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$$- x^{3} \sin{\left(x \right)} + 3 x^{2} \cos{\left(x \right)} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = -28.3796522911214$$
$$x_{2} = -72.2981021067071$$
$$x_{3} = -40.913898225293$$
$$x_{4} = -62.8795272030449$$
$$x_{5} = 97.4201569811411$$
$$x_{6} = 34.6438990396267$$
$$x_{7} = -9.72402747617551$$
$$x_{8} = -47.1873806732917$$
$$x_{9} = 78.5779764426249$$
$$x_{10} = 59.7404355133729$$
$$x_{11} = -50.325024483292$$
$$x_{12} = 72.2981021067071$$
$$x_{13} = -91.1390917936668$$
$$x_{14} = 69.1583898858035$$
$$x_{15} = 3.80876221919969$$
$$x_{16} = 47.1873806732917$$
$$x_{17} = -6.70395577578075$$
$$x_{18} = -44.0502961191214$$
$$x_{19} = 100.560788770886$$
$$x_{20} = -19.0061082873963$$
$$x_{21} = -81.7181040853573$$
$$x_{22} = 37.7783560989567$$
$$x_{23} = 66.0188560490172$$
$$x_{24} = -25.2509941253717$$
$$x_{25} = 44.0502961191214$$
$$x_{26} = 56.6016202331048$$
$$x_{27} = 91.1390917936668$$
$$x_{28} = 1.19245882933643$$
$$x_{29} = 87.9986725257711$$
$$x_{30} = -94.2795891235637$$
$$x_{31} = 81.7181040853573$$
$$x_{32} = -12.7966483902814$$
$$x_{33} = 40.913898225293$$
$$x_{34} = 31.510845756676$$
$$x_{35} = 19.0061082873963$$
$$x_{36} = -56.6016202331048$$
$$x_{37} = -59.7404355133729$$
$$x_{38} = -87.9986725257711$$
$$x_{39} = -75.4379705139506$$
$$x_{40} = 28.3796522911214$$
$$x_{41} = -66.0188560490172$$
$$x_{42} = 0$$
$$x_{43} = -34.6438990396267$$
$$x_{44} = -37.7783560989567$$
$$x_{45} = -97.4201569811411$$
$$x_{46} = -3.80876221919969$$
$$x_{47} = 25.2509941253717$$
$$x_{48} = -84.8583399660622$$
$$x_{49} = 15.8945130636842$$
$$x_{50} = -100.560788770886$$
$$x_{51} = 75.4379705139506$$
$$x_{52} = 9.72402747617551$$
$$x_{53} = -15.8945130636842$$
$$x_{54} = -31.510845756676$$
$$x_{55} = 6.70395577578075$$
$$x_{56} = -1.19245882933643$$
$$x_{57} = -22.12591435735$$
$$x_{58} = -69.1583898858035$$
$$x_{59} = -53.4631297645908$$
$$x_{60} = 62.8795272030449$$
$$x_{61} = 22.12591435735$$
$$x_{62} = 53.4631297645908$$
$$x_{63} = 12.7966483902814$$
$$x_{64} = 94.2795891235637$$
$$x_{65} = 50.325024483292$$
$$x_{66} = 84.8583399660622$$
$$x_{67} = -78.5779764426249$$
Signos de extremos en los puntos:
(-28.37965229112142, 22730.4563261038)
(-72.29810210670713, 377578.383339478)
(-40.91389822529297, 68304.326534245)
(-62.87952720304487, -248332.79602616)
(97.42015698114113, -924146.136898602)
(34.64389903962671, -41424.5724319187)
(-9.72402747617551, 878.608875900237)
(-47.18738067329166, 104858.027361626)
(78.57797644262494, -484826.373587257)
(59.74043551337287, -212940.488750329)
(-50.32502448329199, -127227.703282192)
(72.29810210670713, -377578.383339478)
(-91.13909179366682, 756621.948790976)
(69.15838988580347, 330465.705562301)
(3.808762219199689, -43.4050129540828)
(47.18738067329166, -104858.027361626)
(-6.703955775780748, -275.015342086354)
(-44.05029611912139, -85278.9144731517)
(100.56078877088648, 1016465.96298217)
(-19.006108287396344, -6781.65561120486)
(-81.71810408535728, -545333.761493627)
(37.77835609895673, 53748.2253256845)
(66.01885604901719, -287445.855707585)
(-25.25099412537165, -15987.9141234403)
(44.05029611912139, 85278.9144731517)
(56.60162023310481, 181082.896088805)
(91.13909179366682, -756621.948790976)
(1.1924588293364287, 0.626323798219316)
(87.99867252577111, 681045.511399255)
(-94.27958912356374, -837593.47806229)
(81.71810408535728, 545333.761493627)
(-12.796648390281426, -2040.19006584704)
(40.91389822529297, -68304.326534245)
(31.51084575667604, 31147.3291476214)
(19.006108287396344, 6781.65561120486)
(-56.60162023310481, -181082.896088805)
(-59.74043551337287, 212940.488750329)
(-87.99867252577111, -681045.511399255)
(-75.43797051395065, -428969.926773577)
(28.37965229112142, -22730.4563261038)
(-66.01885604901719, 287445.855707585)
(0, 0)
(-34.64389903962671, 41424.5724319187)
(-37.77835609895673, -53748.2253256845)
(-97.42015698114113, 924146.136898602)
(-3.808762219199689, 43.4050129540828)
(25.25099412537165, 15987.9141234403)
(-84.85833996606219, 610678.128197996)
(15.894513063684203, -3945.84968737938)
(-100.56078877088648, -1016465.96298217)
(75.43797051395065, 428969.926773577)
(9.72402747617551, -878.608875900237)
(-15.894513063684203, 3945.84968737938)
(-31.51084575667604, -31147.3291476214)
(6.703955775780748, 275.015342086354)
(-1.1924588293364287, -0.626323798219316)
(-22.125914357349984, 10733.6615463961)
(-69.15838988580347, -330465.705562301)
(-53.463129764590846, 152573.980219896)
(62.87952720304487, 248332.79602616)
(22.125914357349984, -10733.6615463961)
(53.463129764590846, -152573.980219896)
(12.796648390281426, 2040.19006584704)
(94.27958912356374, 837593.47806229)
(50.32502448329199, 127227.703282192)
(84.85833996606219, -610678.128197996)
(-78.57797644262494, 484826.373587257)
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:
$$x_{1} = -62.8795272030449$$
$$x_{2} = 97.4201569811411$$
$$x_{3} = 34.6438990396267$$
$$x_{4} = 78.5779764426249$$
$$x_{5} = 59.7404355133729$$
$$x_{6} = -50.325024483292$$
$$x_{7} = 72.2981021067071$$
$$x_{8} = 3.80876221919969$$
$$x_{9} = 47.1873806732917$$
$$x_{10} = -6.70395577578075$$
$$x_{11} = -44.0502961191214$$
$$x_{12} = -19.0061082873963$$
$$x_{13} = -81.7181040853573$$
$$x_{14} = 66.0188560490172$$
$$x_{15} = -25.2509941253717$$
$$x_{16} = 91.1390917936668$$
$$x_{17} = -94.2795891235637$$
$$x_{18} = -12.7966483902814$$
$$x_{19} = 40.913898225293$$
$$x_{20} = -56.6016202331048$$
$$x_{21} = -87.9986725257711$$
$$x_{22} = -75.4379705139506$$
$$x_{23} = 28.3796522911214$$
$$x_{24} = -37.7783560989567$$
$$x_{25} = 15.8945130636842$$
$$x_{26} = -100.560788770886$$
$$x_{27} = 9.72402747617551$$
$$x_{28} = -31.510845756676$$
$$x_{29} = -1.19245882933643$$
$$x_{30} = -69.1583898858035$$
$$x_{31} = 22.12591435735$$
$$x_{32} = 53.4631297645908$$
$$x_{33} = 84.8583399660622$$
Puntos máximos de la función:
$$x_{33} = -28.3796522911214$$
$$x_{33} = -72.2981021067071$$
$$x_{33} = -40.913898225293$$
$$x_{33} = -9.72402747617551$$
$$x_{33} = -47.1873806732917$$
$$x_{33} = -91.1390917936668$$
$$x_{33} = 69.1583898858035$$
$$x_{33} = 100.560788770886$$
$$x_{33} = 37.7783560989567$$
$$x_{33} = 44.0502961191214$$
$$x_{33} = 56.6016202331048$$
$$x_{33} = 1.19245882933643$$
$$x_{33} = 87.9986725257711$$
$$x_{33} = 81.7181040853573$$
$$x_{33} = 31.510845756676$$
$$x_{33} = 19.0061082873963$$
$$x_{33} = -59.7404355133729$$
$$x_{33} = -66.0188560490172$$
$$x_{33} = -34.6438990396267$$
$$x_{33} = -97.4201569811411$$
$$x_{33} = -3.80876221919969$$
$$x_{33} = 25.2509941253717$$
$$x_{33} = -84.8583399660622$$
$$x_{33} = 75.4379705139506$$
$$x_{33} = -15.8945130636842$$
$$x_{33} = 6.70395577578075$$
$$x_{33} = -22.12591435735$$
$$x_{33} = -53.4631297645908$$
$$x_{33} = 62.8795272030449$$
$$x_{33} = 12.7966483902814$$
$$x_{33} = 94.2795891235637$$
$$x_{33} = 50.325024483292$$
$$x_{33} = -78.5779764426249$$
Decrece en los intervalos
$$\left[97.4201569811411, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -100.560788770886\right]$$