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$$- \frac{2 x \operatorname{acot}{\left(x \right)}}{\left(x^{2} + 12\right)^{2}} - \frac{1}{\left(x^{2} + 1\right) \left(x^{2} + 12\right)} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = 8965.34784579568$$
$$x_{2} = 7440.41162911171$$
$$x_{3} = -2842.00680980548$$
$$x_{4} = -8060.14976299884$$
$$x_{5} = 7876.0462386779$$
$$x_{6} = 2013.44968090002$$
$$x_{7} = 7222.61720395967$$
$$x_{8} = -7624.49165612708$$
$$x_{9} = -5447.35794262058$$
$$x_{10} = 2659.36286508512$$
$$x_{11} = 3091.90025757797$$
$$x_{12} = 9836.95649742935$$
$$x_{13} = -1980.25355880048$$
$$x_{14} = 5045.98181329941$$
$$x_{15} = -5229.80791014177$$
$$x_{16} = 3742.26936893857$$
$$x_{17} = 6569.34363131717$$
$$x_{18} = -4360.12657634809$$
$$x_{19} = -9585.29845337467$$
$$x_{20} = -4577.45433935364$$
$$x_{21} = -9367.39329693573$$
$$x_{22} = -6971.1175151447$$
$$x_{23} = 2228.22064612239$$
$$x_{24} = 8093.88396230433$$
$$x_{25} = -6753.36209877807$$
$$x_{26} = 2443.57156345037$$
$$x_{27} = -6317.91635442995$$
$$x_{28} = -5882.57343170759$$
$$x_{29} = 10490.7362956907$$
$$x_{30} = -8713.7273580381$$
$$x_{31} = -3925.71750319763$$
$$x_{32} = 7658.22174077254$$
$$x_{33} = -7842.31401126387$$
$$x_{34} = -2194.91880172545$$
$$x_{35} = 3308.52226956504$$
$$x_{36} = -4142.87666974265$$
$$x_{37} = -2410.19055521587$$
$$x_{38} = 1799.46311093739$$
$$x_{39} = 2875.49617961712$$
$$x_{40} = 2232.19615359684$$
$$x_{41} = 10054.8768711649$$
$$x_{42} = 6351.62860216233$$
$$x_{43} = -5012.30301905213$$
$$x_{44} = -1342.73562537559$$
$$x_{45} = -7188.89200183468$$
$$x_{46} = -9149.49600262774$$
$$x_{47} = 9619.04291324894$$
$$x_{48} = -4794.8493990003$$
$$x_{49} = 5916.2769517365$$
$$x_{50} = -9803.21094796885$$
$$x_{51} = -8277.99785459491$$
$$x_{52} = 4393.77640207287$$
$$x_{53} = -10892.8680863725$$
$$x_{54} = -10239.05607852$$
$$x_{55} = -3708.66495835059$$
$$x_{56} = 4176.51369520699$$
$$x_{57} = 6133.93878270854$$
$$x_{58} = 10708.6745866586$$
$$x_{59} = 5481.05057077953$$
$$x_{60} = -10674.9253243983$$
$$x_{61} = -2625.92110341184$$
$$x_{62} = 1586.57194298976$$
$$x_{63} = 7004.83992698629$$
$$x_{64} = 3959.33956225455$$
$$x_{65} = -3058.37290655895$$
$$x_{66} = -7406.68387842315$$
$$x_{67} = -8495.8573375382$$
$$x_{68} = -5664.94792508273$$
$$x_{69} = 4611.11519703223$$
$$x_{70} = -6535.62765662719$$
$$x_{71} = 10926.6181406867$$
$$x_{72} = 1375.27106210798$$
$$x_{73} = 8747.46661379663$$
$$x_{74} = 8529.5950365092$$
$$x_{75} = 4828.51983103122$$
$$x_{76} = -1766.41238180394$$
$$x_{77} = 9183.23804574845$$
$$x_{78} = 6787.08144708697$$
$$x_{79} = -10021.13030227$$
$$x_{80} = -3274.9641438687$$
$$x_{81} = -6100.23066632744$$
$$x_{82} = 5698.64631137486$$
$$x_{83} = -1553.72836418746$$
$$x_{84} = 5263.49405044165$$
$$x_{85} = 8311.73387260043$$
$$x_{86} = 3525.32223851161$$
$$x_{87} = 9401.13659048815$$
$$x_{88} = -8931.60714549778$$
$$x_{89} = -10456.9878754167$$
$$x_{90} = 10272.8036025903$$
$$x_{91} = -3491.7388369394$$
The values of the extrema at the points:
(8965.347845795677, 1.38770930450604e-12)
(7440.411629111706, 2.4277782404329e-12)
(-2842.0068098054826, -4.35636233207266e-11)
(-8060.149762998841, -1.9097239007911e-12)
(7876.046238677896, 2.04679874620175e-12)
(2013.449680900016, 1.22511353374969e-10)
(7222.617203959666, 2.65409279639476e-12)
(-7624.491656127081, -2.25614614056736e-12)
(-5447.357942620581, -6.18645814721903e-12)
(2659.362865085115, 5.31699733328918e-11)
(3091.9002575779677, 3.38316370774726e-11)
(9836.956497429353, 1.05055233478143e-12)
(-1980.2535588004805, -1.28776384194482e-10)
(5045.981813299411, 7.78328166042216e-12)
(-5229.807910141773, -6.99105294795593e-12)
(3742.2693689385724, 1.90807078936892e-11)
(6569.343631317167, 3.52723127704661e-12)
(-4360.1265763480915, -1.20643082912952e-11)
(-9585.298453374668, -1.13548923722249e-12)
(-4577.4543393536405, -1.04262391389471e-11)
(-9367.393296935734, -1.2165884094823e-12)
(-6971.117515144699, -2.95183907969577e-12)
(2228.2206461223905, 9.03908227166096e-11)
(8093.883962304328, 1.88594488478865e-12)
(-6753.36209877807, -3.24668215594303e-12)
(2443.571563450374, 6.85368145608751e-11)
(-6317.916354429945, -3.96532009064325e-12)
(-5882.573431707593, -4.91244582289644e-12)
(10490.736295690673, 8.66127923502696e-13)
(-8713.727358038097, -1.51142996342871e-12)
(-3925.717503197635, -1.65288454726794e-11)
(7658.221740772542, 2.22646615451123e-12)
(-7842.314011263869, -2.07332426134036e-12)
(-2194.9188017254487, -9.45678492708277e-11)
(3308.522269565041, 2.76119656242492e-11)
(-4142.876669742651, -1.40635082824767e-11)
(-2410.1905552158696, -7.14241264017831e-11)
(1799.463110937393, 1.71620633901507e-10)
(2875.496179617116, 4.20591960888666e-11)
(2232.196153596841, 8.99087290840881e-11)
(10054.87687116495, 9.83715867804223e-13)
(6351.628602162328, 3.90251499131334e-12)
(-5012.303019052133, -7.94123104204964e-12)
(-1342.7356253755852, -4.13071749373307e-10)
(-7188.892001834681, -2.69162168476282e-12)
(-9149.496002627737, -1.30559482742556e-12)
(9619.042913248937, 1.12358091924907e-12)
(-4794.849399000296, -9.07141120814539e-12)
(5916.276951736497, 4.82896835285547e-12)
(-9803.210947968846, -1.0614386568335e-12)
(-8277.997854594912, -1.76288504400633e-12)
(4393.77640207287, 1.17892415604732e-11)
(-10892.868086372458, -7.73701113524408e-13)
(-10239.056078519952, -9.31580059987144e-13)
(-3708.6649583505905, -1.96040954358131e-11)
(4176.5136952069915, 1.37264414968015e-11)
(6133.938782708542, 4.33292872918898e-12)
(10708.674586658624, 8.14315663845166e-13)
(5481.050570779533, 6.07307150260347e-12)
(-10674.925324398342, -8.22063595276857e-13)
(-2625.921103411845, -5.52273497614387e-11)
(1586.5719429897629, 2.50390912190865e-10)
(7004.839926986288, 2.90941212768906e-12)
(3959.339562254549, 1.61113306217276e-11)
(-3058.3729065589478, -3.4956513810457e-11)
(-7406.6838784231495, -2.461095549377e-12)
(-8495.8573375382, -1.63071585001502e-12)
(-5664.947925082734, -5.50062536519614e-12)
(4611.115197032229, 1.01995692694973e-11)
(-6535.627656627194, -3.58210217098777e-12)
(10926.61814068673, 7.66553829825206e-13)
(1375.2710621079846, 3.84443261947859e-10)
(8747.46661379663, 1.49400843410532e-12)
(8529.595036509203, 1.61144203807887e-12)
(4828.519831031217, 8.88296011286378e-12)
(-1766.4123818039427, -1.81435381616251e-10)
(9183.238045748449, 1.29125617100504e-12)
(6787.081447086972, 3.1985320076139e-12)
(-10021.13030227002, -9.93687482614462e-13)
(-3274.9641438687017, -2.84695008114753e-11)
(-6100.230666327439, -4.40515388835828e-12)
(5698.646311374858, 5.40361906866529e-12)
(-1553.728364187457, -2.66607583751994e-10)
(5263.494050441651, 6.85768287055537e-12)
(8311.733872600435, 1.74150623725827e-12)
(3525.322238511607, 2.28245964190028e-11)
(9401.136590488146, 1.2035353517068e-12)
(-8931.607145497775, -1.403495729115e-12)
(-10456.98787541667, -8.74540925179594e-13)
(10272.803602590348, 9.22429096145035e-13)
(-3491.7388369394016, -2.34895285625075e-11)
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
Decreasing at the entire real axis