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$$- \frac{2 x \cos{\left(x \right)}}{\left(x^{2} + 1\right)^{2}} - \frac{\sin{\left(x \right)}}{x^{2} + 1} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = -2.54373214752609$$
$$x_{2} = -94.2265597456126$$
$$x_{3} = 50.2256989863186$$
$$x_{4} = 37.6460727029451$$
$$x_{5} = -21.9002665401996$$
$$x_{6} = -15.5808165061202$$
$$x_{7} = 0$$
$$x_{8} = 9.21343494397267$$
$$x_{9} = -37.6460727029451$$
$$x_{10} = 53.3696312768345$$
$$x_{11} = -56.513303694752$$
$$x_{12} = -84.7994242303256$$
$$x_{13} = 94.2265597456126$$
$$x_{14} = -5.96808139239822$$
$$x_{15} = 12.4075674897868$$
$$x_{16} = 25.0532062442974$$
$$x_{17} = -53.3696312768345$$
$$x_{18} = 78.5143529265667$$
$$x_{19} = 72.2289536816917$$
$$x_{20} = -25.0532062442974$$
$$x_{21} = -106.7954266585$$
$$x_{22} = -34.4996609189666$$
$$x_{23} = 15.5808165061202$$
$$x_{24} = -87.9418588604656$$
$$x_{25} = 87.9418588604656$$
$$x_{26} = 97.3688368618863$$
$$x_{27} = -75.3716994196716$$
$$x_{28} = -100.511071203627$$
$$x_{29} = -62.8000247758447$$
$$x_{30} = -43.9368321750172$$
$$x_{31} = 31.3522862210969$$
$$x_{32} = -65.9431328237524$$
$$x_{33} = 163.350575451696$$
$$x_{34} = -47.0814548776037$$
$$x_{35} = 21.9002665401996$$
$$x_{36} = -103.653266658919$$
$$x_{37} = 100.511071203627$$
$$x_{38} = -91.0842354305333$$
$$x_{39} = -97.3688368618863$$
$$x_{40} = 84.7994242303256$$
$$x_{41} = 2.54373214752609$$
$$x_{42} = -81.6569248421486$$
$$x_{43} = 47.0814548776037$$
$$x_{44} = 43.9368321750172$$
$$x_{45} = 69.0861031389786$$
$$x_{46} = 34.4996609189666$$
$$x_{47} = 18.7435542483014$$
$$x_{48} = -122.505790268738$$
$$x_{49} = 75.3716994196716$$
$$x_{50} = -69.0861031389786$$
$$x_{51} = 65.9431328237524$$
$$x_{52} = 62.8000247758447$$
$$x_{53} = 56.513303694752$$
$$x_{54} = 40.7917435749351$$
$$x_{55} = -31.3522862210969$$
$$x_{56} = -78.5143529265667$$
$$x_{57} = -40.7917435749351$$
$$x_{58} = -9.21343494397267$$
$$x_{59} = -72.2289536816917$$
$$x_{60} = -113.079652107775$$
$$x_{61} = 81.6569248421486$$
$$x_{62} = 5.96808139239822$$
$$x_{63} = 91.0842354305333$$
$$x_{64} = -12.4075674897868$$
$$x_{65} = -59.656757255627$$
$$x_{66} = -50.2256989863186$$
$$x_{67} = 59.656757255627$$
$$x_{68} = -18.7435542483014$$
$$x_{69} = -210.477206074369$$
$$x_{70} = -28.203628119338$$
$$x_{71} = 28.203628119338$$
Signos de extremos en los puntos:
(-2.5437321475260917, -0.110639672191836)
(-94.22655974561256, 0.000112591766511704)
(50.22569898631863, 0.000395942499274958)
(37.64607270294512, 0.000704114293701762)
(-21.90026654019963, -0.00207205193264381)
(-15.580816506120234, -0.00406924940329345)
(0, 1)
(9.213434943972674, -0.011384094242491)
(-37.64607270294512, 0.000704114293701762)
(53.36963127683454, -0.000350715231486932)
(-56.51330369475196, 0.000312817485971633)
(-84.79942423032556, -0.00013900585084881)
(94.22655974561256, 0.000112591766511704)
(-5.968081392398221, 0.0259643971802455)
(12.40756748978677, 0.00637258289495849)
(25.053206244297428, 0.00158564848443144)
(-53.36963127683454, -0.000350715231486932)
(78.51435292656672, -0.000162140173318783)
(72.22895368169175, -0.000191570111140939)
(-25.053206244297428, 0.00158564848443144)
(-106.79542665849998, 8.76557646972633e-5)
(-34.49966091896661, -0.000838066136358659)
(15.580816506120234, -0.00406924940329345)
(-87.94185886046559, 0.0001292529049009)
(87.94185886046559, 0.0001292529049009)
(97.3688368618863, -0.000105444189250915)
(-75.37169941967161, 0.00017593577340359)
(-100.51107120362654, 9.89562584809543e-5)
(-62.80002477584475, 0.000253367116057383)
(-43.936832175017194, 0.000517212000046997)
(31.352286221096882, 0.00101423808278872)
(-65.94313282375245, -0.000229806033389755)
(163.35057545169616, 3.74722559859326e-5)
(-47.08145487760369, -0.000450519125938963)
(21.90026654019963, -0.00207205193264381)
(-103.65326665891925, -9.30492289536518e-5)
(100.51107120362654, 9.89562584809543e-5)
(-91.08423543053327, -0.000120491545810595)
(-97.3688368618863, -0.000105444189250915)
(84.79942423032556, -0.00013900585084881)
(2.5437321475260917, -0.110639672191836)
(-81.6569248421486, 0.000149905871666022)
(47.08145487760369, -0.000450519125938963)
(43.936832175017194, 0.000517212000046997)
(69.0861031389786, 0.000209385109224912)
(34.49966091896661, -0.000838066136358659)
(18.7435542483014, 0.00282239086745388)
(-122.50579026873812, -6.66192853304118e-5)
(75.37169941967161, 0.00017593577340359)
(-69.0861031389786, 0.000209385109224912)
(65.94313282375245, -0.000229806033389755)
(62.80002477584475, 0.000253367116057383)
(56.51330369475196, 0.000312817485971633)
(40.79174357493512, -0.000599892999132703)
(-31.352286221096882, 0.00101423808278872)
(-78.51435292656672, -0.000162140173318783)
(-40.79174357493512, -0.000599892999132703)
(-9.213434943972674, -0.011384094242491)
(-72.22895368169175, -0.000191570111140939)
(-113.07965210777498, 7.81860375912636e-5)
(81.6569248421486, 0.000149905871666022)
(5.968081392398221, 0.0259643971802455)
(91.08423543053327, -0.000120491545810595)
(-12.40756748978677, 0.00637258289495849)
(-59.65675725562702, -0.000280746865913829)
(-50.22569898631863, 0.000395942499274958)
(59.65675725562702, -0.000280746865913829)
(-18.7435542483014, 0.00282239086745388)
(-210.47720607436906, -2.25715015693393e-5)
(-28.203628119338006, -0.00125244284383629)
(28.203628119338006, -0.00125244284383629)
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:
$$x_{1} = -2.54373214752609$$
$$x_{2} = -21.9002665401996$$
$$x_{3} = -15.5808165061202$$
$$x_{4} = 9.21343494397267$$
$$x_{5} = 53.3696312768345$$
$$x_{6} = -84.7994242303256$$
$$x_{7} = -53.3696312768345$$
$$x_{8} = 78.5143529265667$$
$$x_{9} = 72.2289536816917$$
$$x_{10} = -34.4996609189666$$
$$x_{11} = 15.5808165061202$$
$$x_{12} = 97.3688368618863$$
$$x_{13} = -65.9431328237524$$
$$x_{14} = -47.0814548776037$$
$$x_{15} = 21.9002665401996$$
$$x_{16} = -103.653266658919$$
$$x_{17} = -91.0842354305333$$
$$x_{18} = -97.3688368618863$$
$$x_{19} = 84.7994242303256$$
$$x_{20} = 2.54373214752609$$
$$x_{21} = 47.0814548776037$$
$$x_{22} = 34.4996609189666$$
$$x_{23} = -122.505790268738$$
$$x_{24} = 65.9431328237524$$
$$x_{25} = 40.7917435749351$$
$$x_{26} = -78.5143529265667$$
$$x_{27} = -40.7917435749351$$
$$x_{28} = -9.21343494397267$$
$$x_{29} = -72.2289536816917$$
$$x_{30} = 91.0842354305333$$
$$x_{31} = -59.656757255627$$
$$x_{32} = 59.656757255627$$
$$x_{33} = -210.477206074369$$
$$x_{34} = -28.203628119338$$
$$x_{35} = 28.203628119338$$
Puntos máximos de la función:
$$x_{35} = -94.2265597456126$$
$$x_{35} = 50.2256989863186$$
$$x_{35} = 37.6460727029451$$
$$x_{35} = 0$$
$$x_{35} = -37.6460727029451$$
$$x_{35} = -56.513303694752$$
$$x_{35} = 94.2265597456126$$
$$x_{35} = -5.96808139239822$$
$$x_{35} = 12.4075674897868$$
$$x_{35} = 25.0532062442974$$
$$x_{35} = -25.0532062442974$$
$$x_{35} = -106.7954266585$$
$$x_{35} = -87.9418588604656$$
$$x_{35} = 87.9418588604656$$
$$x_{35} = -75.3716994196716$$
$$x_{35} = -100.511071203627$$
$$x_{35} = -62.8000247758447$$
$$x_{35} = -43.9368321750172$$
$$x_{35} = 31.3522862210969$$
$$x_{35} = 163.350575451696$$
$$x_{35} = 100.511071203627$$
$$x_{35} = -81.6569248421486$$
$$x_{35} = 43.9368321750172$$
$$x_{35} = 69.0861031389786$$
$$x_{35} = 18.7435542483014$$
$$x_{35} = 75.3716994196716$$
$$x_{35} = -69.0861031389786$$
$$x_{35} = 62.8000247758447$$
$$x_{35} = 56.513303694752$$
$$x_{35} = -31.3522862210969$$
$$x_{35} = -113.079652107775$$
$$x_{35} = 81.6569248421486$$
$$x_{35} = 5.96808139239822$$
$$x_{35} = -12.4075674897868$$
$$x_{35} = -50.2256989863186$$
$$x_{35} = -18.7435542483014$$
Decrece en los intervalos
$$\left[97.3688368618863, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -210.477206074369\right]$$