Para hallar los extremos hay que resolver la ecuación
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
$$\frac{d}{d x} f{\left(x \right)} = $$
primera derivada$$- \frac{2}{\left(x - 1\right) \left(4 x^{2} + 1\right)} - \frac{\operatorname{acot}{\left(2 x \right)}}{\left(x - 1\right)^{2}} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = 10023.3059886798$$
$$x_{2} = -16248.99951668$$
$$x_{3} = 11718.8349580713$$
$$x_{4} = -28115.9864417731$$
$$x_{5} = 13414.2811900563$$
$$x_{6} = -38287.431577539$$
$$x_{7} = -31506.4819421501$$
$$x_{8} = -18791.980335574$$
$$x_{9} = 26976.7635661688$$
$$x_{10} = 39691.0813598296$$
$$x_{11} = 28672.0191082501$$
$$x_{12} = -12858.2578651732$$
$$x_{13} = 17652.6936745703$$
$$x_{14} = 31214.8927402376$$
$$x_{15} = -37439.8151451701$$
$$x_{16} = 10871.0832955839$$
$$x_{17} = 25281.5016345177$$
$$x_{18} = 41386.3118402928$$
$$x_{19} = -29811.2363998511$$
$$x_{20} = -10315.0664908658$$
$$x_{21} = 15957.3540863923$$
$$x_{22} = -15401.3271925303$$
$$x_{23} = -11162.8154801548$$
$$x_{24} = 18500.3543120234$$
$$x_{25} = -21334.9217457585$$
$$x_{26} = -26420.7312291888$$
$$x_{27} = 34605.3768635164$$
$$x_{28} = -32354.1032719495$$
$$x_{29} = 33757.7571094617$$
$$x_{30} = 38843.4653933966$$
$$x_{31} = 23586.2319179501$$
$$x_{32} = 21890.9525830866$$
$$x_{33} = 12566.5662777284$$
$$x_{34} = -25573.1013303947$$
$$x_{35} = 30367.2693415002$$
$$x_{36} = 32062.515115973$$
$$x_{37} = 40538.6968319284$$
$$x_{38} = 27824.3920621207$$
$$x_{39} = 42233.9264130482$$
$$x_{40} = 14261.9826775785$$
$$x_{41} = 16805.0272459016$$
$$x_{42} = -20487.2780980658$$
$$x_{43} = 36300.6141788641$$
$$x_{44} = -12010.5446730764$$
$$x_{45} = 26129.1334776658$$
$$x_{46} = -42525.5065046146$$
$$x_{47} = -39135.0474861716$$
$$x_{48} = -17096.6652306606$$
$$x_{49} = 15109.673041667$$
$$x_{50} = -19639.6311091475$$
$$x_{51} = -36592.1981530122$$
$$x_{52} = 0.299468985102263$$
$$x_{53} = -39982.6629040901$$
$$x_{54} = 24433.8678518666$$
$$x_{55} = -14553.6471327275$$
$$x_{56} = -40830.2778615979$$
$$x_{57} = -33201.7237379436$$
$$x_{58} = -27268.359552755$$
$$x_{59} = -34049.343403984$$
$$x_{60} = 37148.2318410501$$
$$x_{61} = -13705.9579412476$$
$$x_{62} = -28963.6120206595$$
$$x_{63} = 20195.661172063$$
$$x_{64} = -24725.4696966835$$
$$x_{65} = 19348.0099308358$$
$$x_{66} = 32910.1365484833$$
$$x_{67} = -35744.5805616267$$
$$x_{68} = 29519.6448307279$$
$$x_{69} = 21043.3085720763$$
$$x_{70} = -23877.8361460348$$
$$x_{71} = -22182.5624290172$$
$$x_{72} = -17944.3252518746$$
$$x_{73} = -41677.892386554$$
$$x_{74} = -30658.8596777143$$
$$x_{75} = -23030.2004700538$$
$$x_{76} = 35452.9958690241$$
$$x_{77} = -34896.9623277814$$
$$x_{78} = -9467.2926099517$$
$$x_{79} = 37995.8488992771$$
$$x_{80} = 22738.5935890233$$
Signos de extremos en los puntos:
(10023.30598867983, 4.97727179998685e-9)
(-16248.999516679962, 1.89360776601023e-9)
(11718.834958071282, 3.64114680508727e-9)
(-28115.986441773104, 6.32481617412191e-10)
(13414.2811900563, 2.77886577091334e-9)
(-38287.431577539006, 3.41072099113022e-10)
(-31506.481942150105, 5.03681959987282e-10)
(-18791.980335574022, 1.41579963358164e-9)
(26976.763566168815, 6.87078584294831e-10)
(39691.0813598296, 3.17391348115915e-10)
(28672.019108250137, 6.08231063639072e-10)
(-12858.257865173153, 3.02393163075937e-9)
(17652.693674570288, 1.60462161135569e-9)
(31214.892740237614, 5.13168776007676e-10)
(-37439.81514517007, 3.56690065091881e-10)
(10871.083295583927, 4.23120735206181e-9)
(25281.501634517706, 7.82314628168441e-10)
(41386.311840292765, 2.91922148402099e-10)
(-29811.236399851103, 5.62594470647764e-10)
(-10315.06649086584, 4.69876623793991e-9)
(15957.354086392263, 1.96370143326721e-9)
(-15401.327192530276, 2.10778112918953e-9)
(-11162.815480154843, 4.01220955995748e-9)
(18500.35431202337, 1.46094339039383e-9)
(-21334.92174575853, 1.09841774463209e-9)
(-26420.73122918883, 7.1624878901441e-10)
(34605.37686351635, 4.17537405235419e-10)
(-32354.103271949505, 4.77636873128713e-10)
(33757.7571094617, 4.38768762856835e-10)
(38843.465393396575, 3.3139446354388e-10)
(23586.23191795006, 8.98817022075509e-10)
(21890.95258308659, 1.04342328253237e-9)
(12566.56627772839, 3.16644036216381e-9)
(-25573.101330394697, 7.64515373854258e-10)
(30367.269341500167, 5.42216617057071e-10)
(32062.515115973005, 4.86394186318491e-10)
(40538.696831928406, 3.04257400376883e-10)
(27824.39206212073, 6.45853837877993e-10)
(42233.92641304821, 2.80322120504772e-10)
(14261.98267757852, 2.45833263849057e-9)
(16805.02724590162, 1.77058755052948e-9)
(-20487.278098065755, 1.19118792177403e-9)
(36300.61417886415, 3.79449532632734e-10)
(-12010.544673076394, 3.46583944632635e-9)
(26129.133477665782, 7.32380227407848e-10)
(-42525.506504614576, 2.76478141728713e-10)
(-39135.04748617162, 3.26457895680577e-10)
(-17096.665230660605, 1.71049491297e-9)
(15109.673041667047, 2.19022440802021e-9)
(-19639.631109147533, 1.296227524837e-9)
(-36592.19815301222, 3.7340586744676e-10)
(0.29946898510226344, -1.47196637230352)
(-39982.66290409008, 3.12763245855605e-10)
(24433.86785186661, 8.37535718469471e-10)
(-14553.647132727501, 2.36045911551622e-9)
(-40830.27786159789, 2.99912589764952e-10)
(-33201.723737943634, 4.53560954733933e-10)
(-27268.35955275504, 6.72412927209326e-10)
(-34049.343403984, 4.31260583883113e-10)
(37148.23184105011, 3.62330923326709e-10)
(-13705.957941247598, 2.66145746157632e-9)
(-28963.612020659526, 5.96004539352621e-10)
(20195.66117206302, 1.22595733811094e-9)
(-24725.46969668346, 8.17830568060985e-10)
(19348.009930835764, 1.33573359189659e-9)
(32910.13654848328, 4.6166168504241e-10)
(-35744.580561626666, 3.91324864896379e-10)
(29519.644830727913, 5.73802532598462e-10)
(21043.30857207633, 1.12917848775663e-9)
(-23877.83614603476, 8.7692384847109e-10)
(-22182.56242901715, 1.01607789110859e-9)
(-17944.325251874623, 1.55271424577534e-9)
(-41677.89238655404, 2.87837972075299e-10)
(-30658.859677714296, 5.31916967142086e-10)
(-23030.200470053816, 9.42661370342373e-10)
(35452.99586902411, 3.97810621184273e-10)
(-34896.96232778141, 4.10565365367217e-10)
(-9467.292609951697, 5.57792322985131e-9)
(37995.84889927713, 3.46345166149953e-10)
(22738.593589023312, 9.67078893909473e-10)
Intervalos de crecimiento y decrecimiento de la función:Hallemos los intervalos donde la función crece y decrece y también los puntos mínimos y máximos de la función, para lo cual miramos cómo se comporta la función en los extremos con desviación mínima del extremo:
La función no tiene puntos mínimos
Puntos máximos de la función:
$$x_{80} = 0.299468985102263$$
Decrece en los intervalos
$$\left(-\infty, 0.299468985102263\right]$$
Crece en los intervalos
$$\left[0.299468985102263, \infty\right)$$