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$$\frac{- \cos{\left(x \right)} + \frac{2 \sin{\left(x \right)}}{x} + \frac{4 \cos{\left(x \right)}}{x^{2}} - \frac{4 \sin{\left(x \right)}}{x^{3}}}{x} - \frac{\left(- \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{x}\right) + \frac{2 \sin{\left(x \right)}}{x^{2}}}{x^{2}} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = -13.9205214266357$$
$$x_{2} = -3.87023858022217$$
$$x_{3} = -98.9298409677395$$
$$x_{4} = 36.0450220033785$$
$$x_{5} = -89.5018675871561$$
$$x_{6} = -10.7130109882558$$
$$x_{7} = 7.44308705395446$$
$$x_{8} = -83.2161494163012$$
$$x_{9} = 98.9298409677395$$
$$x_{10} = 26.5905550258527$$
$$x_{11} = -42.3406072405316$$
$$x_{12} = 51.7783178293904$$
$$x_{13} = -23.4336891696933$$
$$x_{14} = -39.1933145376163$$
$$x_{15} = -17.1027407890472$$
$$x_{16} = 67.499787704256$$
$$x_{17} = -80.0731410729394$$
$$x_{18} = 89.5018675871561$$
$$x_{19} = -36.0450220033785$$
$$x_{20} = -92.644597693687$$
$$x_{21} = 86.3590546280651$$
$$x_{22} = -26.5905550258527$$
$$x_{23} = -61.2120336791686$$
$$x_{24} = 249.744603496217$$
$$x_{25} = -48.6329734602127$$
$$x_{26} = 70.6433593524237$$
$$x_{27} = -64.356022448055$$
$$x_{28} = -212.043355752614$$
$$x_{29} = 73.7867621864701$$
$$x_{30} = -95.7872531120814$$
$$x_{31} = 45.4871087849823$$
$$x_{32} = 10.7130109882558$$
$$x_{33} = 58.0677849921727$$
$$x_{34} = 23.4336891696933$$
$$x_{35} = -70.6433593524237$$
$$x_{36} = -73.7867621864701$$
$$x_{37} = 83.2161494163012$$
$$x_{38} = 3.87023858022217$$
$$x_{39} = 39.1933145376163$$
$$x_{40} = 20.2720010891386$$
$$x_{41} = 17.1027407890472$$
$$x_{42} = -20.2720010891386$$
$$x_{43} = 61.2120336791686$$
$$x_{44} = -7.44308705395446$$
$$x_{45} = 92.644597693687$$
$$x_{46} = 76.9300169373264$$
$$x_{47} = 48.6329734602127$$
$$x_{48} = -51.7783178293904$$
$$x_{49} = -67.499787704256$$
$$x_{50} = -54.9232316104305$$
$$x_{51} = 42.3406072405316$$
$$x_{52} = 95.7872531120814$$
$$x_{53} = 54.9232316104305$$
$$x_{54} = -86.3590546280651$$
$$x_{55} = -45.4871087849823$$
$$x_{56} = 29.7441555138788$$
$$x_{57} = -29.7441555138788$$
$$x_{58} = 1148.2495022125$$
$$x_{59} = 64.356022448055$$
$$x_{60} = 80.0731410729394$$
$$x_{61} = -58.0677849921727$$
$$x_{62} = 13.9205214266357$$
$$x_{63} = 32.8954402073454$$
$$x_{64} = -76.9300169373264$$
$$x_{65} = 230.894066823744$$
$$x_{66} = -32.8954402073454$$
Signos de extremos en los puntos:
(-13.920521426635718, -0.0716515891912541)
(-3.870238580222165, 0.248692236445789)
(-98.92984096773947, 0.0101076571692611)
(36.04502200337846, 0.027732411296845)
(-89.5018675871561, -0.0111722538908926)
(-10.713010988255775, 0.0929394035698334)
(7.443087053954458, -0.133143202113607)
(-83.21614941630125, -0.012016030684702)
(98.92984096773947, 0.0101076571692611)
(26.590555025852712, -0.0375807705734432)
(-42.34060724053156, 0.0236114046136127)
(51.77831782939038, -0.0193095024183706)
(-23.433689169693317, 0.0426347989465758)
(-39.19331453761631, -0.0255062545726498)
(-17.102740789047186, 0.0583704306894909)
(67.499787704256, 0.0148132358906579)
(-80.07314107293935, 0.012487608370541)
(89.5018675871561, -0.0111722538908926)
(-36.04502200337846, 0.027732411296845)
(-92.64459769368703, 0.0107933087887867)
(86.35905462806514, 0.0115787854311861)
(-26.590555025852712, -0.0375807705734432)
(-61.212033679168634, 0.016334477440491)
(249.7446034962173, 0.00400405842499421)
(-48.63297346021267, 0.0205578354154553)
(70.64335935242372, -0.0141541941579985)
(-64.35602244805503, -0.0155366857700825)
(-212.04335575261447, 0.00471596422453044)
(73.78676218647006, 0.0135513220378694)
(-95.7872531120814, -0.0104392335849293)
(45.48710878498235, -0.02197893994184)
(10.713010988255775, 0.0929394035698334)
(58.06778499217275, -0.0172186996097649)
(23.433689169693317, 0.0426347989465758)
(-70.64335935242372, -0.0141541941579985)
(-73.78676218647006, 0.0135513220378694)
(83.21614941630125, -0.012016030684702)
(3.870238580222165, 0.248692236445789)
(39.19331453761631, -0.0255062545726498)
(20.272001089138612, -0.0492692013251242)
(17.102740789047186, 0.0583704306894909)
(-20.272001089138612, -0.0492692013251242)
(61.212033679168634, 0.016334477440491)
(-7.443087053954458, -0.133143202113607)
(92.64459769368703, 0.0107933087887867)
(76.93001693732643, -0.0129977291788139)
(48.63297346021267, 0.0205578354154553)
(-51.77831782939038, -0.0193095024183706)
(-67.499787704256, 0.0148132358906579)
(-54.92323161043051, 0.0182042144946165)
(42.34060724053156, 0.0236114046136127)
(95.7872531120814, -0.0104392335849293)
(54.92323161043051, 0.0182042144946165)
(-86.35905462806514, 0.0115787854311861)
(-45.48710878498235, -0.02197893994184)
(29.744155513878802, 0.0336010649596176)
(-29.744155513878802, 0.0336010649596176)
(1148.2495022124986, 0.000870890532804869)
(64.35602244805503, -0.0155366857700825)
(80.07314107293935, 0.012487608370541)
(-58.06778499217275, -0.0172186996097649)
(13.920521426635718, -0.0716515891912541)
(32.89544020734542, -0.0303853130040648)
(-76.93001693732643, -0.0129977291788139)
(230.8940668237441, 0.0043309498383777)
(-32.89544020734542, -0.0303853130040648)
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:
$$x_{1} = -13.9205214266357$$
$$x_{2} = -89.5018675871561$$
$$x_{3} = 7.44308705395446$$
$$x_{4} = -83.2161494163012$$
$$x_{5} = 26.5905550258527$$
$$x_{6} = 51.7783178293904$$
$$x_{7} = -39.1933145376163$$
$$x_{8} = 89.5018675871561$$
$$x_{9} = -26.5905550258527$$
$$x_{10} = 70.6433593524237$$
$$x_{11} = -64.356022448055$$
$$x_{12} = -95.7872531120814$$
$$x_{13} = 45.4871087849823$$
$$x_{14} = 58.0677849921727$$
$$x_{15} = -70.6433593524237$$
$$x_{16} = 83.2161494163012$$
$$x_{17} = 39.1933145376163$$
$$x_{18} = 20.2720010891386$$
$$x_{19} = -20.2720010891386$$
$$x_{20} = -7.44308705395446$$
$$x_{21} = 76.9300169373264$$
$$x_{22} = -51.7783178293904$$
$$x_{23} = 95.7872531120814$$
$$x_{24} = -45.4871087849823$$
$$x_{25} = 64.356022448055$$
$$x_{26} = -58.0677849921727$$
$$x_{27} = 13.9205214266357$$
$$x_{28} = 32.8954402073454$$
$$x_{29} = -76.9300169373264$$
$$x_{30} = -32.8954402073454$$
Puntos máximos de la función:
$$x_{30} = -3.87023858022217$$
$$x_{30} = -98.9298409677395$$
$$x_{30} = 36.0450220033785$$
$$x_{30} = -10.7130109882558$$
$$x_{30} = 98.9298409677395$$
$$x_{30} = -42.3406072405316$$
$$x_{30} = -23.4336891696933$$
$$x_{30} = -17.1027407890472$$
$$x_{30} = 67.499787704256$$
$$x_{30} = -80.0731410729394$$
$$x_{30} = -36.0450220033785$$
$$x_{30} = -92.644597693687$$
$$x_{30} = 86.3590546280651$$
$$x_{30} = -61.2120336791686$$
$$x_{30} = 249.744603496217$$
$$x_{30} = -48.6329734602127$$
$$x_{30} = -212.043355752614$$
$$x_{30} = 73.7867621864701$$
$$x_{30} = 10.7130109882558$$
$$x_{30} = 23.4336891696933$$
$$x_{30} = -73.7867621864701$$
$$x_{30} = 3.87023858022217$$
$$x_{30} = 17.1027407890472$$
$$x_{30} = 61.2120336791686$$
$$x_{30} = 92.644597693687$$
$$x_{30} = 48.6329734602127$$
$$x_{30} = -67.499787704256$$
$$x_{30} = -54.9232316104305$$
$$x_{30} = 42.3406072405316$$
$$x_{30} = 54.9232316104305$$
$$x_{30} = -86.3590546280651$$
$$x_{30} = 29.7441555138788$$
$$x_{30} = -29.7441555138788$$
$$x_{30} = 1148.2495022125$$
$$x_{30} = 80.0731410729394$$
$$x_{30} = 230.894066823744$$
Decrece en los intervalos
$$\left[95.7872531120814, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -95.7872531120814\right]$$