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$$- \frac{\operatorname{acot}{\left(x \right)}}{\left(x - 3\right)^{2}} - \frac{1}{\left(x - 3\right) \left(x^{2} + 1\right)} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = -31921.8835434698$$
$$x_{2} = 24848.9527856772$$
$$x_{3} = 22304.9178115051$$
$$x_{4} = 35871.0864641394$$
$$x_{5} = 38414.3867016121$$
$$x_{6} = -32769.6940726206$$
$$x_{7} = -25986.8382213314$$
$$x_{8} = 37566.6273450369$$
$$x_{9} = -14960.8860813988$$
$$x_{10} = -37008.6063243624$$
$$x_{11} = 15518.8336328721$$
$$x_{12} = -24290.9504985193$$
$$x_{13} = -12414.7034718108$$
$$x_{14} = -18354.3754214744$$
$$x_{15} = 14670.2260130038$$
$$x_{16} = -29378.3813978389$$
$$x_{17} = -16657.783408065$$
$$x_{18} = -14112.2895735892$$
$$x_{19} = -30226.22818334$$
$$x_{20} = -38704.1170790913$$
$$x_{21} = -27682.6438443875$$
$$x_{22} = -35313.0668342955$$
$$x_{23} = -26834.7503365653$$
$$x_{24} = -40399.6026837339$$
$$x_{25} = 41805.3607928303$$
$$x_{26} = -28530.5203814017$$
$$x_{27} = 33327.7111964611$$
$$x_{28} = -37856.3650546609$$
$$x_{29} = -36160.8404213184$$
$$x_{30} = 1.26241821842168$$
$$x_{31} = -20050.7379682616$$
$$x_{32} = 24000.9702280612$$
$$x_{33} = 34175.5123312146$$
$$x_{34} = 25696.9101211369$$
$$x_{35} = 19760.5640406041$$
$$x_{36} = 16367.3357554312$$
$$x_{37} = 30784.242139886$$
$$x_{38} = -20898.8503044486$$
$$x_{39} = 20608.7247596784$$
$$x_{40} = -17506.1121672217$$
$$x_{41} = 39262.1393115611$$
$$x_{42} = 17215.7481743741$$
$$x_{43} = 23152.9596471974$$
$$x_{44} = -23442.9703742917$$
$$x_{45} = 40957.6259784778$$
$$x_{46} = -19202.5816602195$$
$$x_{47} = -42095.0661510489$$
$$x_{48} = 28240.6537163389$$
$$x_{49} = -33617.4943084304$$
$$x_{50} = -34465.2850023106$$
$$x_{51} = -15809.3788845165$$
$$x_{52} = 27392.7585806149$$
$$x_{53} = 18912.3527533172$$
$$x_{54} = 35023.3038583412$$
$$x_{55} = 26544.8446743911$$
$$x_{56} = 29936.3941106863$$
$$x_{57} = 12123.544841056$$
$$x_{58} = -13263.5701308315$$
$$x_{59} = -41247.3370154355$$
$$x_{60} = -21746.9237165814$$
$$x_{61} = 29088.5317374502$$
$$x_{62} = 36718.860771076$$
$$x_{63} = -39551.8628248821$$
$$x_{64} = 18064.0836828544$$
$$x_{65} = 12972.6101323084$$
$$x_{66} = -31074.0618885764$$
$$x_{67} = 21456.8409729567$$
$$x_{68} = -25138.9056447972$$
$$x_{69} = 40109.8856055808$$
$$x_{70} = 13821.4931681849$$
$$x_{71} = -22594.9625089587$$
$$x_{72} = 32479.8996955549$$
$$x_{73} = 31632.0769879872$$
$$x_{74} = -42942.7903957194$$
The values of the extrema at the points:
-3.13264722713223e-5
(-31921.8835434698, ---------------------)
-31921.8835434698 - 3
4.02431445737447e-5
(24848.9527856772, ---------------------)
-3 + 24848.9527856772
4.48331622551052e-5
(22304.9178115051, ---------------------)
-3 + 22304.9178115051
2.78776055679455e-5
(35871.0864641394, ---------------------)
-3 + 35871.0864641394
2.60319137083125e-5
(38414.3867016121, ---------------------)
-3 + 38414.3867016121
-3.05160004690157e-5
(-32769.6940726206, ---------------------)
-32769.6940726206 - 3
-3.84810183905155e-5
(-25986.8382213314, ---------------------)
-25986.8382213314 - 3
2.66193712461632e-5
(37566.6273450369, ---------------------)
-3 + 37566.6273450369
-6.6840960693771e-5
(-14960.8860813988, ---------------------)
-14960.8860813988 - 3
-2.70207419050616e-5
(-37008.6063243624, ---------------------)
-37008.6063243624 - 3
6.44378322670923e-5
(15518.8336328721, ---------------------)
-3 + 15518.8336328721
-4.11675944708724e-5
(-24290.9504985193, ---------------------)
-24290.9504985193 - 3
-8.05496482544173e-5
(-12414.7034718108, ---------------------)
-12414.7034718108 - 3
-5.44829217038106e-5
(-18354.3754214744, ---------------------)
-18354.3754214744 - 3
6.81652755427732e-5
(14670.2260130038, ---------------------)
-3 + 14670.2260130038
-3.40386349428819e-5
(-29378.3813978389, ---------------------)
-29378.3813978389 - 3
-6.00319967129938e-5
(-16657.783408065, ---------------------)
-16657.783408065 - 3
-7.08602238574916e-5
(-14112.2895735892, ---------------------)
-14112.2895735892 - 3
-3.30838500116375e-5
(-30226.22818334, ---------------------)
-30226.22818334 - 3
-2.58370446155379e-5
(-38704.1170790913, ---------------------)
-38704.1170790913 - 3
-3.6123717271599e-5
(-27682.6438443875, ---------------------)
-27682.6438443875 - 3
-2.83181295021794e-5
(-35313.0668342955, ---------------------)
-35313.0668342955 - 3
-3.72651128478916e-5
(-26834.7503365653, ---------------------)
-26834.7503365653 - 3
-2.47527186745923e-5
(-40399.6026837339, ---------------------)
-40399.6026837339 - 3
2.39203772158515e-5
(41805.3607928303, ---------------------)
-3 + 41805.3607928303
-3.50501843717638e-5
(-28530.5203814017, ---------------------)
-28530.5203814017 - 3
3.00050607677518e-5
(33327.7111964611, ---------------------)
-3 + 33327.7111964611
-2.64156370619182e-5
(-37856.3650546609, ---------------------)
-37856.3650546609 - 3
-2.76542245172913e-5
(-36160.8404213184, ---------------------)
-36160.8404213184 - 3
0.669924024969721
(1.26241821842168, ---------------------)
-3 + 1.26241821842168
-4.98734760163832e-5
(-20050.7379682616, ---------------------)
-20050.7379682616 - 3
4.16649822869317e-5
(24000.9702280612, ---------------------)
-3 + 24000.9702280612
2.92607171480862e-5
(34175.5123312146, ---------------------)
-3 + 34175.5123312146
3.89151845409093e-5
(25696.9101211369, ---------------------)
-3 + 25696.9101211369
5.06058428844209e-5
(19760.5640406041, ---------------------)
-3 + 19760.5640406041
6.10972985278851e-5
(16367.3357554312, ---------------------)
-3 + 16367.3357554312
3.24841519600899e-5
(30784.242139886, --------------------)
-3 + 30784.242139886
-4.78495220870569e-5
(-20898.8503044486, ---------------------)
-20898.8503044486 - 3
4.85231381794035e-5
(20608.7247596784, ---------------------)
-3 + 20608.7247596784
-5.71229059519402e-5
(-17506.1121672217, ---------------------)
-17506.1121672217 - 3
2.54698296455105e-5
(39262.1393115611, ---------------------)
-3 + 39262.1393115611
5.80863514467433e-5
(17215.7481743741, ---------------------)
-3 + 17215.7481743741
4.31910224272008e-5
(23152.9596471974, ---------------------)
-3 + 23152.9596471974
-4.26567104521063e-5
(-23442.9703742917, ---------------------)
-23442.9703742917 - 3
2.44154776042627e-5
(40957.6259784778, ---------------------)
-3 + 40957.6259784778
-5.20763310262411e-5
(-19202.5816602195, ---------------------)
-19202.5816602195 - 3
-2.3755753137995e-5
(-42095.0661510489, ---------------------)
-42095.0661510489 - 3
3.54099451672214e-5
(28240.6537163389, ---------------------)
-3 + 28240.6537163389
-2.97464168664788e-5
(-33617.4943084304, ---------------------)
-33617.4943084304 - 3
-2.90147027553186e-5
(-34465.2850023106, ---------------------)
-34465.2850023106 - 3
-6.32535918059195e-5
(-15809.3788845165, ---------------------)
-15809.3788845165 - 3
3.65059983503612e-5
(27392.7585806149, ---------------------)
-3 + 27392.7585806149
5.28754942397455e-5
(18912.3527533172, ---------------------)
-3 + 18912.3527533172
2.85524176637634e-5
(35023.3038583412, ---------------------)
-3 + 35023.3038583412
3.76720983600886e-5
(26544.8446743911, ---------------------)
-3 + 26544.8446743911
3.34041566907047e-5
(29936.3941106863, ---------------------)
-3 + 29936.3941106863
8.24841257934442e-5
(12123.544841056, --------------------)
-3 + 12123.544841056
-7.53944818959944e-5
(-13263.5701308315, ---------------------)
-13263.5701308315 - 3
-2.42439893617825e-5
(-41247.3370154355, ---------------------)
-41247.3370154355 - 3
-4.59835152929101e-5
(-21746.9237165814, ---------------------)
-21746.9237165814 - 3
3.43778093934731e-5
(29088.5317374502, ---------------------)
-3 + 29088.5317374502
2.72339603885665e-5
(36718.860771076, --------------------)
-3 + 36718.860771076
-2.52832592036049e-5
(-39551.8628248821, ---------------------)
-39551.8628248821 - 3
5.53584680261216e-5
(18064.0836828544, ---------------------)
-3 + 18064.0836828544
7.70854891822246e-5
(12972.6101323084, ---------------------)
-3 + 12972.6101323084
-3.21811806657441e-5
(-31074.0618885764, ---------------------)
-31074.0618885764 - 3
4.66051829594275e-5
(21456.8409729567, ---------------------)
-3 + 21456.8409729567
-3.97789789898633e-5
(-25138.9056447972, ---------------------)
-25138.9056447972 - 3
2.49315096439385e-5
(40109.8856055808, ---------------------)
-3 + 40109.8856055808
7.23510829175073e-5
(13821.4931681849, ---------------------)
-3 + 13821.4931681849
-4.42576525165994e-5
(-22594.9625089587, ---------------------)
-22594.9625089587 - 3
3.07882724102404e-5
(32479.8996955549, ---------------------)
-3 + 32479.8996955549
3.16134789393257e-5
(31632.0769879872, ---------------------)
-3 + 31632.0769879872
-2.32867960047357e-5
(-42942.7903957194, ---------------------)
-42942.7903957194 - 3
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
Maxima of the function at points:
$$x_{74} = 1.26241821842168$$
Decreasing at intervals
$$\left(-\infty, 1.26241821842168\right]$$
Increasing at intervals
$$\left[1.26241821842168, \infty\right)$$