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$$\frac{2 x^{3} \cos{\left(x \right)}}{\sqrt{x^{4} + 1}} - \sqrt{x^{4} + 1} \sin{\left(x \right)} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = -37.7520396086384$$
$$x_{2} = -18.9546809625376$$
$$x_{3} = -3.64145590371878$$
$$x_{4} = 62.8636572266689$$
$$x_{5} = 25.2119028697101$$
$$x_{6} = -12.7222929857609$$
$$x_{7} = 0$$
$$x_{8} = -84.8465692428547$$
$$x_{9} = 34.6152330151878$$
$$x_{10} = 18.9546809625376$$
$$x_{11} = -72.2842925026702$$
$$x_{12} = 78.565267383932$$
$$x_{13} = -69.1439554752287$$
$$x_{14} = -47.1662675942319$$
$$x_{15} = 87.9873209343097$$
$$x_{16} = -44.0276918871959$$
$$x_{17} = -34.6152330151878$$
$$x_{18} = 56.5839987344219$$
$$x_{19} = -40.8895777486063$$
$$x_{20} = 31.4793748560042$$
$$x_{21} = 66.0037377692333$$
$$x_{22} = 94.2689923090381$$
$$x_{23} = 37.7520396086384$$
$$x_{24} = -81.7058821474877$$
$$x_{25} = -100.55085272523$$
$$x_{26} = -75.4247339737051$$
$$x_{27} = 47.1662675942319$$
$$x_{28} = 84.8465692428547$$
$$x_{29} = 100.55085272523$$
$$x_{30} = -6.57819120703837$$
$$x_{31} = 15.8336094522557$$
$$x_{32} = -62.8636572266689$$
$$x_{33} = 81.7058821474877$$
$$x_{34} = 53.4444796651863$$
$$x_{35} = 9.6295376590189$$
$$x_{36} = -9.6295376590189$$
$$x_{37} = -22.0814753909444$$
$$x_{38} = -56.5839987344219$$
$$x_{39} = -87.9873209343097$$
$$x_{40} = 44.0276918871959$$
$$x_{41} = 40.8895777486063$$
$$x_{42} = -59.7237354298028$$
$$x_{43} = -50.3052188301361$$
$$x_{44} = -53.4444796651863$$
$$x_{45} = -31.4793748560042$$
$$x_{46} = 75.4247339737051$$
$$x_{47} = 50.3052188301361$$
$$x_{48} = -25.2119028697101$$
$$x_{49} = 3.64145590371878$$
$$x_{50} = 91.1281305508213$$
$$x_{51} = -78.565267383932$$
$$x_{52} = 59.7237354298028$$
$$x_{53} = -66.0037377692333$$
$$x_{54} = -97.4099011704444$$
$$x_{55} = -91.1281305508213$$
$$x_{56} = 12.7222929857609$$
$$x_{57} = 6.57819120703837$$
$$x_{58} = -94.2689923090381$$
$$x_{59} = 97.4099011704444$$
$$x_{60} = -28.3447767612856$$
$$x_{61} = -15.8336094522557$$
$$x_{62} = 22.0814753909444$$
$$x_{63} = 28.3447767612856$$
$$x_{64} = 72.2842925026702$$
$$x_{65} = 69.1439554752287$$
Signos de extremos en los puntos:
(-37.752039608638356, 1423.2210469704)
(-18.954680962537612, 357.297891480163)
(-3.6414559037187755, -11.6708368992285)
(62.8636572266689, 3949.84104362804)
(25.21190286971014, 633.650230339188)
(-12.72229298576089, 159.896260189732)
(0, 1)
(-84.84656924285468, -7196.9412148653)
(34.61523301518781, -1196.21976962219)
(18.954680962537612, 357.297891480163)
(-72.28429250267018, -5223.02018600386)
(78.56526738393204, -6170.50229172045)
(-69.1439554752287, 4778.88793759756)
(-47.16626759423188, -2222.65971713915)
(87.98732093430974, 7739.76948451511)
(-44.02769188719591, 1936.44100161224)
(-34.61523301518781, -1196.21976962219)
(56.58399873442193, 3199.75094126066)
(-40.889577748606285, -1669.9614500469)
(31.479374856004206, 988.957583418349)
(66.0037377692333, -4354.49489064025)
(94.2689923090381, 8884.6436421811)
(37.752039608638356, 1423.2210469704)
(-81.70588214748767, 6673.85215077647)
(-100.55085272522955, 10108.4746265022)
(-75.42473397370513, 5686.89163705697)
(47.16626759423188, -2222.65971713915)
(84.84656924285468, -7196.9412148653)
(100.55085272522955, 10108.4746265022)
(-6.5781912070383655, 41.4142957561914)
(15.833609452255745, -248.728847949114)
(-62.8636572266689, 3949.84104362804)
(81.70588214748767, 6673.85215077647)
(53.44447966518631, -2854.31468026233)
(9.6295376590189, -90.7961753166522)
(-9.6295376590189, -90.7961753166522)
(-22.08147539094437, -485.604815184633)
(-56.58399873442193, 3199.75094126066)
(-87.98732093430974, 7739.76948451511)
(44.02769188719591, 1936.44100161224)
(40.889577748606285, -1669.9614500469)
(-59.72373542980282, -3564.92639465213)
(-50.305218830136084, 2528.61760744185)
(-53.44447966518631, -2854.31468026233)
(-31.479374856004206, 988.957583418349)
(75.42473397370513, 5686.89163705697)
(50.305218830136084, 2528.61760744185)
(-25.21190286971014, 633.650230339188)
(3.6414559037187755, -11.6708368992285)
(91.12813055082128, -8302.33696016471)
(-78.56526738393204, -6170.50229172045)
(59.72373542980282, -3564.92639465213)
(-66.0037377692333, -4354.49489064025)
(-97.40990117044437, -9486.68953087313)
(-91.12813055082128, -8302.33696016471)
(12.72229298576089, 159.896260189732)
(6.5781912070383655, 41.4142957561914)
(-94.2689923090381, 8884.6436421811)
(97.40990117044437, -9486.68953087313)
(-28.344776761285594, -801.43443375393)
(-15.833609452255745, -248.728847949114)
(22.08147539094437, -485.604815184633)
(28.344776761285594, -801.43443375393)
(72.28429250267018, -5223.02018600386)
(69.1439554752287, 4778.88793759756)
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:
$$x_{1} = -3.64145590371878$$
$$x_{2} = -84.8465692428547$$
$$x_{3} = 34.6152330151878$$
$$x_{4} = -72.2842925026702$$
$$x_{5} = 78.565267383932$$
$$x_{6} = -47.1662675942319$$
$$x_{7} = -34.6152330151878$$
$$x_{8} = -40.8895777486063$$
$$x_{9} = 66.0037377692333$$
$$x_{10} = 47.1662675942319$$
$$x_{11} = 84.8465692428547$$
$$x_{12} = 15.8336094522557$$
$$x_{13} = 53.4444796651863$$
$$x_{14} = 9.6295376590189$$
$$x_{15} = -9.6295376590189$$
$$x_{16} = -22.0814753909444$$
$$x_{17} = 40.8895777486063$$
$$x_{18} = -59.7237354298028$$
$$x_{19} = -53.4444796651863$$
$$x_{20} = 3.64145590371878$$
$$x_{21} = 91.1281305508213$$
$$x_{22} = -78.565267383932$$
$$x_{23} = 59.7237354298028$$
$$x_{24} = -66.0037377692333$$
$$x_{25} = -97.4099011704444$$
$$x_{26} = -91.1281305508213$$
$$x_{27} = 97.4099011704444$$
$$x_{28} = -28.3447767612856$$
$$x_{29} = -15.8336094522557$$
$$x_{30} = 22.0814753909444$$
$$x_{31} = 28.3447767612856$$
$$x_{32} = 72.2842925026702$$
Puntos máximos de la función:
$$x_{32} = -37.7520396086384$$
$$x_{32} = -18.9546809625376$$
$$x_{32} = 62.8636572266689$$
$$x_{32} = 25.2119028697101$$
$$x_{32} = -12.7222929857609$$
$$x_{32} = 0$$
$$x_{32} = 18.9546809625376$$
$$x_{32} = -69.1439554752287$$
$$x_{32} = 87.9873209343097$$
$$x_{32} = -44.0276918871959$$
$$x_{32} = 56.5839987344219$$
$$x_{32} = 31.4793748560042$$
$$x_{32} = 94.2689923090381$$
$$x_{32} = 37.7520396086384$$
$$x_{32} = -81.7058821474877$$
$$x_{32} = -100.55085272523$$
$$x_{32} = -75.4247339737051$$
$$x_{32} = 100.55085272523$$
$$x_{32} = -6.57819120703837$$
$$x_{32} = -62.8636572266689$$
$$x_{32} = 81.7058821474877$$
$$x_{32} = -56.5839987344219$$
$$x_{32} = -87.9873209343097$$
$$x_{32} = 44.0276918871959$$
$$x_{32} = -50.3052188301361$$
$$x_{32} = -31.4793748560042$$
$$x_{32} = 75.4247339737051$$
$$x_{32} = 50.3052188301361$$
$$x_{32} = -25.2119028697101$$
$$x_{32} = 12.7222929857609$$
$$x_{32} = 6.57819120703837$$
$$x_{32} = -94.2689923090381$$
$$x_{32} = 69.1439554752287$$
Decrece en los intervalos
$$\left[97.4099011704444, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -97.4099011704444\right]$$