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$$\frac{2 \sin{\left(\frac{x}{3} \right)} \cos{\left(\frac{x}{3} \right)}}{3} + \frac{\sin{\left(\frac{x}{5} \right)}}{5} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = -79.8725525823422$$
$$x_{2} = -22.6166405439692$$
$$x_{3} = -14.3752270253516$$
$$x_{4} = -43.08439670267$$
$$x_{5} = 79.8725525823422$$
$$x_{6} = 60.9228693229623$$
$$x_{7} = -5.5648558902089$$
$$x_{8} = 65.5070766722175$$
$$x_{9} = 85.7797994162419$$
$$x_{10} = -748.417380971341$$
$$x_{11} = 99.8126354979027$$
$$x_{12} = -57.3893306393696$$
$$x_{13} = 33.3249102847315$$
$$x_{14} = -99.8126354979027$$
$$x_{15} = -36.8584489683242$$
$$x_{16} = -19.4855370618986$$
$$x_{17} = 94.2477796076938$$
$$x_{18} = -60.9228693229623$$
$$x_{19} = 0$$
$$x_{20} = -71.6311390637246$$
$$x_{21} = 57.3893306393696$$
$$x_{22} = -85.7797994162419$$
$$x_{23} = -65.5070766722175$$
$$x_{24} = -51.1633829050238$$
$$x_{25} = -74.7622425457952$$
$$x_{26} = 28.7407029354763$$
$$x_{27} = 71.6311390637246$$
$$x_{28} = -28.7407029354763$$
$$x_{29} = -88.6829237174849$$
$$x_{30} = -47.1238898038469$$
$$x_{31} = 74.7622425457952$$
$$x_{32} = 14.3752270253516$$
$$x_{33} = 19.4855370618986$$
$$x_{34} = 36.8584489683242$$
$$x_{35} = -8.46798019145186$$
$$x_{36} = -33.3249102847315$$
$$x_{37} = 471.238898038469$$
$$x_{38} = 8.46798019145186$$
$$x_{39} = 22.6166405439692$$
$$x_{40} = 51.1633829050238$$
$$x_{41} = 5.5648558902089$$
$$x_{42} = 47.1238898038469$$
$$x_{43} = -94.2477796076938$$
$$x_{44} = 43.08439670267$$
$$x_{45} = 88.6829237174849$$
Signos de extremos en los puntos:
(-79.87255258234218, 3.95840235727146)
(-22.616640543969176, 3.09189055929575)
(-14.375227025351618, 3.95840235727146)
(-43.08439670267002, 3.64154709336703)
(79.87255258234218, 3.95840235727146)
(60.92286932296227, 2.05934919395075)
(-5.564855890208903, 2.47940725500054)
(65.50707667221747, 1.16372679891789)
(85.77979941624194, 2.22080689513273)
(-748.4173809713415, 2.47940725500051)
(99.8126354979027, 2.47940725500054)
(-57.3893306393696, 1.61267904410199)
(33.32491028473153, 2.05934919395075)
(-99.8126354979027, 2.47940725500054)
(-36.858448968324204, 1.61267904410199)
(-19.4855370618986, 2.77219080296186)
(94.2477796076938, 1)
(-60.92286932296227, 2.05934919395075)
(0, 1)
(-71.63113906372462, 3.09189055929575)
(57.3893306393696, 1.61267904410199)
(-85.77979941624194, 2.22080689513273)
(-65.50707667221747, 1.16372679891789)
(-51.16338290502378, 3.64154709336703)
(-74.76224254579519, 2.77219080296186)
(28.740702935476335, 1.16372679891789)
(71.63113906372462, 3.09189055929575)
(-28.740702935476335, 1.16372679891789)
(-88.6829237174849, 2.47940725500054)
(-47.1238898038469, 3)
(74.76224254579519, 2.77219080296186)
(14.375227025351618, 3.95840235727146)
(19.4855370618986, 2.77219080296186)
(36.858448968324204, 1.61267904410199)
(-8.467980191451856, 2.22080689513273)
(-33.32491028473153, 2.05934919395075)
(471.23889803846896, 1)
(8.467980191451856, 2.22080689513273)
(22.616640543969176, 3.09189055929575)
(51.16338290502378, 3.64154709336703)
(5.564855890208903, 2.47940725500054)
(47.1238898038469, 3)
(-94.2477796076938, 1)
(43.08439670267002, 3.64154709336703)
(88.6829237174849, 2.47940725500054)
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:
$$x_{1} = 65.5070766722175$$
$$x_{2} = 85.7797994162419$$
$$x_{3} = -57.3893306393696$$
$$x_{4} = -36.8584489683242$$
$$x_{5} = -19.4855370618986$$
$$x_{6} = 94.2477796076938$$
$$x_{7} = 0$$
$$x_{8} = 57.3893306393696$$
$$x_{9} = -85.7797994162419$$
$$x_{10} = -65.5070766722175$$
$$x_{11} = -74.7622425457952$$
$$x_{12} = 28.7407029354763$$
$$x_{13} = -28.7407029354763$$
$$x_{14} = -47.1238898038469$$
$$x_{15} = 74.7622425457952$$
$$x_{16} = 19.4855370618986$$
$$x_{17} = 36.8584489683242$$
$$x_{18} = -8.46798019145186$$
$$x_{19} = 471.238898038469$$
$$x_{20} = 8.46798019145186$$
$$x_{21} = 47.1238898038469$$
$$x_{22} = -94.2477796076938$$
Puntos máximos de la función:
$$x_{22} = -79.8725525823422$$
$$x_{22} = -22.6166405439692$$
$$x_{22} = -14.3752270253516$$
$$x_{22} = -43.08439670267$$
$$x_{22} = 79.8725525823422$$
$$x_{22} = 60.9228693229623$$
$$x_{22} = -5.5648558902089$$
$$x_{22} = -748.417380971341$$
$$x_{22} = 99.8126354979027$$
$$x_{22} = 33.3249102847315$$
$$x_{22} = -99.8126354979027$$
$$x_{22} = -60.9228693229623$$
$$x_{22} = -71.6311390637246$$
$$x_{22} = -51.1633829050238$$
$$x_{22} = 71.6311390637246$$
$$x_{22} = -88.6829237174849$$
$$x_{22} = 14.3752270253516$$
$$x_{22} = -33.3249102847315$$
$$x_{22} = 22.6166405439692$$
$$x_{22} = 51.1633829050238$$
$$x_{22} = 5.5648558902089$$
$$x_{22} = 43.08439670267$$
$$x_{22} = 88.6829237174849$$
Decrece en los intervalos
$$\left[471.238898038469, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -94.2477796076938\right]$$