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$$- \frac{\left(\frac{x}{8} - \frac{1}{8}\right) \cos{\left(\frac{x}{2} \right)}}{4} + \frac{3 \sin{\left(\frac{x}{2} \right)}}{16} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = -72.0928234997439$$
$$x_{2} = 40.5395084538546$$
$$x_{3} = 21.4195644604296$$
$$x_{4} = 97.2648772499214$$
$$x_{5} = -84.6831790656061$$
$$x_{6} = 53.1780992694566$$
$$x_{7} = -53.1865162868496$$
$$x_{8} = -46.8745347430149$$
$$x_{9} = 34.1999323668518$$
$$x_{10} = -0.485362558276441$$
$$x_{11} = 14.8925876523829$$
$$x_{12} = -14.9900112041524$$
$$x_{13} = 46.8637225983274$$
$$x_{14} = 90.9730108075959$$
$$x_{15} = 179.003392323252$$
$$x_{16} = 65.7887551882277$$
$$x_{17} = -78.3889486119439$$
$$x_{18} = 72.0882258411896$$
$$x_{19} = -21.469262706264$$
$$x_{20} = 84.6798429343937$$
$$x_{21} = 59.4857976886522$$
$$x_{22} = -65.7942704913902$$
$$x_{23} = 27.834382800265$$
$$x_{24} = -8.2765436209072$$
$$x_{25} = 8.00876143267673$$
$$x_{26} = -59.4925355374796$$
$$x_{27} = 103.555681188122$$
$$x_{28} = -34.2200439239142$$
$$x_{29} = -90.9759025986992$$
$$x_{30} = 78.3850572624003$$
$$x_{31} = -40.5539051427507$$
$$x_{32} = -97.2674079064454$$
$$x_{33} = 1416.84981134451$$
$$x_{34} = -27.8644346285563$$
Signos de extremos en los puntos:
(-72.09282349974393, 4.59389354372914)
(40.53950845385463, -2.51826357281154)
(21.419564460429562, 1.36541620628308)
(97.26487724992135, 6.03600591986278)
(-84.68317906560608, 5.37704419460649)
(53.17809926945661, -3.29690091496593)
(-53.18651628684961, -3.42111264145565)
(-46.87453474301491, 3.03111013420797)
(34.19993236685178, 2.13083942229892)
(-0.4853625582764406, -0.507657432693553)
(14.89258765238292, -0.995365828004467)
(-14.990011204152419, -1.11133040564379)
(46.863722598327406, 2.90712242054539)
(90.97301080759586, -5.64412041401817)
(179.00339232325229, -11.1357413416032)
(65.78875518822768, -4.07815101169087)
(-78.38894861194393, -4.98538014145867)
(72.08822584118958, 4.46932432786362)
(-21.46926270626403, 1.4857842065916)
(84.67984293439375, 5.25235682700065)
(59.48579768865222, 3.68730412130719)
(-65.79427049139015, -4.20263430613656)
(27.834382800265022, -1.74583718643573)
(-8.276543620907198, 0.758375279900881)
(8.008761432676732, 0.657927652486149)
(-59.49253553747962, 3.81167297570818)
(103.55568118812153, -6.42799097104595)
(-34.220043923914226, 2.25395858951351)
(-90.97590259869924, -5.76884940626111)
(78.38505726240027, -4.86074477885135)
(-40.55390514275072, -2.64191627024295)
(-97.2674079064454, 6.16076874563619)
(1416.8498113445055, 88.491937493712)
(-27.86443462855629, -1.86803002902737)
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:
$$x_{1} = 40.5395084538546$$
$$x_{2} = 53.1780992694566$$
$$x_{3} = -53.1865162868496$$
$$x_{4} = -0.485362558276441$$
$$x_{5} = 14.8925876523829$$
$$x_{6} = -14.9900112041524$$
$$x_{7} = 90.9730108075959$$
$$x_{8} = 179.003392323252$$
$$x_{9} = 65.7887551882277$$
$$x_{10} = -78.3889486119439$$
$$x_{11} = -65.7942704913902$$
$$x_{12} = 27.834382800265$$
$$x_{13} = 103.555681188122$$
$$x_{14} = -90.9759025986992$$
$$x_{15} = 78.3850572624003$$
$$x_{16} = -40.5539051427507$$
$$x_{17} = -27.8644346285563$$
Puntos máximos de la función:
$$x_{17} = -72.0928234997439$$
$$x_{17} = 21.4195644604296$$
$$x_{17} = 97.2648772499214$$
$$x_{17} = -84.6831790656061$$
$$x_{17} = -46.8745347430149$$
$$x_{17} = 34.1999323668518$$
$$x_{17} = 46.8637225983274$$
$$x_{17} = 72.0882258411896$$
$$x_{17} = -21.469262706264$$
$$x_{17} = 84.6798429343937$$
$$x_{17} = 59.4857976886522$$
$$x_{17} = -8.2765436209072$$
$$x_{17} = 8.00876143267673$$
$$x_{17} = -59.4925355374796$$
$$x_{17} = -34.2200439239142$$
$$x_{17} = -97.2674079064454$$
$$x_{17} = 1416.84981134451$$
Decrece en los intervalos
$$\left[179.003392323252, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -90.9759025986992\right]$$