60} = -8420.37201352502$$
$$x_{61} = 41600.2364126928$$
$$x_{62} = 27192.0307619732$$
$$x_{63} = -42316.5735052903$$
$$x_{64} = -18586.1809993501$$
$$x_{65} = -26213.3083727677$$
$$x_{66} = -13502.2521178288$$
$$x_{67} = 13633.4257117535$$
$$x_{68} = -33841.0266214391$$
$$x_{69} = -10113.913790961$$
$$x_{70} = -40621.447587593$$
$$x_{71} = -16044.0821523617$$
$$x_{72} = 32277.1711035644$$
$$x_{73} = 17022.6195482377$$
$$x_{74} = 18717.3825921039$$
$$x_{75} = -41469.0097027937$$
$$x_{76} = 22107.1135839482$$
$$x_{77} = 21259.6603274632$$
$$x_{78} = -11807.9338157736$$
$$x_{79} = 29734.5801929714$$
$$x_{80} = -32145.9485975863$$
$$x_{81} = 10245.0412006996$$
$$x_{82} = 9398.17409812154$$
The values of the extrema at the points:
(-36383.66644802879, 1.00000000025181)
(30582.106398549957, 1.00000000035641)
(-16891.424559122082, 1.00000000116828)
(33972.25015572168, 1.00000000028882)
(19564.79305500968, 1.00000000087082)
(8551.453030227216, 1.00000000455826)
(42447.8004663952, 1.000000000185)
(28887.058566260628, 1.00000000039946)
(39905.113426091, 1.00000000020933)
(-32993.48593400886, 1.00000000030621)
(35667.3415095204, 1.00000000026202)
(40752.67403042468, 1.00000000020071)
(-35536.11709028476, 1.00000000026396)
(-34688.570414259906, 1.00000000027702)
(-17738.791981858223, 1.00000000105933)
(31429.636812584253, 1.00000000033744)
(-25365.810848258036, 1.00000000051806)
(24649.53564585282, 1.00000000054861)
(16175.273031426721, 1.00000000127402)
(-38078.77249242158, 1.00000000022989)
(39057.55471543576, 1.00000000021851)
(-15196.768945621745, 1.00000000144336)
(34819.79440720148, 1.00000000027493)
(-19433.58878695641, 1.00000000088262)
(-39773.887267552374, 1.00000000021071)
(-21975.903058039054, 1.00000000069022)
(-10960.87787644252, 1.00000000277452)
(-38926.32885992955, 1.00000000021998)
(20412.219608134823, 1.00000000080002)
(36514.89126416845, 1.00000000025)
(26344.525578382065, 1.00000000048028)
(22954.57798907731, 1.00000000063262)
(-22823.365825763038, 1.00000000063991)
(14480.670489780925, 1.00000000158965)
(38209.99802447215, 1.00000000022831)
(-27060.81258149867, 1.00000000045519)
(-20281.012992024975, 1.0000000008104)
(23802.052352061375, 1.00000000058837)
(-14349.490110877094, 1.00000000161885)
(-12655.063168038458, 1.00000000208137)
(-31298.41488439077, 1.00000000034028)
(-24518.320701073833, 1.00000000055449)
(11092.020896021957, 1.0000000027093)
(25497.02697993837, 1.00000000051274)
(-29603.35957241684, 1.00000000038036)
(-37231.21830435275, 1.00000000024047)
(15327.955011312279, 1.00000000141876)
(11939.089225727435, 1.00000000233849)
(-21128.451639271025, 1.0000000007467)
(28039.541933610733, 1.00000000042397)
(-30450.88509694595, 1.00000000035948)
(17869.990508320032, 1.00000000104383)
(12786.228576781117, 1.00000000203889)
(-27908.322865528775, 1.00000000042797)
(33124.7089739066, 1.00000000030379)
(37362.44349064715, 1.00000000023879)
(-23670.838723567595, 1.00000000059491)
(-9267.06673722176, 1.00000000388145)
(-28755.838687714244, 1.00000000040311)
(-8420.37201352502, 1.00000000470128)
(41600.23641269277, 1.00000000019261)
(27192.03076197315, 1.00000000045081)
(-42316.573505290275, 1.00000000018615)
(-18586.180999350086, 1.00000000096494)
(-26213.308372767653, 1.0000000004851)
(-13502.25211782884, 1.00000000182838)
(13633.425711753476, 1.00000000179337)
(-33841.02662143908, 1.00000000029107)
(-10113.913790961042, 1.00000000325867)
(-40621.447587593044, 1.00000000020201)
(-16044.082152361707, 1.00000000129494)
(32277.171103564393, 1.00000000031995)
(17022.619548237748, 1.00000000115034)
(18717.38259210387, 1.00000000095146)
(-41469.00970279369, 1.00000000019383)
(22107.113583948245, 1.00000000068205)
(21259.66032746322, 1.00000000073751)
(-11807.933815773578, 1.00000000239073)
(29734.580192971367, 1.00000000037701)
(-32145.948597586266, 1.00000000032257)
(10245.041200699563, 1.00000000317579)
(9398.174098121537, 1.00000000377391)
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:
$$x_{1} = 39057.5547154358$$
$$x_{2} = 34819.7944072015$$
$$x_{3} = -37231.2183043527$$
$$x_{4} = 11939.0892257274$$
$$x_{5} = 37362.4434906472$$
$$x_{6} = 18717.3825921039$$
$$x_{7} = 22107.1135839482$$
Maxima of the function at points:
$$x_{7} = -32993.4859340089$$
$$x_{7} = 31429.6368125843$$
$$x_{7} = -38078.7724924216$$
$$x_{7} = 14480.6704897809$$
$$x_{7} = -14349.4901108771$$
$$x_{7} = 25497.0269799384$$
$$x_{7} = -42316.5735052903$$
$$x_{7} = -26213.3083727677$$
Decreasing at intervals
$$\left[39057.5547154358, \infty\right)$$
Increasing at intervals
$$\left(-\infty, -37231.2183043527\right]$$