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$$\operatorname{acot}{\left(x \right)} - \frac{x}{x^{2} + 1} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = -40621.447587593$$
$$x_{2} = -31298.4148843908$$
$$x_{3} = -34688.5704142599$$
$$x_{4} = -30450.8850969459$$
$$x_{5} = -38926.3288599295$$
$$x_{6} = 16175.2730314267$$
$$x_{7} = 30582.10639855$$
$$x_{8} = 33972.2501557217$$
$$x_{9} = 17022.6195482377$$
$$x_{10} = -21128.451639271$$
$$x_{11} = 39905.113426091$$
$$x_{12} = -27908.3228655288$$
$$x_{13} = -19433.5887869564$$
$$x_{14} = 33124.7089739066$$
$$x_{15} = -16891.4245591221$$
$$x_{16} = 22107.1135839482$$
$$x_{17} = -13502.2521178288$$
$$x_{18} = 17869.99050832$$
$$x_{19} = -33841.0266214391$$
$$x_{20} = 36514.8912641685$$
$$x_{21} = 9398.17409812154$$
$$x_{22} = 29734.5801929714$$
$$x_{23} = -32145.9485975863$$
$$x_{24} = 8551.45303022722$$
$$x_{25} = -17738.7919818582$$
$$x_{26} = -29603.3595724168$$
$$x_{27} = -11807.9338157736$$
$$x_{28} = -35536.1170902848$$
$$x_{29} = -32993.4859340089$$
$$x_{30} = 38209.9980244722$$
$$x_{31} = -42316.5735052903$$
$$x_{32} = 19564.7930550097$$
$$x_{33} = -22823.365825763$$
$$x_{34} = 40752.6740304247$$
$$x_{35} = -20281.012992025$$
$$x_{36} = -37231.2183043527$$
$$x_{37} = -10960.8778764425$$
$$x_{38} = -18586.1809993501$$
$$x_{39} = 14480.6704897809$$
$$x_{40} = 11939.0892257274$$
$$x_{41} = -24518.3207010738$$
$$x_{42} = 15327.9550113123$$
$$x_{43} = 10245.0412006996$$
$$x_{44} = 26344.5255783821$$
$$x_{45} = -16044.0821523617$$
$$x_{46} = -14349.4901108771$$
$$x_{47} = 37362.4434906472$$
$$x_{48} = -26213.3083727677$$
$$x_{49} = -25365.810848258$$
$$x_{50} = 27192.0307619732$$
$$x_{51} = 18717.3825921039$$
$$x_{52} = -15196.7689456217$$
$$x_{53} = 28039.5419336107$$
$$x_{54} = -21975.9030580391$$
$$x_{55} = 24649.5356458528$$
$$x_{56} = -39773.8872675524$$
$$x_{57} = 21259.6603274632$$
$$x_{58} = -23670.8387235676$$
$$x_{59} = 42447.8004663952$$
$$x_{60} = 34819.7944072015$$
$$x_{61} = -41469.0097027937$$
$$x_{62} = -36383.6664480288$$
$$x_{63} = 41600.2364126928$$
$$x_{64} = 20412.2196081348$$
$$x_{65} = 35667.3415095204$$
$$x_{66} = -28755.8386877142$$
$$x_{67} = -9267.06673722176$$
$$x_{68} = -38078.7724924216$$
$$x_{69} = 32277.1711035644$$
$$x_{70} = -8420.37201352502$$
$$x_{71} = 11092.020896022$$
$$x_{72} = 22954.5779890773$$
$$x_{73} = 25497.0269799384$$
$$x_{74} = -12655.0631680385$$
$$x_{75} = 23802.0523520614$$
$$x_{76} = 39057.5547154358$$
$$x_{77} = 28887.0585662606$$
$$x_{78} = 12786.2285767811$$
$$x_{79} = -27060.8125814987$$
$$x_{80} = 31429.6368125843$$
$$x_{81} = 13633.4257117535$$
$$x_{82} = -10113.913790961$$
The values of the extrema at the points:
(-40621.447587593, 0.999999999797992)
(-31298.4148843908, 0.999999999659722)
(-34688.5704142599, 0.999999999722983)
(-30450.8850969459, 0.999999999640517)
(-38926.3288599295, 0.999999999780016)
(16175.2730314267, 0.999999998725982)
(30582.10639855, 0.999999999643595)
(33972.2501557217, 0.999999999711178)
(17022.6195482377, 0.999999998849661)
(-21128.451639271, 0.999999999253305)
(39905.113426091, 0.999999999790675)
(-27908.3228655288, 0.999999999572032)
(-19433.5887869564, 0.999999999117382)
(33124.7089739066, 0.999999999696209)
(-16891.4245591221, 0.999999998831722)
(22107.1135839482, 0.999999999317952)
(-13502.2521178288, 0.999999998171621)
(17869.99050832, 0.999999998956169)
(-33841.0266214391, 0.999999999708934)
(36514.8912641685, 0.999999999750001)
(9398.17409812154, 0.999999996226088)
(29734.5801929714, 0.999999999622988)
(-32145.9485975863, 0.999999999677428)
(8551.45303022722, 0.999999995441742)
(-17738.7919818582, 0.999999998940671)
(-29603.3595724168, 0.999999999619638)
(-11807.9338157736, 0.999999997609268)
(-35536.1170902848, 0.99999999973604)
(-32993.4859340089, 0.999999999693788)
(38209.9980244722, 0.99999999977169)
(-42316.5735052903, 0.999999999813852)
(19564.7930550097, 0.99999999912918)
(-22823.365825763, 0.999999999360089)
(40752.6740304247, 0.999999999799291)
(-20281.012992025, 0.9999999991896)
(-37231.2183043527, 0.999999999759528)
(-10960.8778764425, 0.999999997225479)
(-18586.1809993501, 0.999999999035064)
(14480.6704897809, 0.99999999841035)
(11939.0892257274, 0.999999997661506)
(-24518.3207010738, 0.999999999445505)
(15327.9550113123, 0.999999998581235)
(10245.0412006996, 0.999999996824213)
(26344.5255783821, 0.999999999519716)
(-16044.0821523617, 0.999999998705062)
(-14349.4901108771, 0.999999998381153)
(37362.4434906472, 0.999999999761214)
(-26213.3083727677, 0.999999999514896)
(-25365.810848258, 0.999999999481939)
(27192.0307619732, 0.999999999549188)
(18717.3825921039, 0.999999999048545)
(-15196.7689456217, 0.999999998556635)
(28039.5419336107, 0.999999999576028)
(-21975.9030580391, 0.999999999309784)
(24649.5356458528, 0.999999999451393)
(-39773.8872675524, 0.999999999789291)
(21259.6603274632, 0.999999999262493)
(-23670.8387235676, 0.99999999940509)
(42447.8004663952, 0.999999999815001)
(34819.7944072015, 0.999999999725067)
(-41469.0097027937, 0.999999999806165)
(-36383.6664480288, 0.999999999748194)
(41600.2364126928, 0.999999999807386)
(20412.2196081348, 0.999999999199985)
(35667.3415095204, 0.999999999737978)
(-28755.8386877142, 0.999999999596887)
(-9267.06673722176, 0.999999996118549)
(-38078.7724924216, 0.999999999770114)
(32277.1711035644, 0.999999999680046)
(-8420.37201352502, 0.999999995298719)
(11092.020896022, 0.999999997290698)
(22954.5779890773, 0.999999999367384)
(25497.0269799384, 0.999999999487257)
(-12655.0631680385, 0.999999997918626)
(23802.0523520614, 0.999999999411631)
(39057.5547154358, 0.999999999781491)
(28887.0585662606, 0.999999999600541)
(12786.2285767811, 0.99999999796111)
(-27060.8125814987, 0.999999999544805)
(31429.6368125843, 0.999999999662557)
(13633.4257117535, 0.999999998206635)
(-10113.913790961, 0.999999996741331)
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:
$$x_{1} = -40621.447587593$$
$$x_{2} = 17022.6195482377$$
$$x_{3} = -32993.4859340089$$
$$x_{4} = -42316.5735052903$$
$$x_{5} = -22823.365825763$$
$$x_{6} = 40752.6740304247$$
$$x_{7} = 14480.6704897809$$
$$x_{8} = -14349.4901108771$$
$$x_{9} = -26213.3083727677$$
$$x_{10} = 24649.5356458528$$
$$x_{11} = -38078.7724924216$$
$$x_{12} = 32277.1711035644$$
$$x_{13} = 25497.0269799384$$
$$x_{14} = 31429.6368125843$$
Maxima of the function at points:
$$x_{14} = -16891.4245591221$$
$$x_{14} = 22107.1135839482$$
$$x_{14} = -32145.9485975863$$
$$x_{14} = 38209.9980244722$$
$$x_{14} = -37231.2183043527$$
$$x_{14} = 11939.0892257274$$
$$x_{14} = 10245.0412006996$$
$$x_{14} = -16044.0821523617$$
$$x_{14} = 18717.3825921039$$
$$x_{14} = 34819.7944072015$$
$$x_{14} = 39057.5547154358$$
Decreasing at intervals
$$\left[40752.6740304247, \infty\right)$$
Increasing at intervals
$$\left(-\infty, -42316.5735052903\right]$$