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$$- \frac{1}{x \left(x^{2} + 1\right)} - \frac{\operatorname{acot}{\left(x \right)}}{x^{2}} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = -28754.5720832173$$
$$x_{2} = 36513.8938244144$$
$$x_{3} = 32276.0426843694$$
$$x_{4} = -36382.6654280841$$
$$x_{5} = -41468.131451574$$
$$x_{6} = 17020.4793835477$$
$$x_{7} = 28885.7976883074$$
$$x_{8} = -11804.8475421597$$
$$x_{9} = 34818.7484021429$$
$$x_{10} = 20410.4350191541$$
$$x_{11} = -14346.9510241183$$
$$x_{12} = -34687.5204711935$$
$$x_{13} = 12783.3785763737$$
$$x_{14} = 39056.6222184191$$
$$x_{15} = 17867.9518916845$$
$$x_{16} = -18584.2210520308$$
$$x_{17} = 26343.1429783643$$
$$x_{18} = -30449.6890121582$$
$$x_{19} = -25364.3749223954$$
$$x_{20} = -13499.5535498777$$
$$x_{21} = 35666.3203642305$$
$$x_{22} = 14478.1543122982$$
$$x_{23} = 41599.3609185449$$
$$x_{24} = -32144.815594025$$
$$x_{25} = -16889.2678433082$$
$$x_{26} = -26211.9188841492$$
$$x_{27} = -27059.4666206439$$
$$x_{28} = 11936.0367344929$$
$$x_{29} = 19562.9311293969$$
$$x_{30} = 33123.6094321483$$
$$x_{31} = 37361.4686807589$$
$$x_{32} = 22952.9911333361$$
$$x_{33} = -21126.7276262566$$
$$x_{34} = -22821.7698890071$$
$$x_{35} = -37230.2400751649$$
$$x_{36} = 24648.0579430374$$
$$x_{37} = -12652.1837404312$$
$$x_{38} = 25495.5984095976$$
$$x_{39} = -19431.7143467142$$
$$x_{40} = -27907.0177880916$$
$$x_{41} = 38209.044840371$$
$$x_{42} = -38077.8160393051$$
$$x_{43} = 22105.4658726159$$
$$x_{44} = -42315.7128464392$$
$$x_{45} = 28038.2429350559$$
$$x_{46} = 42446.9424554335$$
$$x_{47} = -38925.3932344872$$
$$x_{48} = 15325.5780406013$$
$$x_{49} = -23669.2999443148$$
$$x_{50} = 13630.7530068279$$
$$x_{51} = 33971.1780502568$$
$$x_{52} = -40620.5510097953$$
$$x_{53} = 27190.691265887$$
$$x_{54} = 30580.9154216729$$
$$x_{55} = -33839.9503785982$$
$$x_{56} = 29733.3552619912$$
$$x_{57} = 18715.4363230897$$
$$x_{58} = -29602.129237463$$
$$x_{59} = -16041.8114526469$$
$$x_{60} = 40751.780325776$$
$$x_{61} = -17736.7383530475$$
$$x_{62} = -31297.2511944755$$
$$x_{63} = 31428.4779581971$$
$$x_{64} = 16173.0206711654$$
$$x_{65} = -21974.2455536639$$
$$x_{66} = 21257.9469066706$$
$$x_{67} = -20279.2169096663$$
$$x_{68} = -32992.3820399739$$
$$x_{69} = -24516.8351266811$$
$$x_{70} = -15194.3715408$$
$$x_{71} = -35535.0921922843$$
$$x_{72} = 39904.2007372773$$
$$x_{73} = 23800.5220167199$$
$$x_{74} = -39772.9715820244$$
Signos de extremos en los puntos:
(-28754.57208321733, 1.20944516293517e-9)
(36513.89382441435, 7.50038753065707e-10)
(32276.042684369393, 9.59929719294004e-10)
(-36382.66542808411, 7.55459130506513e-10)
(-41468.13145157402, 5.81528584332591e-10)
(17020.47938354767, 3.45188583500149e-9)
(28885.797688307437, 1.19848132029617e-9)
(-11804.84754215968, 7.17594718693526e-9)
(34818.74840214285, 8.2484754877614e-10)
(20410.435019154116, 2.40046554818317e-9)
(-14346.951024118276, 4.85826016207709e-9)
(-34687.5204711935, 8.3110039044608e-10)
(12783.378576373669, 6.11939796664849e-9)
(39056.62221841914, 6.55557270003213e-10)
(17867.951891684497, 3.13220696217645e-9)
(-18584.221052030778, 2.89541802966794e-9)
(26343.14297836428, 1.44100279019725e-9)
(-30449.689012158226, 1.07853508385086e-9)
(-25364.374922395407, 1.554360214983e-9)
(-13499.553549877683, 5.48733137015487e-9)
(35666.320364230516, 7.86110120503169e-10)
(14478.154312298198, 4.77060651575018e-9)
(41599.36091854488, 5.77865387842876e-10)
(-32144.815594025018, 9.67783297380155e-10)
(-16889.26784330823, 3.50572909699655e-9)
(-26211.918884149185, 1.45546702043703e-9)
(-27059.466620643874, 1.36571958114469e-9)
(11936.036734492856, 7.01907213279033e-9)
(19562.931129396926, 2.61295630664365e-9)
(33123.60943214828, 9.11432875514495e-10)
(37361.468680758895, 7.16394296902737e-10)
(22952.99113333609, 1.89811019523146e-9)
(-21126.727626256612, 2.24045142860024e-9)
(-22821.769889007064, 1.92000056082403e-9)
(-37230.24007516492, 7.21453469740323e-10)
(24648.057943037446, 1.64601802701379e-9)
(-12652.183740431175, 6.24696421999817e-9)
(25495.59840959757, 1.53840109445342e-9)
(-19431.714346714212, 2.64836454119158e-9)
(-27907.017788091554, 1.28402400029464e-9)
(38209.044840371025, 6.84963827783871e-10)
(-38077.816039305086, 6.89693188941407e-10)
(22105.46587261586, 2.04644772560039e-9)
(-42315.71284643916, 5.58465921890688e-10)
(28038.24293505593, 1.27203309639058e-9)
(42446.94245543354, 5.55018136381889e-10)
(-38925.39323448724, 6.59984874728492e-10)
(15325.57804060126, 4.25761386490084e-9)
(-23669.2999443148, 1.78496281577192e-9)
(13630.753006827903, 5.38220583289574e-9)
(33971.17805025678, 8.66520386476298e-10)
(-40620.55100979533, 6.06049893789816e-10)
(27190.691265887024, 1.35256922689075e-9)
(30580.9154216729, 1.06929869452076e-9)
(-33839.95037859824, 8.73253962135754e-10)
(29733.355261991168, 1.13112905968158e-9)
(18715.4363230897, 2.85496038596343e-9)
(-29602.12923746302, 1.14117986183505e-9)
(-16041.811452646887, 3.88591399481171e-9)
(40751.780325776024, 6.02152961766238e-10)
(-17736.73835304749, 3.17872150865334e-9)
(-31297.25119447549, 1.02091035462809e-9)
(31428.477958197083, 1.01240271575178e-9)
(16173.020671165448, 3.82311811900536e-9)
(-21974.245553663874, 2.07096163295244e-9)
(21257.946906670644, 2.21287745209561e-9)
(-20279.216909666342, 2.43163081627016e-9)
(-32992.38203997393, 9.18697754050756e-10)
(-24516.835126681057, 1.66368532879824e-9)
(-15194.371540799999, 4.33146207083914e-9)
(-35535.09219228433, 7.91926921190877e-10)
(39904.200737277286, 6.28004516156724e-10)
(23800.52201671992, 1.76533460543662e-9)
(-39772.971582024365, 6.32155498949648e-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
La función no tiene puntos máximos
Crece en todo el eje numérico