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$$- \frac{1}{\left(x - 3\right) \left(x^{2} + 1\right)} - \frac{\operatorname{acot}{\left(x \right)}}{\left(x - 3\right)^{2}} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = -35313.0668342955$$
$$x_{2} = 15518.8336328721$$
$$x_{3} = -38704.1170790913$$
$$x_{4} = 41805.3607928303$$
$$x_{5} = 34175.5123312146$$
$$x_{6} = -42942.7903957194$$
$$x_{7} = -16657.783408065$$
$$x_{8} = -27682.6438443875$$
$$x_{9} = 29936.3941106863$$
$$x_{10} = 26544.8446743911$$
$$x_{11} = -14960.8860813988$$
$$x_{12} = -25986.8382213314$$
$$x_{13} = -33617.4943084304$$
$$x_{14} = -34465.2850023106$$
$$x_{15} = -20050.7379682616$$
$$x_{16} = 35871.0864641394$$
$$x_{17} = 23152.9596471974$$
$$x_{18} = -21746.9237165814$$
$$x_{19} = 40957.6259784778$$
$$x_{20} = 12972.6101323084$$
$$x_{21} = 40109.8856055808$$
$$x_{22} = -39551.8628248821$$
$$x_{23} = 27392.7585806149$$
$$x_{24} = -29378.3813978389$$
$$x_{25} = -13263.5701308315$$
$$x_{26} = -18354.3754214744$$
$$x_{27} = -12414.7034718108$$
$$x_{28} = -42095.0661510489$$
$$x_{29} = -23442.9703742917$$
$$x_{30} = -15809.3788845165$$
$$x_{31} = -37856.3650546609$$
$$x_{32} = -31074.0618885764$$
$$x_{33} = 36718.860771076$$
$$x_{34} = -37008.6063243624$$
$$x_{35} = 20608.7247596784$$
$$x_{36} = -22594.9625089587$$
$$x_{37} = 18912.3527533172$$
$$x_{38} = 21456.8409729567$$
$$x_{39} = -41247.3370154355$$
$$x_{40} = 28240.6537163389$$
$$x_{41} = 30784.242139886$$
$$x_{42} = -24290.9504985193$$
$$x_{43} = -31921.8835434698$$
$$x_{44} = 31632.0769879872$$
$$x_{45} = 37566.6273450369$$
$$x_{46} = 22304.9178115051$$
$$x_{47} = -17506.1121672217$$
$$x_{48} = 19760.5640406041$$
$$x_{49} = -30226.22818334$$
$$x_{50} = 14670.2260130038$$
$$x_{51} = 24000.9702280612$$
$$x_{52} = -20898.8503044486$$
$$x_{53} = 32479.8996955549$$
$$x_{54} = -14112.2895735892$$
$$x_{55} = 35023.3038583412$$
$$x_{56} = -19202.5816602195$$
$$x_{57} = 25696.9101211369$$
$$x_{58} = 24848.9527856772$$
$$x_{59} = 39262.1393115611$$
$$x_{60} = 18064.0836828544$$
$$x_{61} = 16367.3357554312$$
$$x_{62} = 33327.7111964611$$
$$x_{63} = -28530.5203814017$$
$$x_{64} = 29088.5317374502$$
$$x_{65} = 17215.7481743741$$
$$x_{66} = -25138.9056447972$$
$$x_{67} = -32769.6940726206$$
$$x_{68} = 12123.544841056$$
$$x_{69} = 38414.3867016121$$
$$x_{70} = 1.26241821842168$$
$$x_{71} = -36160.8404213184$$
$$x_{72} = -26834.7503365653$$
$$x_{73} = 13821.4931681849$$
$$x_{74} = -40399.6026837339$$
Signos de extremos en los puntos:
(-35313.06683429549, 8.01848338181294e-10)
(15518.833632872089, 4.15303707114861e-9)
(-38704.11707909125, 6.67501135843943e-10)
(41805.36079283031, 5.72225509807911e-10)
(34175.512331214624, 8.56264733025152e-10)
(-42942.790395719414, 5.42236987377855e-10)
(-16657.78340806499, 3.60319171329807e-9)
(-27682.64384438751, 1.30478154940659e-9)
(29936.394110686302, 1.11594951668977e-9)
(26544.844674391126, 1.41934740490877e-9)
(-14960.886081398789, 4.46681833383236e-9)
(-25986.8382213314, 1.48061785005386e-9)
(-33617.49430843041, 8.84770360411383e-10)
(-34465.285002310586, 8.41779704251999e-10)
(-20050.737968261594, 2.48699150728489e-9)
(35871.08646413939, 7.77225893994018e-10)
(23152.959647197393, 1.86570616473751e-9)
(-21746.923716581445, 2.11419202623933e-9)
(40957.625978477816, 5.96159213298478e-10)
(12972.610132308384, 5.94354713795122e-9)
(40109.885605580756, 6.21626667528365e-10)
(-39551.86282488213, 6.39194713316016e-10)
(27392.758580614878, 1.33283388544353e-9)
(-29378.38139783892, 1.15851036688784e-9)
(-13263.570130831458, 5.68304250099865e-9)
(-18354.375421474422, 2.96790366013197e-9)
(-12414.703471810783, 6.48667834896136e-9)
(-42095.066151048944, 5.64295591459207e-10)
(-23442.970374291708, 1.81936212368839e-9)
(-15809.37888451648, 4.00025778966488e-9)
(-37856.365054660906, 6.97730588555291e-10)
(-31074.061888576398, 1.03552841581763e-9)
(36718.86077107599, 7.41749200934459e-10)
(-37008.606324362416, 7.30061312882698e-10)
(20608.7247596784, 2.35483773297575e-9)
(-22594.962508958713, 1.95847977440683e-9)
(18912.352753317246, 2.79626145482265e-9)
(21456.840972956685, 2.17234680811585e-9)
(-41247.33701543549, 5.87728273655331e-10)
(28240.65371633886, 1.25399742921036e-9)
(30784.242139885995, 1.05532297275285e-9)
(-24290.95049851926, 1.69456155240711e-9)
(-31921.883543469776, 9.81255647453413e-10)
(31632.076987987162, 9.99506844645914e-10)
(37566.62734503691, 7.0864751696245e-10)
(22304.91781150513, 2.01028282114719e-9)
(-17506.11216722175, 3.26246730310394e-9)
(19760.564040604117, 2.56134019256726e-9)
(-30226.22818334005, 1.09443250786901e-9)
(14670.226013003838, 4.64745518220953e-9)
(24000.970228061182, 1.73618776467237e-9)
(-20898.85030444859, 2.2892481474175e-9)
(32479.89969555486, 9.48005280641194e-10)
(-14112.289573589214, 5.02010415642315e-9)
(35023.303858341176, 8.15310391916053e-10)
(-19202.581660219523, 2.7115206374669e-9)
(25696.91012113692, 1.51456840774484e-9)
(24848.95278567718, 1.61970623227391e-9)
(39262.1393115611, 6.48761793868724e-10)
(18064.083682854387, 3.06506901790584e-9)
(16367.335755431184, 3.73356422411508e-9)
(33327.71119646112, 9.00384720241422e-10)
(-28530.520381401748, 1.22838625950304e-9)
(29088.531737450157, 1.18195567830066e-9)
(17215.748174374123, 3.37461228493546e-9)
(-25138.905644797193, 1.58217835799154e-9)
(-32769.694072620616, 9.31141040812685e-10)
(12123.544841055986, 6.80531501472155e-9)
(38414.38670161214, 6.77713458004886e-10)
(1.262418218421678, -0.385549636898932)
(-36160.84042131838, 7.64692692897442e-10)
(-26834.750336565336, 1.38853340464679e-9)
(13821.493168184918, 5.23581565927066e-9)
(-40399.602683733865, 6.12651587531458e-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:
$$x_{74} = 1.26241821842168$$
Decrece en los intervalos
$$\left(-\infty, 1.26241821842168\right]$$
Crece en los intervalos
$$\left[1.26241821842168, \infty\right)$$