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$$\frac{\left(- \frac{x}{\sqrt{x^{2} + 5}} - 1\right) \operatorname{atan}{\left(\frac{1}{x} \right)}}{\left(x + \sqrt{x^{2} + 5}\right)^{2}} - \frac{1}{x^{2} \left(1 + \frac{1}{x^{2}}\right) \left(x + \sqrt{x^{2} + 5}\right)} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = 30586.4960317016$$
$$x_{2} = -41473.4250678483$$
$$x_{3} = 18724.5503307565$$
$$x_{4} = 29739.0948477105$$
$$x_{5} = -21984.2320113325$$
$$x_{6} = -20290.0372329335$$
$$x_{7} = 14489.9291942501$$
$$x_{8} = 42450.9635985586$$
$$x_{9} = -9286.77399607226$$
$$x_{10} = 39908.4780233899$$
$$x_{11} = -11823.4164644708$$
$$x_{12} = -38931.0325474025$$
$$x_{13} = -38083.5808415457$$
$$x_{14} = 40755.9686812106$$
$$x_{15} = -16902.2569581704$$
$$x_{16} = -18596.0270685$$
$$x_{17} = 37366.0369535824$$
$$x_{18} = 34823.650131951$$
$$x_{19} = 33128.7618861189$$
$$x_{20} = -39778.49075349$$
$$x_{21} = -17749.107716313$$
$$x_{22} = -32999.0350923798$$
$$x_{23} = -21137.1143172439$$
$$x_{24} = -26220.2919562211$$
$$x_{25} = -8442.04857510117$$
$$x_{26} = -12669.5125313416$$
$$x_{27} = 25502.291220155$$
$$x_{28} = 24654.9806766047$$
$$x_{29} = 41603.4639652552$$
$$x_{30} = 39060.99229278$$
$$x_{31} = 17030.4992728625$$
$$x_{32} = 27196.9671750743$$
$$x_{33} = -13515.7972632899$$
$$x_{34} = -37236.1360768888$$
$$x_{35} = 17877.4974187058$$
$$x_{36} = 23807.6910229565$$
$$x_{37} = -36388.6987394278$$
$$x_{38} = -40625.9550520886$$
$$x_{39} = 33976.2020168722$$
$$x_{40} = -10977.552558557$$
$$x_{41} = 20418.793240764$$
$$x_{42} = -34693.8485179822$$
$$x_{43} = 19571.6509121846$$
$$x_{44} = -10131.9787605055$$
$$x_{45} = -22831.3857984867$$
$$x_{46} = 15336.7034773974$$
$$x_{47} = 32281.330368093$$
$$x_{48} = 31433.908158897$$
$$x_{49} = -35541.2693589076$$
$$x_{50} = -27067.5775930096$$
$$x_{51} = 36518.5680929011$$
$$x_{52} = 28044.3292740089$$
$$x_{53} = -24525.7867030123$$
$$x_{54} = 21265.9723356637$$
$$x_{55} = -42320.9004555853$$
$$x_{56} = -30456.8973738324$$
$$x_{57} = 12796.7093513341$$
$$x_{58} = -31304.2644344512$$
$$x_{59} = 16183.5644707302$$
$$x_{60} = 11104.0940915695$$
$$x_{61} = -15208.8069857835$$
$$x_{62} = 13643.2577105972$$
$$x_{63} = -32151.6439945354$$
$$x_{64} = -19443.0060610182$$
$$x_{65} = -25373.0275918757$$
$$x_{66} = -33846.4368615773$$
$$x_{67} = 35671.1056626215$$
$$x_{68} = 22960.4245689479$$
$$x_{69} = 26349.6206405067$$
$$x_{70} = 22113.1839776482$$
$$x_{71} = -23678.5718081203$$
$$x_{72} = 11950.3101659441$$
$$x_{73} = 28891.705569053$$
$$x_{74} = 10258.1061741605$$
$$x_{75} = 38213.5118170516$$
$$x_{76} = -28762.2051678377$$
$$x_{77} = 9412.40750297377$$
$$x_{78} = -14362.2373797794$$
$$x_{79} = -27914.8825661469$$
$$x_{80} = -29609.5438845902$$
$$x_{81} = -16055.4856324891$$
Signos de extremos en los puntos:
(30586.496031701594, 5.34454267167801e-10)
(-41473.42506784832, -0.400000012279845)
(18724.550330756523, 1.42609089961922e-9)
(29739.094847710516, 5.65346244465958e-10)
(-21984.23201133254, -0.400000007576938)
(-20290.037232933544, -0.399999998420807)
(14489.929194250144, 2.38142810367477e-9)
(42450.96359855857, 2.77456496714578e-10)
(-9286.773996072257, -0.400000004294441)
(39908.47802338989, 3.13934953560287e-10)
(-11823.416464470785, -0.400000003821089)
(-38931.03254740248, -0.399999992247514)
(-38083.58084154573, -0.400000006054859)
(40755.968681210645, 3.01014602577858e-10)
(-16902.256958170354, -0.399999995778915)
(-18596.027068500047, -0.399999994416919)
(37366.03695358239, 3.58109569013238e-10)
(34823.65013195104, 4.12307677711122e-10)
(33128.76188611892, 4.55574694822403e-10)
(-39778.49075349002, -0.400000010580786)
(-17749.107716313, -0.400000004254328)
(-32999.03509237983, -0.399999976167207)
(-21137.114317243886, -0.399999998873498)
(-26220.291956221063, -0.400000005418882)
(-8442.048575101167, -0.400000005681411)
(-12669.512531341603, -0.40000000415797)
(25502.29122015501, 7.68796861392899e-10)
(24654.980676604722, 8.22546900543479e-10)
(41603.4639652552, 2.88875705867705e-10)
(39060.992292779985, 3.27705296254185e-10)
(17030.49927286251, 1.72391259148653e-9)
(27196.96717507431, 6.75972532570977e-10)
(-13515.797263289915, -0.40000000168503)
(-37236.13607688878, -0.400000008200881)
(17877.49741870577, 1.56443150805898e-9)
(23807.691022956493, 8.82135800707402e-10)
(-36388.69873942783, -0.400000007965551)
(-40625.95505208863, -0.399999996278548)
(33976.20201687218, 4.33132072279703e-10)
(-10977.552558556952, -0.400000004788246)
(20418.793240763964, 1.19925036588736e-9)
(-34693.848517982166, -0.400000023398145)
(19571.65091218462, 1.30531425461478e-9)
(-10131.978760505515, -0.400000004657371)
(-22831.38579848668, -0.400000005334315)
(15336.703477397365, 2.12571951496136e-9)
(32281.33036809296, 4.7980763540737e-10)
(31433.908158897022, 5.06026479996294e-10)
(-35541.26935890759, -0.399999993067072)
(-27067.577593009584, -0.399999996708123)
(36518.56809290106, 3.7492337958797e-10)
(28044.32927400888, 6.35740513324481e-10)
(-24525.786703012323, -0.400000002768438)
(21265.972335663708, 1.10560377693177e-9)
(-42320.90045558528, -0.40000000277653)
(-30456.897373832446, -0.399999996926122)
(12796.709351334095, 3.05332749182599e-9)
(-31304.26443445121, -0.400000009825044)
(16183.564470730218, 1.90906905136312e-9)
(11104.094091569514, 4.05512020506996e-9)
(-15208.806985783538, -0.400000000644032)
(13643.257710597203, 2.68617210785303e-9)
(-32151.643994535425, -0.399999998043644)
(-19443.0060610182, -0.399999996480221)
(-25373.027591875747, -0.400000005855039)
(-33846.4368615773, -0.399999999692163)
(35671.10566262152, 3.92949609835289e-10)
(22960.424568947943, 9.48440681818139e-10)
(26349.620640506673, 7.20147188252385e-10)
(22113.183977648223, 1.02250971869938e-9)
(-23678.571808120276, -0.400000008922719)
(11950.310165944054, 3.50115747384076e-9)
(28891.70556905302, 5.98995614500285e-10)
(10258.10617416046, 4.75155344102293e-9)
(38213.51181705159, 3.42401849284767e-10)
(-28762.205167837703, -0.399999995778117)
(9412.407502973765, 5.64376021685755e-9)
(-14362.237379779373, -0.400000002638577)
(-27914.882566146887, -0.400000003445607)
(-29609.543884590195, -0.400000005657061)
(-16055.4856324891, -0.400000000203136)
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:
$$x_{1} = -21984.2320113325$$
$$x_{2} = -11823.4164644708$$
$$x_{3} = -26220.2919562211$$
$$x_{4} = -10977.552558557$$
$$x_{5} = -34693.8485179822$$
$$x_{6} = -10131.9787605055$$
$$x_{7} = -31304.2644344512$$
$$x_{8} = -23678.5718081203$$
Puntos máximos de la función:
$$x_{8} = -18596.0270685$$
Decrece en los intervalos
$$\left[-10131.9787605055, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -34693.8485179822\right]$$