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$$\frac{\cos{\left(x \right)}}{x^{3}} - \frac{3 \sin{\left(x \right)}}{x^{4}} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = -13.924969952549$$
$$x_{2} = -95.7872667660245$$
$$x_{3} = 61.2120859995403$$
$$x_{4} = 13.924969952549$$
$$x_{5} = -26.591193287969$$
$$x_{6} = -23.4346216921802$$
$$x_{7} = -45.4872362867621$$
$$x_{8} = -108.357267428671$$
$$x_{9} = -98.9298533613919$$
$$x_{10} = 29.7446115259422$$
$$x_{11} = 76.930043294192$$
$$x_{12} = 42.3407653325706$$
$$x_{13} = -54.9233040395155$$
$$x_{14} = -32.8957773192946$$
$$x_{15} = 54.9233040395155$$
$$x_{16} = -186.908713650658$$
$$x_{17} = 23.4346216921802$$
$$x_{18} = -64.3560674689022$$
$$x_{19} = 67.4998267230665$$
$$x_{20} = 39.1935138550425$$
$$x_{21} = 17.1051395364267$$
$$x_{22} = -92.6446127847888$$
$$x_{23} = 10.7227710626892$$
$$x_{24} = 86.359073259985$$
$$x_{25} = -51.7784042739041$$
$$x_{26} = 70.6433933906731$$
$$x_{27} = -76.930043294192$$
$$x_{28} = 89.5018843244389$$
$$x_{29} = 48.6330777853047$$
$$x_{30} = -36.0452782424582$$
$$x_{31} = 83.2161702400252$$
$$x_{32} = -67.4998267230665$$
$$x_{33} = 80.0731644462726$$
$$x_{34} = -89.5018843244389$$
$$x_{35} = 32.8957773192946$$
$$x_{36} = 3215.41914794509$$
$$x_{37} = -48.6330777853047$$
$$x_{38} = 98.9298533613919$$
$$x_{39} = -10.7227710626892$$
$$x_{40} = 26.591193287969$$
$$x_{41} = -42.3407653325706$$
$$x_{42} = 4.07814976485137$$
$$x_{43} = 36.0452782424582$$
$$x_{44} = -86.359073259985$$
$$x_{45} = 95.7872667660245$$
$$x_{46} = 7.47219265966058$$
$$x_{47} = -58.0678462801751$$
$$x_{48} = -39.1935138550425$$
$$x_{49} = -20.2734415170608$$
$$x_{50} = -83.2161702400252$$
$$x_{51} = -73.7867920572034$$
$$x_{52} = -70.6433933906731$$
$$x_{53} = 92.6446127847888$$
$$x_{54} = -17.1051395364267$$
$$x_{55} = 64.3560674689022$$
$$x_{56} = 20.2734415170608$$
$$x_{57} = 45.4872362867621$$
$$x_{58} = -7.47219265966058$$
$$x_{59} = 58.0678462801751$$
$$x_{60} = -61.2120859995403$$
$$x_{61} = -29.7446115259422$$
$$x_{62} = 73.7867920572034$$
$$x_{63} = -4.07814976485137$$
$$x_{64} = -80.0731644462726$$
$$x_{65} = 51.7784042739041$$
Signos de extremos en los puntos:
(-13.92496995254897, 0.000362047304723488)
(-95.78726676602449, 1.1372704641271e-6)
(61.21208599954033, -4.35479288166118e-6)
(13.92496995254897, 0.000362047304723488)
(-26.59119328796898, 5.28494009082656e-5)
(-23.4346216921802, -7.70719574291125e-5)
(-45.48723628676209, 1.06020259212848e-5)
(-108.35726742867121, 7.85704936475449e-7)
(-98.9298533613919, -1.03232943885669e-6)
(29.744611525942226, -3.78074454700618e-5)
(76.93004329419203, 2.19473504875366e-6)
(42.34076533257061, -1.31412441116291e-5)
(-54.92330403951548, -6.02674942320184e-6)
(-32.89577731929462, 2.79757033726946e-5)
(54.92330403951548, -6.02674942320184e-6)
(-186.90871365065752, -1.53128285331065e-7)
(23.4346216921802, -7.70719574291125e-5)
(-64.35606746890217, 3.7476597286644e-6)
(67.49982672306646, -3.24835521431348e-6)
(39.193513855042454, 1.65610881918558e-5)
(17.105139536426744, -0.000196807350078387)
(-92.64461278478876, -1.25693237122389e-6)
(10.722771062689203, -0.00078111312666525)
(86.359073259985, -1.55172297524868e-6)
(-51.77840427390411, 7.1916125507828e-6)
(70.64339339067311, 2.83396240646379e-6)
(-76.93004329419203, 2.19473504875366e-6)
(89.50188432443886, 1.39398991793862e-6)
(48.63307778530466, -8.67720806398467e-6)
(-36.04527824245817, -2.12792275158681e-5)
(83.21617024002518, 1.73418247352317e-6)
(-67.49982672306646, -3.24835521431348e-6)
(80.07316444627257, -1.94641047170913e-6)
(-89.50188432443886, 1.39398991793862e-6)
(32.89577731929462, 2.79757033726946e-5)
(3215.41914794509, -3.00806370783767e-11)
(-48.63307778530466, -8.67720806398467e-6)
(98.9298533613919, -1.03232943885669e-6)
(-10.722771062689203, -0.00078111312666525)
(26.59119328796898, 5.28494009082656e-5)
(-42.34076533257061, -1.31412441116291e-5)
(4.078149764851372, -0.0118764951343876)
(36.04527824245817, -2.12792275158681e-5)
(-86.359073259985, -1.55172297524868e-6)
(95.78726676602449, 1.1372704641271e-6)
(7.472192659660579, 0.00222435242575847)
(-58.067846280175104, 5.1005149012716e-6)
(-39.193513855042454, 1.65610881918558e-5)
(-20.27344151706078, 0.000118717289027919)
(-83.21617024002518, 1.73418247352317e-6)
(-73.78679205720341, -2.48716990126569e-6)
(-70.64339339067311, 2.83396240646379e-6)
(92.64461278478876, -1.25693237122389e-6)
(-17.105139536426744, -0.000196807350078387)
(64.35606746890217, 3.7476597286644e-6)
(20.27344151706078, 0.000118717289027919)
(45.48723628676209, 1.06020259212848e-5)
(-7.472192659660579, 0.00222435242575847)
(58.067846280175104, 5.1005149012716e-6)
(-61.21208599954033, -4.35479288166118e-6)
(-29.744611525942226, -3.78074454700618e-5)
(73.78679205720341, -2.48716990126569e-6)
(-4.078149764851372, -0.0118764951343876)
(-80.07316444627257, -1.94641047170913e-6)
(51.77840427390411, 7.1916125507828e-6)
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:
$$x_{1} = 61.2120859995403$$
$$x_{2} = -23.4346216921802$$
$$x_{3} = -98.9298533613919$$
$$x_{4} = 29.7446115259422$$
$$x_{5} = 42.3407653325706$$
$$x_{6} = -54.9233040395155$$
$$x_{7} = 54.9233040395155$$
$$x_{8} = -186.908713650658$$
$$x_{9} = 23.4346216921802$$
$$x_{10} = 67.4998267230665$$
$$x_{11} = 17.1051395364267$$
$$x_{12} = -92.6446127847888$$
$$x_{13} = 10.7227710626892$$
$$x_{14} = 86.359073259985$$
$$x_{15} = 48.6330777853047$$
$$x_{16} = -36.0452782424582$$
$$x_{17} = -67.4998267230665$$
$$x_{18} = 80.0731644462726$$
$$x_{19} = 3215.41914794509$$
$$x_{20} = -48.6330777853047$$
$$x_{21} = 98.9298533613919$$
$$x_{22} = -10.7227710626892$$
$$x_{23} = -42.3407653325706$$
$$x_{24} = 4.07814976485137$$
$$x_{25} = 36.0452782424582$$
$$x_{26} = -86.359073259985$$
$$x_{27} = -73.7867920572034$$
$$x_{28} = 92.6446127847888$$
$$x_{29} = -17.1051395364267$$
$$x_{30} = -61.2120859995403$$
$$x_{31} = -29.7446115259422$$
$$x_{32} = 73.7867920572034$$
$$x_{33} = -4.07814976485137$$
$$x_{34} = -80.0731644462726$$
Puntos máximos de la función:
$$x_{34} = -13.924969952549$$
$$x_{34} = -95.7872667660245$$
$$x_{34} = 13.924969952549$$
$$x_{34} = -26.591193287969$$
$$x_{34} = -45.4872362867621$$
$$x_{34} = -108.357267428671$$
$$x_{34} = 76.930043294192$$
$$x_{34} = -32.8957773192946$$
$$x_{34} = -64.3560674689022$$
$$x_{34} = 39.1935138550425$$
$$x_{34} = -51.7784042739041$$
$$x_{34} = 70.6433933906731$$
$$x_{34} = -76.930043294192$$
$$x_{34} = 89.5018843244389$$
$$x_{34} = 83.2161702400252$$
$$x_{34} = -89.5018843244389$$
$$x_{34} = 32.8957773192946$$
$$x_{34} = 26.591193287969$$
$$x_{34} = 95.7872667660245$$
$$x_{34} = 7.47219265966058$$
$$x_{34} = -58.0678462801751$$
$$x_{34} = -39.1935138550425$$
$$x_{34} = -20.2734415170608$$
$$x_{34} = -83.2161702400252$$
$$x_{34} = -70.6433933906731$$
$$x_{34} = 64.3560674689022$$
$$x_{34} = 20.2734415170608$$
$$x_{34} = 45.4872362867621$$
$$x_{34} = -7.47219265966058$$
$$x_{34} = 58.0678462801751$$
$$x_{34} = 51.7784042739041$$
Decrece en los intervalos
$$\left[3215.41914794509, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -186.908713650658\right]$$