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$$- \frac{2 x \sin{\left(x \right)}}{\left(x^{2} - 4\right)^{2}} + \frac{\cos{\left(x \right)}}{x^{2} - 4} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = 45.5090895943713$$
$$x_{2} = -80.0856290117635$$
$$x_{3} = 23.4763396531088$$
$$x_{4} = 4.15222477298826$$
$$x_{5} = 64.3715597764874$$
$$x_{6} = 70.657514030234$$
$$x_{7} = -23.4763396531088$$
$$x_{8} = 80.0856290117635$$
$$x_{9} = -10.8062270456027$$
$$x_{10} = -42.3642209820001$$
$$x_{11} = -58.0850045720611$$
$$x_{12} = 73.8003139321454$$
$$x_{13} = 89.5130400768468$$
$$x_{14} = -26.6281461112737$$
$$x_{15} = -98.9399487990654$$
$$x_{16} = -13.9922722908994$$
$$x_{17} = -83.2281657410082$$
$$x_{18} = 67.51460149576$$
$$x_{19} = -67.51460149576$$
$$x_{20} = 17.1611578256127$$
$$x_{21} = -64.3715597764874$$
$$x_{22} = -48.6535328151386$$
$$x_{23} = -4.15222477298826$$
$$x_{24} = -45.5090895943713$$
$$x_{25} = 61.2283689056001$$
$$x_{26} = 95.7976925376814$$
$$x_{27} = -39.2188237682302$$
$$x_{28} = -290.590437700776$$
$$x_{29} = 42.3642209820001$$
$$x_{30} = -7.57754404149509$$
$$x_{31} = 7.57754404149509$$
$$x_{32} = -20.3212959521274$$
$$x_{33} = -86.3706336762639$$
$$x_{34} = 83.2281657410082$$
$$x_{35} = 58.0850045720611$$
$$x_{36} = -124.076787974095$$
$$x_{37} = 54.9414368692733$$
$$x_{38} = 48.6535328151386$$
$$x_{39} = -95.7976925376814$$
$$x_{40} = -36.0727582894777$$
$$x_{41} = 10.8062270456027$$
$$x_{42} = 32.9258307018341$$
$$x_{43} = -32.9258307018341$$
$$x_{44} = 98.9399487990654$$
$$x_{45} = -17.1611578256127$$
$$x_{46} = 202.62285496058$$
$$x_{47} = 36.0727582894777$$
$$x_{48} = 92.655391214883$$
$$x_{49} = -73.8003139321454$$
$$x_{50} = -29.777763739304$$
$$x_{51} = -70.657514030234$$
$$x_{52} = -155.495972864649$$
$$x_{53} = 26.6281461112737$$
$$x_{54} = 76.9430150396882$$
$$x_{55} = -76.9430150396882$$
$$x_{56} = -51.7976285839881$$
$$x_{57} = 20.3212959521274$$
$$x_{58} = -61.2283689056001$$
$$x_{59} = 86.3706336762639$$
$$x_{60} = -54.9414368692733$$
$$x_{61} = -92.655391214883$$
$$x_{62} = 51.7976285839881$$
$$x_{63} = 13.9922722908994$$
$$x_{64} = -89.5130400768468$$
$$x_{65} = 39.2188237682302$$
$$x_{66} = 29.777763739304$$
Signos de extremos en los puntos:
(45.50908959437133, 0.000483306558706209)
(-80.08562901176347, 0.000155964659973988)
(23.476339653108777, -0.00182099790998716)
(4.152224772988261, -0.0639807988925772)
(64.37155977648736, 0.000241446798078732)
(70.65751403023398, 0.000200381305938913)
(-23.476339653108777, 0.00182099790998716)
(80.08562901176347, -0.000155964659973988)
(-10.806227045602704, 0.0087087677703681)
(-42.36422098200009, 0.000557808305842402)
(-58.08500457206114, -0.000296571379601852)
(73.80031393214543, -0.000183672018953665)
(89.51304007684683, 0.000124834826472013)
(-26.62814611127368, -0.00141429382702696)
(-98.9399487990654, 0.000102175165394175)
(-13.992272290899354, -0.00515956885743496)
(-83.22816574100821, -0.000144405824378095)
(67.51460149576005, -0.000219480051529099)
(-67.51460149576005, 0.000219480051529099)
(17.16115782561268, -0.00341850130927523)
(-64.37155977648736, -0.000241446798078732)
(-48.653532815138554, 0.000422802805149809)
(-4.152224772988261, 0.0639807988925772)
(-45.50908959437133, -0.000483306558706209)
(61.22836890560009, -0.000266886242151348)
(95.7976925376814, 0.000108989467051541)
(-39.21882376823015, -0.000650990789484618)
(-290.5904377007758, -1.18426160073303e-5)
(42.36422098200009, -0.000557808305842402)
(-7.577544041495088, -0.0180091484531043)
(7.577544041495088, 0.0180091484531043)
(-20.32129595212744, -0.00243326959458963)
(-86.37063367626389, 0.000134086233502956)
(83.22816574100821, 0.000144405824378095)
(58.08500457206114, 0.000296571379601852)
(-124.07678797409463, 6.49643845271334e-5)
(54.94143686927333, -0.000331503052571738)
(48.653532815138554, -0.000422802805149809)
(-95.7976925376814, -0.000108989467051541)
(-36.072758289477676, 0.0007696756958536)
(10.806227045602704, -0.0087087677703681)
(32.92583070183405, 0.000924115453595151)
(-32.92583070183405, -0.000924115453595151)
(98.9399487990654, -0.000102175165394175)
(-17.16115782561268, 0.00341850130927523)
(202.62285496058013, 2.43581496647271e-5)
(36.072758289477676, -0.0007696756958536)
(92.65539121488297, -0.000116509077378612)
(-73.80031393214543, 0.000183672018953665)
(-29.777763739304003, 0.00113029856328114)
(-70.65751403023398, -0.000200381305938913)
(-155.49597286464936, 4.1361628405071e-5)
(26.62814611127368, 0.00141429382702696)
(76.94301503968819, 0.000168969479119145)
(-76.94301503968819, -0.000168969479119145)
(-51.79762858398812, -0.000372995628973321)
(20.32129595212744, 0.00243326959458963)
(-61.22836890560009, 0.000266886242151348)
(86.37063367626389, -0.000134086233502956)
(-54.94143686927333, 0.000331503052571738)
(-92.65539121488297, 0.000116509077378612)
(51.79762858398812, 0.000372995628973321)
(13.992272290899354, 0.00515956885743496)
(-89.51304007684683, -0.000124834826472013)
(39.21882376823015, 0.000650990789484618)
(29.777763739304003, -0.00113029856328114)
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:
$$x_{1} = 23.4763396531088$$
$$x_{2} = 4.15222477298826$$
$$x_{3} = 80.0856290117635$$
$$x_{4} = -58.0850045720611$$
$$x_{5} = 73.8003139321454$$
$$x_{6} = -26.6281461112737$$
$$x_{7} = -13.9922722908994$$
$$x_{8} = -83.2281657410082$$
$$x_{9} = 67.51460149576$$
$$x_{10} = 17.1611578256127$$
$$x_{11} = -64.3715597764874$$
$$x_{12} = -45.5090895943713$$
$$x_{13} = 61.2283689056001$$
$$x_{14} = -39.2188237682302$$
$$x_{15} = -290.590437700776$$
$$x_{16} = 42.3642209820001$$
$$x_{17} = -7.57754404149509$$
$$x_{18} = -20.3212959521274$$
$$x_{19} = 54.9414368692733$$
$$x_{20} = 48.6535328151386$$
$$x_{21} = -95.7976925376814$$
$$x_{22} = 10.8062270456027$$
$$x_{23} = -32.9258307018341$$
$$x_{24} = 98.9399487990654$$
$$x_{25} = 36.0727582894777$$
$$x_{26} = 92.655391214883$$
$$x_{27} = -70.657514030234$$
$$x_{28} = -76.9430150396882$$
$$x_{29} = -51.7976285839881$$
$$x_{30} = 86.3706336762639$$
$$x_{31} = -89.5130400768468$$
$$x_{32} = 29.777763739304$$
Puntos máximos de la función:
$$x_{32} = 45.5090895943713$$
$$x_{32} = -80.0856290117635$$
$$x_{32} = 64.3715597764874$$
$$x_{32} = 70.657514030234$$
$$x_{32} = -23.4763396531088$$
$$x_{32} = -10.8062270456027$$
$$x_{32} = -42.3642209820001$$
$$x_{32} = 89.5130400768468$$
$$x_{32} = -98.9399487990654$$
$$x_{32} = -67.51460149576$$
$$x_{32} = -48.6535328151386$$
$$x_{32} = -4.15222477298826$$
$$x_{32} = 95.7976925376814$$
$$x_{32} = 7.57754404149509$$
$$x_{32} = -86.3706336762639$$
$$x_{32} = 83.2281657410082$$
$$x_{32} = 58.0850045720611$$
$$x_{32} = -124.076787974095$$
$$x_{32} = -36.0727582894777$$
$$x_{32} = 32.9258307018341$$
$$x_{32} = -17.1611578256127$$
$$x_{32} = 202.62285496058$$
$$x_{32} = -73.8003139321454$$
$$x_{32} = -29.777763739304$$
$$x_{32} = -155.495972864649$$
$$x_{32} = 26.6281461112737$$
$$x_{32} = 76.9430150396882$$
$$x_{32} = 20.3212959521274$$
$$x_{32} = -61.2283689056001$$
$$x_{32} = -54.9414368692733$$
$$x_{32} = -92.655391214883$$
$$x_{32} = 51.7976285839881$$
$$x_{32} = 13.9922722908994$$
$$x_{32} = 39.2188237682302$$
Decrece en los intervalos
$$\left[98.9399487990654, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -290.590437700776\right]$$