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$$\frac{x \cos{\left(x \right)}}{2} + \frac{\sin{\left(x \right)}}{2} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = 29.8785865061074$$
$$x_{2} = 26.7409160147873$$
$$x_{3} = 39.295350981473$$
$$x_{4} = 64.4181717218392$$
$$x_{5} = 4.91318043943488$$
$$x_{6} = -83.2642147040886$$
$$x_{7} = -54.9960525574964$$
$$x_{8} = 89.5465575382492$$
$$x_{9} = -2.02875783811043$$
$$x_{10} = 17.3363779239834$$
$$x_{11} = 14.2074367251912$$
$$x_{12} = -17.3363779239834$$
$$x_{13} = 83.2642147040886$$
$$x_{14} = -36.1559664195367$$
$$x_{15} = 86.4053708116885$$
$$x_{16} = -29.8785865061074$$
$$x_{17} = -64.4181717218392$$
$$x_{18} = 61.2773745335697$$
$$x_{19} = -42.4350618814099$$
$$x_{20} = 7.97866571241324$$
$$x_{21} = 58.1366632448992$$
$$x_{22} = -45.57503179559$$
$$x_{23} = 92.687771772017$$
$$x_{24} = 11.085538406497$$
$$x_{25} = -33.0170010333572$$
$$x_{26} = 102.111554139654$$
$$x_{27} = 70.69997803861$$
$$x_{28} = -14.2074367251912$$
$$x_{29} = 42.4350618814099$$
$$x_{30} = 0$$
$$x_{31} = 23.6042847729804$$
$$x_{32} = 76.9820093304187$$
$$x_{33} = -73.8409691490209$$
$$x_{34} = -76.9820093304187$$
$$x_{35} = -7.97866571241324$$
$$x_{36} = -4.91318043943488$$
$$x_{37} = -98.9702722883957$$
$$x_{38} = 73.8409691490209$$
$$x_{39} = 48.7152107175577$$
$$x_{40} = 20.469167402741$$
$$x_{41} = -80.1230928148503$$
$$x_{42} = 51.855560729152$$
$$x_{43} = 80.1230928148503$$
$$x_{44} = -23.6042847729804$$
$$x_{45} = 95.8290108090195$$
$$x_{46} = -86.4053708116885$$
$$x_{47} = -26.7409160147873$$
$$x_{48} = 67.5590428388084$$
$$x_{49} = -89.5465575382492$$
$$x_{50} = 2.02875783811043$$
$$x_{51} = -92.687771772017$$
$$x_{52} = -70.69997803861$$
$$x_{53} = -20.469167402741$$
$$x_{54} = -11.085538406497$$
$$x_{55} = 36.1559664195367$$
$$x_{56} = 54.9960525574964$$
$$x_{57} = 33.0170010333572$$
$$x_{58} = -61.2773745335697$$
$$x_{59} = 45.57503179559$$
$$x_{60} = 98.9702722883957$$
$$x_{61} = -58.1366632448992$$
$$x_{62} = -67.5590428388084$$
$$x_{63} = -95.8290108090195$$
$$x_{64} = -51.855560729152$$
$$x_{65} = -39.295350981473$$
$$x_{66} = -48.7152107175577$$
Signos de extremos en los puntos:
(29.878586506107393, 1.0690669204066)
(26.74091601478731, 29.3611188323487)
(39.295350981472986, 35.6413165034459)
(64.41817172183916, 48.2052056696877)
(4.913180439434884, 13.5927650551439)
(-83.26421470408864, 57.6291051864766)
(-54.99605255749639, -11.4934816248488)
(89.54655753824919, 60.7704871864426)
(-2.028757838110434, 16.9098528705798)
(17.33637792398336, 7.34619569607074)
(14.207436725191188, 23.0861870568871)
(-17.33637792398336, 7.34619569607074)
(83.26421470408864, 57.6291051864766)
(-36.15596641953672, -2.07107268612105)
(86.40537081168854, -27.1997923578054)
(-29.878586506107393, 1.0690669204066)
(-64.41817172183916, 48.2052056696877)
(61.277374533569656, -14.6346082722383)
(-42.43506188140989, -5.21164203862955)
(7.978665712413241, 19.9583636857939)
(58.13666324489916, 45.0640323640429)
(-45.57503179559002, 38.7820324180134)
(92.687771772017, -30.3411888940296)
(11.085538406497022, 10.479645992035)
(-33.017001033357246, 32.5009338654227)
(102.11155413965392, 67.0533289431582)
(70.69997803861, 51.3464534807966)
(-14.207436725191188, 23.0861870568871)
(42.43506188140989, -5.21164203862955)
(0, 16)
(23.604284772980407, 4.20843467518332)
(76.98200933041872, 54.4877575641319)
(-73.8409691490209, -20.9170993857708)
(-76.98200933041872, 54.4877575641319)
(-7.978665712413241, 19.9583636857939)
(-4.913180439434884, 13.5927650551439)
(-98.9702722883957, -33.4826103265593)
(73.8409691490209, -20.9170993857708)
(48.715210717557724, -8.35247511268397)
(20.46916740274095, 26.2223920291261)
(-80.12309281485025, -24.0584265728296)
(51.85556072915197, 41.9229606251007)
(80.12309281485025, -24.0584265728296)
(-23.604284772980407, 4.20843467518332)
(95.82901080901948, 63.9118968042328)
(-86.40537081168854, -27.1997923578054)
(-26.74091601478731, 29.3611188323487)
(67.5590428388084, -17.7758215604863)
(-89.54655753824919, 60.7704871864426)
(2.028757838110434, 16.9098528705798)
(-92.687771772017, -30.3411888940296)
(-70.69997803861, 51.3464534807966)
(-20.46916740274095, 26.2223920291261)
(-11.085538406497022, 10.479645992035)
(36.15596641953672, -2.07107268612105)
(54.99605255749639, -11.4934816248488)
(33.017001033357246, 32.5009338654227)
(-61.277374533569656, -14.6346082722383)
(45.57503179559002, 38.7820324180134)
(98.9702722883957, -33.4826103265593)
(-58.13666324489916, 45.0640323640429)
(-67.5590428388084, -17.7758215604863)
(-95.82901080901948, 63.9118968042328)
(-51.85556072915197, 41.9229606251007)
(-39.295350981472986, 35.6413165034459)
(-48.715210717557724, -8.35247511268397)
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:
$$x_{1} = 29.8785865061074$$
$$x_{2} = 4.91318043943488$$
$$x_{3} = -54.9960525574964$$
$$x_{4} = 17.3363779239834$$
$$x_{5} = -17.3363779239834$$
$$x_{6} = -36.1559664195367$$
$$x_{7} = 86.4053708116885$$
$$x_{8} = -29.8785865061074$$
$$x_{9} = 61.2773745335697$$
$$x_{10} = -42.4350618814099$$
$$x_{11} = 92.687771772017$$
$$x_{12} = 11.085538406497$$
$$x_{13} = 42.4350618814099$$
$$x_{14} = 0$$
$$x_{15} = 23.6042847729804$$
$$x_{16} = -73.8409691490209$$
$$x_{17} = -4.91318043943488$$
$$x_{18} = -98.9702722883957$$
$$x_{19} = 73.8409691490209$$
$$x_{20} = 48.7152107175577$$
$$x_{21} = -80.1230928148503$$
$$x_{22} = 80.1230928148503$$
$$x_{23} = -23.6042847729804$$
$$x_{24} = -86.4053708116885$$
$$x_{25} = 67.5590428388084$$
$$x_{26} = -92.687771772017$$
$$x_{27} = -11.085538406497$$
$$x_{28} = 36.1559664195367$$
$$x_{29} = 54.9960525574964$$
$$x_{30} = -61.2773745335697$$
$$x_{31} = 98.9702722883957$$
$$x_{32} = -67.5590428388084$$
$$x_{33} = -48.7152107175577$$
Puntos máximos de la función:
$$x_{33} = 26.7409160147873$$
$$x_{33} = 39.295350981473$$
$$x_{33} = 64.4181717218392$$
$$x_{33} = -83.2642147040886$$
$$x_{33} = 89.5465575382492$$
$$x_{33} = -2.02875783811043$$
$$x_{33} = 14.2074367251912$$
$$x_{33} = 83.2642147040886$$
$$x_{33} = -64.4181717218392$$
$$x_{33} = 7.97866571241324$$
$$x_{33} = 58.1366632448992$$
$$x_{33} = -45.57503179559$$
$$x_{33} = -33.0170010333572$$
$$x_{33} = 102.111554139654$$
$$x_{33} = 70.69997803861$$
$$x_{33} = -14.2074367251912$$
$$x_{33} = 76.9820093304187$$
$$x_{33} = -76.9820093304187$$
$$x_{33} = -7.97866571241324$$
$$x_{33} = 20.469167402741$$
$$x_{33} = 51.855560729152$$
$$x_{33} = 95.8290108090195$$
$$x_{33} = -26.7409160147873$$
$$x_{33} = -89.5465575382492$$
$$x_{33} = 2.02875783811043$$
$$x_{33} = -70.69997803861$$
$$x_{33} = -20.469167402741$$
$$x_{33} = 33.0170010333572$$
$$x_{33} = 45.57503179559$$
$$x_{33} = -58.1366632448992$$
$$x_{33} = -95.8290108090195$$
$$x_{33} = -51.855560729152$$
$$x_{33} = -39.295350981473$$
Decrece en los intervalos
$$\left[98.9702722883957, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -98.9702722883957\right]$$