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$$\frac{x^{2} \cos{\left(x \right)}}{4} + \frac{x \sin{\left(x \right)}}{4} + \frac{\cos{\left(x \right)}}{4} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = -29.8785491411967$$
$$x_{2} = -11.0848217516286$$
$$x_{3} = -48.7152020786718$$
$$x_{4} = 20.4690516301297$$
$$x_{5} = 4.90565966567415$$
$$x_{6} = -92.6877705166201$$
$$x_{7} = 45.5750212470478$$
$$x_{8} = 48.7152020786718$$
$$x_{9} = 83.264212972531$$
$$x_{10} = -33.0169733260496$$
$$x_{11} = 33.0169733260496$$
$$x_{12} = -64.4181679836523$$
$$x_{13} = 92.6877705166201$$
$$x_{14} = -73.8409666666368$$
$$x_{15} = 39.2953345326852$$
$$x_{16} = 42.4350488166082$$
$$x_{17} = 58.1366581602094$$
$$x_{18} = -89.5465561460837$$
$$x_{19} = 29.8785491411967$$
$$x_{20} = 70.6999752105956$$
$$x_{21} = -67.559039597919$$
$$x_{22} = 11.0848217516286$$
$$x_{23} = -51.8555535655761$$
$$x_{24} = 7.97678462750978$$
$$x_{25} = 54.9960465516394$$
$$x_{26} = -54.9960465516394$$
$$x_{27} = 51.8555535655761$$
$$x_{28} = -61.2773701909402$$
$$x_{29} = 1.95588396372027$$
$$x_{30} = -14.2070931146973$$
$$x_{31} = -42.4350488166082$$
$$x_{32} = -70.6999752105956$$
$$x_{33} = -17.3361878945599$$
$$x_{34} = 95.8290096730487$$
$$x_{35} = 14.2070931146973$$
$$x_{36} = -39.2953345326852$$
$$x_{37} = -80.1230908716212$$
$$x_{38} = 23.6042091422871$$
$$x_{39} = -23.6042091422871$$
$$x_{40} = 64.4181679836523$$
$$x_{41} = -45.5750212470478$$
$$x_{42} = -20.4690516301297$$
$$x_{43} = -26.7408639367464$$
$$x_{44} = 36.1559453106627$$
$$x_{45} = -86.4053692621466$$
$$x_{46} = -1.95588396372027$$
$$x_{47} = -98.9702712571724$$
$$x_{48} = 26.7408639367464$$
$$x_{49} = 73.8409666666368$$
$$x_{50} = 67.559039597919$$
$$x_{51} = 80.1230908716212$$
$$x_{52} = 86.4053692621466$$
$$x_{53} = 61.2773701909402$$
$$x_{54} = -95.8290096730487$$
$$x_{55} = 89.5465561460837$$
$$x_{56} = -7.97678462750978$$
$$x_{57} = 17.3361878945599$$
$$x_{58} = -76.9820071395694$$
$$x_{59} = 98.9702712571724$$
$$x_{60} = -4.90565966567415$$
$$x_{61} = -83.264212972531$$
$$x_{62} = -58.1366581602094$$
$$x_{63} = 76.9820071395694$$
$$x_{64} = -36.1559453106627$$
Signos de extremos en los puntos:
(-29.878549141196686, 222.807727876424)
(-11.084821751628633, 30.3490672449957)
(-48.7152020786718, 592.918030981887)
(20.469051630129695, 104.372225362231)
(4.905659665674149, -5.66879988485142)
(-92.68777051662009, 2147.38078447737)
(45.57502124704777, 518.895986097301)
(48.7152020786718, -592.918030981887)
(83.26421297253098, 1732.85739412349)
(-33.01697332604957, -272.155789935058)
(33.01697332604957, 272.155789935058)
(-64.41816798365227, -1037.05026470801)
(92.68777051662009, -2147.38078447737)
(-73.84096666663683, 1362.74722133438)
(39.29533453268523, 385.656293835577)
(42.43504881660823, -449.808740683475)
(58.13665816020942, 844.592968029985)
(-89.54655614608366, -2004.27151901738)
(29.878549141196686, -222.807727876424)
(70.69997521059557, 1249.24676742635)
(-67.55903959791895, 1140.68111524883)
(11.084821751628633, -30.3490672449957)
(-51.855553565576145, -671.874875977163)
(7.97678462750978, 15.5432002945774)
(54.99604655163941, -755.766521546023)
(-54.99604655163941, 755.766521546023)
(51.855553565576145, 671.874875977163)
(-61.277370190940225, 938.354215685465)
(1.9558839637202659, 0.702653980972551)
(-14.207093114697331, -50.0888971088226)
(-42.43504881660823, 449.808740683475)
(-70.69997521059557, -1249.24676742635)
(-17.336187894559934, 74.7632271836238)
(95.82900967304874, 2295.42485197891)
(14.207093114697331, 50.0888971088226)
(-39.29533453268523, -385.656293835577)
(-80.12309087162116, 1604.55253462789)
(23.60420914228713, -138.915957372853)
(-23.60420914228713, 138.915957372853)
(64.41816798365227, 1037.05026470801)
(-45.57502124704777, -518.895986097301)
(-20.469051630129695, -104.372225362231)
(-26.740863936746408, -178.394453146302)
(36.15594531066269, -326.438644240618)
(-86.40536926214665, 1866.09705557544)
(-1.9558839637202659, -0.702653980972551)
(-98.9702712571724, 2448.40372154187)
(26.740863936746408, 178.394453146302)
(73.84096666663683, -1362.74722133438)
(67.55903959791895, -1140.68111524883)
(80.12309087162116, -1604.55253462789)
(86.40536926214665, -1866.09705557544)
(61.277370190940225, -938.354215685465)
(-95.82900967304874, -2295.42485197891)
(89.54655614608366, 2004.27151901738)
(-7.97678462750978, -15.5432002945774)
(17.336187894559934, -74.7632271836238)
(-76.98200713956943, -1481.182477048)
(98.9702712571724, -2448.40372154187)
(-4.905659665674149, 5.66879988485142)
(-83.26421297253098, -1732.85739412349)
(-58.13665816020942, -844.592968029985)
(76.98200713956943, 1481.182477048)
(-36.15594531066269, 326.438644240618)
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:
$$x_{1} = 4.90565966567415$$
$$x_{2} = 48.7152020786718$$
$$x_{3} = -33.0169733260496$$
$$x_{4} = -64.4181679836523$$
$$x_{5} = 92.6877705166201$$
$$x_{6} = 42.4350488166082$$
$$x_{7} = -89.5465561460837$$
$$x_{8} = 29.8785491411967$$
$$x_{9} = 11.0848217516286$$
$$x_{10} = -51.8555535655761$$
$$x_{11} = 54.9960465516394$$
$$x_{12} = -14.2070931146973$$
$$x_{13} = -70.6999752105956$$
$$x_{14} = -39.2953345326852$$
$$x_{15} = 23.6042091422871$$
$$x_{16} = -45.5750212470478$$
$$x_{17} = -20.4690516301297$$
$$x_{18} = -26.7408639367464$$
$$x_{19} = 36.1559453106627$$
$$x_{20} = -1.95588396372027$$
$$x_{21} = 73.8409666666368$$
$$x_{22} = 67.559039597919$$
$$x_{23} = 80.1230908716212$$
$$x_{24} = 86.4053692621466$$
$$x_{25} = 61.2773701909402$$
$$x_{26} = -95.8290096730487$$
$$x_{27} = -7.97678462750978$$
$$x_{28} = 17.3361878945599$$
$$x_{29} = -76.9820071395694$$
$$x_{30} = 98.9702712571724$$
$$x_{31} = -83.264212972531$$
$$x_{32} = -58.1366581602094$$
Puntos máximos de la función:
$$x_{32} = -29.8785491411967$$
$$x_{32} = -11.0848217516286$$
$$x_{32} = -48.7152020786718$$
$$x_{32} = 20.4690516301297$$
$$x_{32} = -92.6877705166201$$
$$x_{32} = 45.5750212470478$$
$$x_{32} = 83.264212972531$$
$$x_{32} = 33.0169733260496$$
$$x_{32} = -73.8409666666368$$
$$x_{32} = 39.2953345326852$$
$$x_{32} = 58.1366581602094$$
$$x_{32} = 70.6999752105956$$
$$x_{32} = -67.559039597919$$
$$x_{32} = 7.97678462750978$$
$$x_{32} = -54.9960465516394$$
$$x_{32} = 51.8555535655761$$
$$x_{32} = -61.2773701909402$$
$$x_{32} = 1.95588396372027$$
$$x_{32} = -42.4350488166082$$
$$x_{32} = -17.3361878945599$$
$$x_{32} = 95.8290096730487$$
$$x_{32} = 14.2070931146973$$
$$x_{32} = -80.1230908716212$$
$$x_{32} = -23.6042091422871$$
$$x_{32} = 64.4181679836523$$
$$x_{32} = -86.4053692621466$$
$$x_{32} = -98.9702712571724$$
$$x_{32} = 26.7408639367464$$
$$x_{32} = 89.5465561460837$$
$$x_{32} = -4.90565966567415$$
$$x_{32} = 76.9820071395694$$
$$x_{32} = -36.1559453106627$$
Decrece en los intervalos
$$\left[98.9702712571724, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -95.8290096730487\right]$$