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$$- 4 x^{3} \sin{\left(x \right)} + 12 x^{2} \cos{\left(x \right)} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = -9.72402747617551$$
$$x_{2} = -72.2981021067071$$
$$x_{3} = -87.9986725257711$$
$$x_{4} = -84.8583399660622$$
$$x_{5} = 84.8583399660622$$
$$x_{6} = 81.7181040853573$$
$$x_{7} = 56.6016202331048$$
$$x_{8} = 40.913898225293$$
$$x_{9} = 3.80876221919969$$
$$x_{10} = 0$$
$$x_{11} = -75.4379705139506$$
$$x_{12} = -97.4201569811411$$
$$x_{13} = 97.4201569811411$$
$$x_{14} = 22.12591435735$$
$$x_{15} = 100.560788770886$$
$$x_{16} = -15.8945130636842$$
$$x_{17} = -66.0188560490172$$
$$x_{18} = 66.0188560490172$$
$$x_{19} = -22.12591435735$$
$$x_{20} = 69.1583898858035$$
$$x_{21} = -28.3796522911214$$
$$x_{22} = -34.6438990396267$$
$$x_{23} = -78.5779764426249$$
$$x_{24} = 37.7783560989567$$
$$x_{25} = -25.2509941253717$$
$$x_{26} = -59.7404355133729$$
$$x_{27} = 12.7966483902814$$
$$x_{28} = -62.8795272030449$$
$$x_{29} = -6.70395577578075$$
$$x_{30} = -1.19245882933643$$
$$x_{31} = 91.1390917936668$$
$$x_{32} = 47.1873806732917$$
$$x_{33} = 44.0502961191214$$
$$x_{34} = 1.19245882933643$$
$$x_{35} = -56.6016202331048$$
$$x_{36} = 15.8945130636842$$
$$x_{37} = -53.4631297645908$$
$$x_{38} = 62.8795272030449$$
$$x_{39} = -40.913898225293$$
$$x_{40} = 19.0061082873963$$
$$x_{41} = 6.70395577578075$$
$$x_{42} = 78.5779764426249$$
$$x_{43} = 72.2981021067071$$
$$x_{44} = -31.510845756676$$
$$x_{45} = 31.510845756676$$
$$x_{46} = -44.0502961191214$$
$$x_{47} = -50.325024483292$$
$$x_{48} = 75.4379705139506$$
$$x_{49} = 50.325024483292$$
$$x_{50} = 59.7404355133729$$
$$x_{51} = -94.2795891235637$$
$$x_{52} = -19.0061082873963$$
$$x_{53} = -100.560788770886$$
$$x_{54} = -37.7783560989567$$
$$x_{55} = -3.80876221919969$$
$$x_{56} = 87.9986725257711$$
$$x_{57} = 9.72402747617551$$
$$x_{58} = 53.4631297645908$$
$$x_{59} = -12.7966483902814$$
$$x_{60} = -81.7181040853573$$
$$x_{61} = 25.2509941253717$$
$$x_{62} = 28.3796522911214$$
$$x_{63} = 34.6438990396267$$
$$x_{64} = -91.1390917936668$$
$$x_{65} = 94.2795891235637$$
$$x_{66} = -69.1583898858035$$
$$x_{67} = -47.1873806732917$$
Signos de extremos en los puntos:
(-9.72402747617551, 3514.43550360095)
(-72.29810210670713, 1510313.53335791)
(-87.99867252577111, -2724182.04559702)
(-84.85833996606219, 2442712.51279198)
(84.85833996606219, -2442712.51279198)
(81.71810408535728, 2181335.04597451)
(56.60162023310481, 724331.58435522)
(40.91389822529297, -273217.30613698)
(3.808762219199689, -173.620051816331)
(0, 0)
(-75.43797051395065, -1715879.70709431)
(-97.42015698114113, 3696584.54759441)
(97.42015698114113, -3696584.54759441)
(22.125914357349984, -42934.6461855845)
(100.56078877088648, 4065863.85192867)
(-15.894513063684203, 15783.3987495175)
(-66.01885604901719, 1149783.42283034)
(66.01885604901719, -1149783.42283034)
(-22.125914357349984, 42934.6461855845)
(69.15838988580347, 1321862.8222492)
(-28.37965229112142, 90921.8253044151)
(-34.64389903962671, 165698.289727675)
(-78.57797644262494, 1939305.49434903)
(37.77835609895673, 214992.901302738)
(-25.25099412537165, -63951.6564937612)
(-59.74043551337287, 851761.955001316)
(12.796648390281426, 8160.76026338815)
(-62.87952720304487, -993331.184104639)
(-6.703955775780748, -1100.06136834542)
(-1.1924588293364287, -2.50529519287726)
(91.13909179366682, -3026487.7951639)
(47.18738067329166, -419432.109446505)
(44.05029611912139, 341115.657892607)
(1.1924588293364287, 2.50529519287726)
(-56.60162023310481, -724331.58435522)
(15.894513063684203, -15783.3987495175)
(-53.463129764590846, 610295.920879583)
(62.87952720304487, 993331.184104639)
(-40.91389822529297, 273217.30613698)
(19.006108287396344, 27126.6224448194)
(6.703955775780748, 1100.06136834542)
(78.57797644262494, -1939305.49434903)
(72.29810210670713, -1510313.53335791)
(-31.51084575667604, -124589.316590486)
(31.51084575667604, 124589.316590486)
(-44.05029611912139, -341115.657892607)
(-50.32502448329199, -508910.813128767)
(75.43797051395065, 1715879.70709431)
(50.32502448329199, 508910.813128767)
(59.74043551337287, -851761.955001316)
(-94.27958912356374, -3350373.91224916)
(-19.006108287396344, -27126.6224448194)
(-100.56078877088648, -4065863.85192867)
(-37.77835609895673, -214992.901302738)
(-3.808762219199689, 173.620051816331)
(87.99867252577111, 2724182.04559702)
(9.72402747617551, -3514.43550360095)
(53.463129764590846, -610295.920879583)
(-12.796648390281426, -8160.76026338815)
(-81.71810408535728, -2181335.04597451)
(25.25099412537165, 63951.6564937612)
(28.37965229112142, -90921.8253044151)
(34.64389903962671, -165698.289727675)
(-91.13909179366682, 3026487.7951639)
(94.27958912356374, 3350373.91224916)
(-69.15838988580347, -1321862.8222492)
(-47.18738067329166, 419432.109446505)
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:
$$x_{1} = -87.9986725257711$$
$$x_{2} = 84.8583399660622$$
$$x_{3} = 40.913898225293$$
$$x_{4} = 3.80876221919969$$
$$x_{5} = -75.4379705139506$$
$$x_{6} = 97.4201569811411$$
$$x_{7} = 22.12591435735$$
$$x_{8} = 66.0188560490172$$
$$x_{9} = -25.2509941253717$$
$$x_{10} = -62.8795272030449$$
$$x_{11} = -6.70395577578075$$
$$x_{12} = -1.19245882933643$$
$$x_{13} = 91.1390917936668$$
$$x_{14} = 47.1873806732917$$
$$x_{15} = -56.6016202331048$$
$$x_{16} = 15.8945130636842$$
$$x_{17} = 78.5779764426249$$
$$x_{18} = 72.2981021067071$$
$$x_{19} = -31.510845756676$$
$$x_{20} = -44.0502961191214$$
$$x_{21} = -50.325024483292$$
$$x_{22} = 59.7404355133729$$
$$x_{23} = -94.2795891235637$$
$$x_{24} = -19.0061082873963$$
$$x_{25} = -100.560788770886$$
$$x_{26} = -37.7783560989567$$
$$x_{27} = 9.72402747617551$$
$$x_{28} = 53.4631297645908$$
$$x_{29} = -12.7966483902814$$
$$x_{30} = -81.7181040853573$$
$$x_{31} = 28.3796522911214$$
$$x_{32} = 34.6438990396267$$
$$x_{33} = -69.1583898858035$$
Puntos máximos de la función:
$$x_{33} = -9.72402747617551$$
$$x_{33} = -72.2981021067071$$
$$x_{33} = -84.8583399660622$$
$$x_{33} = 81.7181040853573$$
$$x_{33} = 56.6016202331048$$
$$x_{33} = -97.4201569811411$$
$$x_{33} = 100.560788770886$$
$$x_{33} = -15.8945130636842$$
$$x_{33} = -66.0188560490172$$
$$x_{33} = -22.12591435735$$
$$x_{33} = 69.1583898858035$$
$$x_{33} = -28.3796522911214$$
$$x_{33} = -34.6438990396267$$
$$x_{33} = -78.5779764426249$$
$$x_{33} = 37.7783560989567$$
$$x_{33} = -59.7404355133729$$
$$x_{33} = 12.7966483902814$$
$$x_{33} = 44.0502961191214$$
$$x_{33} = 1.19245882933643$$
$$x_{33} = -53.4631297645908$$
$$x_{33} = 62.8795272030449$$
$$x_{33} = -40.913898225293$$
$$x_{33} = 19.0061082873963$$
$$x_{33} = 6.70395577578075$$
$$x_{33} = 31.510845756676$$
$$x_{33} = 75.4379705139506$$
$$x_{33} = 50.325024483292$$
$$x_{33} = -3.80876221919969$$
$$x_{33} = 87.9986725257711$$
$$x_{33} = 25.2509941253717$$
$$x_{33} = -91.1390917936668$$
$$x_{33} = 94.2795891235637$$
$$x_{33} = -47.1873806732917$$
Decrece en los intervalos
$$\left[97.4201569811411, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -100.560788770886\right]$$