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$$\frac{4 x}{x^{3} \left(x^{2} + 1\right)} - \frac{3 \log{\left(\frac{\left(x^{2} + 1\right)^{2}}{5} \right)}}{x^{4}} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = 6518.06419316066$$
$$x_{2} = -11851.6998120867$$
$$x_{3} = 3399.05359434548$$
$$x_{4} = 4444.99631763352$$
$$x_{5} = -9813.26997388103$$
$$x_{6} = -12360.0298581425$$
$$x_{7} = -3095.489171382$$
$$x_{8} = 11382.3012710817$$
$$x_{9} = 7290.71220421772$$
$$x_{10} = 3661.29254612632$$
$$x_{11} = -11597.3560596516$$
$$x_{12} = -12614.0225337288$$
$$x_{13} = -10068.5627021184$$
$$x_{14} = 5484.14302919119$$
$$x_{15} = -5444.02039378729$$
$$x_{16} = 12145.2794322771$$
$$x_{17} = 6001.68280602692$$
$$x_{18} = 5743.06582175864$$
$$x_{19} = 11636.7493233136$$
$$x_{20} = 8830.15156373676$$
$$x_{21} = 6775.85744526435$$
$$x_{22} = -11088.2939056672$$
$$x_{23} = 9086.06540077348$$
$$x_{24} = -6220.02683861568$$
$$x_{25} = 11127.7264977828$$
$$x_{26} = 8574.06158530194$$
$$x_{27} = -8021.59448266171$$
$$x_{28} = -8278.08788897982$$
$$x_{29} = -5702.99212038224$$
$$x_{30} = -6993.53773800257$$
$$x_{31} = -10578.7071403181$$
$$x_{32} = 2872.80087895029$$
$$x_{33} = -9046.44833748345$$
$$x_{34} = 10363.2030364676$$
$$x_{35} = -11342.8886291357$$
$$x_{36} = -4925.0749937381$$
$$x_{37} = 10873.0212263762$$
$$x_{38} = 7033.40292555665$$
$$x_{39} = -7764.90165153377$$
$$x_{40} = -3620.72063043173$$
$$x_{41} = -3882.51019172682$$
$$x_{42} = 9852.81206483734$$
$$x_{43} = 10618.1814741767$$
$$x_{44} = -10833.5680813767$$
$$x_{45} = -4404.64434229941$$
$$x_{46} = 7804.66395559646$$
$$x_{47} = 12907.2083322547$$
$$x_{48} = -6735.95468103015$$
$$x_{49} = -9557.82362360971$$
$$x_{50} = -7250.8828646363$$
$$x_{51} = -5184.72155271381$$
$$x_{52} = 13160.9655718628$$
$$x_{53} = 12399.368141897$$
$$x_{54} = -4143.80592051648$$
$$x_{55} = -3358.39588959699$$
$$x_{56} = 3923.00306647649$$
$$x_{57} = 9341.80949354275$$
$$x_{58} = 4184.22571182609$$
$$x_{59} = 11891.074240728$$
$$x_{60} = 4705.34662360077$$
$$x_{61} = -12105.9233220089$$
$$x_{62} = -6478.12197186495$$
$$x_{63} = -10323.7068418273$$
$$x_{64} = 8317.78862814313$$
$$x_{65} = 5224.89612538554$$
$$x_{66} = 6260.01056836151$$
$$x_{67} = -9302.21830634859$$
$$x_{68} = -7508.00085000034$$
$$x_{69} = -8790.50766114978$$
$$x_{70} = -13121.678040193$$
$$x_{71} = 10108.0814683842$$
$$x_{72} = -8534.3898147497$$
$$x_{73} = 8061.32537042608$$
$$x_{74} = 7547.79593529315$$
$$x_{75} = -1.61966878843573$$
$$x_{76} = 9597.38983821362$$
$$x_{77} = 12653.3434597092$$
$$x_{78} = 3136.24029749988$$
$$x_{79} = 4965.3048375654$$
$$x_{80} = -5961.65532289636$$
$$x_{81} = -12867.9043169035$$
$$x_{82} = -4665.05779296199$$
Signos de extremos en los puntos:
(6518.064193160658, 1.21044963357984e-10)
(-11851.6998120867, -2.1572040913802e-11)
(3399.053594345479, 7.87232912264856e-10)
(4444.996317633525, 3.64235580016607e-10)
(-9813.269973881028, -3.7201864419083e-11)
(-12360.02985814245, -1.91073916367717e-11)
(-3095.489171382003, -1.0296760402958e-9)
(11382.301271081735, 2.42428487001415e-11)
(7290.71220421772, 8.76516226797997e-11)
(3661.2925461263185, 6.35960449220625e-10)
(-11597.356059651616, -2.29670644928876e-11)
(-12614.022533728794, -1.80167917149272e-11)
(-10068.562702118408, -3.45438465575271e-11)
(5484.143029191189, 1.9903602409585e-10)
(-5444.020393787294, -2.03287202743133e-10)
(12145.27943227713, 2.00998443311055e-11)
(6001.682806026924, 1.53527002578723e-10)
(5743.065821758638, 1.74285366777242e-10)
(11636.749323313585, 2.27432134697838e-11)
(8830.15156373676, 5.04491352207206e-11)
(6775.857445264345, 1.08246890126596e-10)
(-11088.293905667222, -2.61460552826252e-11)
(9086.065400773476, 4.64576406580142e-11)
(-6220.026838615678, -1.38513932393409e-10)
(11127.726497782784, 2.58793882164634e-11)
(8574.061585301943, 5.49191718295356e-11)
(-8021.594482661712, -6.65497306159415e-11)
(-8278.087888979817, -6.0775307787533e-11)
(-5702.992120382243, -1.77834237736223e-10)
(-6993.537738002567, -9.88201668904932e-11)
(-10578.707140318073, -2.99504717035545e-11)
(2872.800878950293, 1.27557049464138e-9)
(-9046.448337483454, -4.70470634194096e-11)
(10363.203036467585, 3.17840963729678e-11)
(-11342.888629135738, -2.44869274167472e-11)
(-4925.074993738103, -2.71201617451594e-10)
(10873.021226376193, 2.76689786559982e-11)
(7033.402925556651, 9.72146877563512e-11)
(-7764.901651533772, -7.30924659800414e-11)
(-3620.7206304317315, -6.56640592230445e-10)
(-3882.5101917268184, -5.37339124728599e-10)
(9852.812064837335, 3.67725722400657e-11)
(10618.181474176656, 2.96301234594712e-11)
(-10833.568081376741, -2.79609355716693e-11)
(-4404.6443422994125, -3.73911236685375e-10)
(7804.663955596462, 7.20239743151431e-11)
(12907.208332254668, 1.68594623367515e-11)
(-6735.954681030153, -1.10104721777148e-10)
(-9557.823623609715, -4.01442794398931e-11)
(-7250.882864636299, -8.90465122139599e-11)
(-5184.72155271381, -2.33937873255817e-10)
(13160.965571862756, 1.59371024220174e-11)
(12399.368141897032, 1.89327750035414e-11)
(-4143.805920516483, -4.45626701109751e-10)
(-3358.3958895969886, -8.14901099295812e-10)
(3923.0030664764877, 5.21558595535981e-10)
(9341.809493542745, 4.2881821572017e-11)
(4184.225711826089, 4.33366764973401e-10)
(11891.074240728041, 2.13663484494429e-11)
(4705.346623600766, 3.09244431858387e-10)
(-12105.923322008866, -2.02891971586852e-11)
(-6478.121971864946, -1.23207341133632e-10)
(-10323.706841827317, -3.2136408610236e-11)
(8317.788628143135, 5.99424703019038e-11)
(5224.8961253855405, 2.28799426898079e-10)
(6260.010568361509, 1.35981194181301e-10)
(-9302.218306348592, -4.34105781777473e-11)
(-7508.000850000335, -8.05371435413554e-11)
(-8790.50766114978, -5.11082753217141e-11)
(-13121.678040192954, -1.60753899129819e-11)
(10108.081468384169, 3.41554403434072e-11)
(-8534.389814749704, -5.56587630741449e-11)
(8061.325370426076, 6.56083107640302e-11)
(7547.795935293155, 7.93191496137724e-11)
(-1.6196687884357346, -0.227198256848027)
(9597.389838213616, 3.96685194304599e-11)
(12653.343459709155, 1.78554939981933e-11)
(3136.2402974998754, 9.91753542121228e-10)
(4965.3048375654025, 2.64928722921356e-10)
(-5961.655322896363, -1.56513897456787e-10)
(-12867.904316903538, -1.70086970816499e-11)
(-4665.057792961993, -3.16987204621213e-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:
Puntos mínimos de la función:
$$x_{1} = -1.61966878843573$$
La función no tiene puntos máximos
Decrece en los intervalos
$$\left[-1.61966878843573, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -1.61966878843573\right]$$