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$$- x \sin{\left(x \right)} \cos{\left(\frac{1}{2} \right)} + \cos{\left(\frac{1}{2} \right)} \cos{\left(x \right)} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = 6.43729817917195$$
$$x_{2} = -12.6452872238566$$
$$x_{3} = 18.90240995686$$
$$x_{4} = 75.4114834888481$$
$$x_{5} = -31.4477146375462$$
$$x_{6} = 9.52933440536196$$
$$x_{7} = 25.1724463266467$$
$$x_{8} = 78.5525459842429$$
$$x_{9} = 100.540910786842$$
$$x_{10} = 91.1171613944647$$
$$x_{11} = -22.0364967279386$$
$$x_{12} = 34.5864242152889$$
$$x_{13} = 3.42561845948173$$
$$x_{14} = 37.7256128277765$$
$$x_{15} = -56.5663442798215$$
$$x_{16} = -91.1171613944647$$
$$x_{17} = 72.270467060309$$
$$x_{18} = -0.86033358901938$$
$$x_{19} = 44.0050179208308$$
$$x_{20} = -94.2583883450399$$
$$x_{21} = -28.309642854452$$
$$x_{22} = -75.4114834888481$$
$$x_{23} = -37.7256128277765$$
$$x_{24} = 84.8347887180423$$
$$x_{25} = 94.2583883450399$$
$$x_{26} = -59.7070073053355$$
$$x_{27} = -84.8347887180423$$
$$x_{28} = 62.8477631944545$$
$$x_{29} = -18.90240995686$$
$$x_{30} = -34.5864242152889$$
$$x_{31} = 65.9885986984904$$
$$x_{32} = -69.1295029738953$$
$$x_{33} = 15.7712848748159$$
$$x_{34} = -25.1724463266467$$
$$x_{35} = 87.9759605524932$$
$$x_{36} = -81.6936492356017$$
$$x_{37} = -47.145097736761$$
$$x_{38} = 50.2853663377737$$
$$x_{39} = -65.9885986984904$$
$$x_{40} = -100.540910786842$$
$$x_{41} = -62.8477631944545$$
$$x_{42} = -3.42561845948173$$
$$x_{43} = 0.86033358901938$$
$$x_{44} = -87.9759605524932$$
$$x_{45} = -53.4257904773947$$
$$x_{46} = -15.7712848748159$$
$$x_{47} = -40.8651703304881$$
$$x_{48} = -9.52933440536196$$
$$x_{49} = -50.2853663377737$$
$$x_{50} = -116.247530303932$$
$$x_{51} = 59.7070073053355$$
$$x_{52} = 40.8651703304881$$
$$x_{53} = 69.1295029738953$$
$$x_{54} = -44.0050179208308$$
$$x_{55} = -6.43729817917195$$
$$x_{56} = 28.309642854452$$
$$x_{57} = 22.0364967279386$$
$$x_{58} = -78.5525459842429$$
$$x_{59} = 97.3996388790738$$
$$x_{60} = 31.4477146375462$$
$$x_{61} = 12.6452872238566$$
$$x_{62} = -72.270467060309$$
$$x_{63} = 53.4257904773947$$
$$x_{64} = 56.5663442798215$$
$$x_{65} = 81.6936492356017$$
$$x_{66} = -97.3996388790738$$
$$x_{67} = -147.661626855354$$
$$x_{68} = 47.145097736761$$
Signos de extremos en los puntos:
(6.437298179171947, 6.36100394483385*cos(1/2))
(-12.645287223856643, -12.6059312978927*cos(1/2))
(18.902409956860023, 18.876013697969*cos(1/2))
(75.41148348884815, 75.4048540732019*cos(1/2))
(-31.447714637546234, -31.4318272785346*cos(1/2))
(9.529334405361963, -9.47729425947979*cos(1/2))
(25.172446326646664, 25.1526068178715*cos(1/2))
(78.55254598424293, -78.5461815917343*cos(1/2))
(100.54091078684232, 100.535938055826*cos(1/2))
(91.11716139446474, -91.1116744496469*cos(1/2))
(-22.036496727938566, 22.0138420791585*cos(1/2))
(34.58642421528892, -34.5719767335884*cos(1/2))
(3.4256184594817283, -3.2883713955909*cos(1/2))
(37.7256128277765, 37.71236621281*cos(1/2))
(-56.56634427982152, -56.5575071728762*cos(1/2))
(-91.11716139446474, 91.1116744496469*cos(1/2))
(72.27046706030896, -72.2635495982494*cos(1/2))
(-0.8603335890193797, -0.561096338191045*cos(1/2))
(44.005017920830845, 43.9936599791065*cos(1/2))
(-94.25838834503986, -94.2530842251087*cos(1/2))
(-28.30964285445201, 28.2919975390943*cos(1/2))
(-75.41148348884815, -75.4048540732019*cos(1/2))
(-37.7256128277765, -37.71236621281*cos(1/2))
(84.83478871804229, -84.8288955236568*cos(1/2))
(94.25838834503986, 94.2530842251087*cos(1/2))
(-59.70700730533546, 59.6986348402658*cos(1/2))
(-84.83478871804229, 84.8288955236568*cos(1/2))
(62.84776319445445, 62.8398089721545*cos(1/2))
(-18.902409956860023, -18.876013697969*cos(1/2))
(-34.58642421528892, 34.5719767335884*cos(1/2))
(65.98859869849039, -65.9810229367917*cos(1/2))
(-69.12950297389526, -69.1222713069218*cos(1/2))
(15.771284874815882, -15.7396769621337*cos(1/2))
(-25.172446326646664, -25.1526068178715*cos(1/2))
(87.97596055249322, 87.9702777324248*cos(1/2))
(-81.69364923560168, -81.6875294965246*cos(1/2))
(-47.14509773676103, 47.1344957575419*cos(1/2))
(50.28536633777365, 50.2754260353972*cos(1/2))
(-65.98859869849039, 65.9810229367917*cos(1/2))
(-100.54091078684232, -100.535938055826*cos(1/2))
(-62.84776319445445, -62.8398089721545*cos(1/2))
(-3.4256184594817283, 3.2883713955909*cos(1/2))
(0.8603335890193797, 0.561096338191045*cos(1/2))
(-87.97596055249322, -87.9702777324248*cos(1/2))
(-53.42579047739466, 53.4164341598961*cos(1/2))
(-15.771284874815882, 15.7396769621337*cos(1/2))
(-40.86517033048807, 40.8529404645174*cos(1/2))
(-9.529334405361963, 9.47729425947979*cos(1/2))
(-50.28536633777365, -50.2754260353972*cos(1/2))
(-116.2475303039321, 116.243229375987*cos(1/2))
(59.70700730533546, -59.6986348402658*cos(1/2))
(40.86517033048807, -40.8529404645174*cos(1/2))
(69.12950297389526, 69.1222713069218*cos(1/2))
(-44.005017920830845, -43.9936599791065*cos(1/2))
(-6.437298179171947, -6.36100394483385*cos(1/2))
(28.30964285445201, -28.2919975390943*cos(1/2))
(22.036496727938566, -22.0138420791585*cos(1/2))
(-78.55254598424293, 78.5461815917343*cos(1/2))
(97.39963887907376, -97.3945057956234*cos(1/2))
(31.447714637546234, 31.4318272785346*cos(1/2))
(12.645287223856643, 12.6059312978927*cos(1/2))
(-72.27046706030896, 72.2635495982494*cos(1/2))
(53.42579047739466, -53.4164341598961*cos(1/2))
(56.56634427982152, 56.5575071728762*cos(1/2))
(81.69364923560168, 81.6875294965246*cos(1/2))
(-97.39963887907376, 97.3945057956234*cos(1/2))
(-147.66162685535437, 147.658240851742*cos(1/2))
(47.14509773676103, -47.1344957575419*cos(1/2))
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:
$$x_{1} = -12.6452872238566$$
$$x_{2} = -31.4477146375462$$
$$x_{3} = 9.52933440536196$$
$$x_{4} = 78.5525459842429$$
$$x_{5} = 91.1171613944647$$
$$x_{6} = 34.5864242152889$$
$$x_{7} = 3.42561845948173$$
$$x_{8} = -56.5663442798215$$
$$x_{9} = 72.270467060309$$
$$x_{10} = -0.86033358901938$$
$$x_{11} = -94.2583883450399$$
$$x_{12} = -75.4114834888481$$
$$x_{13} = -37.7256128277765$$
$$x_{14} = 84.8347887180423$$
$$x_{15} = -18.90240995686$$
$$x_{16} = 65.9885986984904$$
$$x_{17} = -69.1295029738953$$
$$x_{18} = 15.7712848748159$$
$$x_{19} = -25.1724463266467$$
$$x_{20} = -81.6936492356017$$
$$x_{21} = -100.540910786842$$
$$x_{22} = -62.8477631944545$$
$$x_{23} = -87.9759605524932$$
$$x_{24} = -50.2853663377737$$
$$x_{25} = 59.7070073053355$$
$$x_{26} = 40.8651703304881$$
$$x_{27} = -44.0050179208308$$
$$x_{28} = -6.43729817917195$$
$$x_{29} = 28.309642854452$$
$$x_{30} = 22.0364967279386$$
$$x_{31} = 97.3996388790738$$
$$x_{32} = 53.4257904773947$$
$$x_{33} = 47.145097736761$$
Puntos máximos de la función:
$$x_{33} = 6.43729817917195$$
$$x_{33} = 18.90240995686$$
$$x_{33} = 75.4114834888481$$
$$x_{33} = 25.1724463266467$$
$$x_{33} = 100.540910786842$$
$$x_{33} = -22.0364967279386$$
$$x_{33} = 37.7256128277765$$
$$x_{33} = -91.1171613944647$$
$$x_{33} = 44.0050179208308$$
$$x_{33} = -28.309642854452$$
$$x_{33} = 94.2583883450399$$
$$x_{33} = -59.7070073053355$$
$$x_{33} = -84.8347887180423$$
$$x_{33} = 62.8477631944545$$
$$x_{33} = -34.5864242152889$$
$$x_{33} = 87.9759605524932$$
$$x_{33} = -47.145097736761$$
$$x_{33} = 50.2853663377737$$
$$x_{33} = -65.9885986984904$$
$$x_{33} = -3.42561845948173$$
$$x_{33} = 0.86033358901938$$
$$x_{33} = -53.4257904773947$$
$$x_{33} = -15.7712848748159$$
$$x_{33} = -40.8651703304881$$
$$x_{33} = -9.52933440536196$$
$$x_{33} = -116.247530303932$$
$$x_{33} = 69.1295029738953$$
$$x_{33} = -78.5525459842429$$
$$x_{33} = 31.4477146375462$$
$$x_{33} = 12.6452872238566$$
$$x_{33} = -72.270467060309$$
$$x_{33} = 56.5663442798215$$
$$x_{33} = 81.6936492356017$$
$$x_{33} = -97.3996388790738$$
$$x_{33} = -147.661626855354$$
Decrece en los intervalos
$$\left[97.3996388790738, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -100.540910786842\right]$$