In order to find the extrema, we need to solve the equation
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative$$- \frac{x \sin{\left(x \right)}}{6} + \frac{\cos{\left(x \right)}}{6} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = 53.4257904773947$$
$$x_{2} = -0.86033358901938$$
$$x_{3} = -84.8347887180423$$
$$x_{4} = -75.4114834888481$$
$$x_{5} = 56.5663442798215$$
$$x_{6} = -40.8651703304881$$
$$x_{7} = 34.5864242152889$$
$$x_{8} = -44.0050179208308$$
$$x_{9} = -53.4257904773947$$
$$x_{10} = -12.6452872238566$$
$$x_{11} = -78.5525459842429$$
$$x_{12} = 75.4114834888481$$
$$x_{13} = 47.145097736761$$
$$x_{14} = -3.42561845948173$$
$$x_{15} = -100.540910786842$$
$$x_{16} = -147.661626855354$$
$$x_{17} = 97.3996388790738$$
$$x_{18} = -94.2583883450399$$
$$x_{19} = -15.7712848748159$$
$$x_{20} = 87.9759605524932$$
$$x_{21} = 91.1171613944647$$
$$x_{22} = -69.1295029738953$$
$$x_{23} = 15.7712848748159$$
$$x_{24} = 69.1295029738953$$
$$x_{25} = -31.4477146375462$$
$$x_{26} = -18.90240995686$$
$$x_{27} = 22.0364967279386$$
$$x_{28} = 18.90240995686$$
$$x_{29} = -72.270467060309$$
$$x_{30} = -22.0364967279386$$
$$x_{31} = -9.52933440536196$$
$$x_{32} = 28.309642854452$$
$$x_{33} = 81.6936492356017$$
$$x_{34} = -47.145097736761$$
$$x_{35} = 25.1724463266467$$
$$x_{36} = 3.42561845948173$$
$$x_{37} = 78.5525459842429$$
$$x_{38} = 37.7256128277765$$
$$x_{39} = -81.6936492356017$$
$$x_{40} = 100.540910786842$$
$$x_{41} = 62.8477631944545$$
$$x_{42} = -91.1171613944647$$
$$x_{43} = 31.4477146375462$$
$$x_{44} = -97.3996388790738$$
$$x_{45} = -65.9885986984904$$
$$x_{46} = 40.8651703304881$$
$$x_{47} = 65.9885986984904$$
$$x_{48} = -56.5663442798215$$
$$x_{49} = -116.247530303932$$
$$x_{50} = -59.7070073053355$$
$$x_{51} = -34.5864242152889$$
$$x_{52} = 50.2853663377737$$
$$x_{53} = 94.2583883450399$$
$$x_{54} = -25.1724463266467$$
$$x_{55} = -6.43729817917195$$
$$x_{56} = -62.8477631944545$$
$$x_{57} = 9.52933440536196$$
$$x_{58} = -37.7256128277765$$
$$x_{59} = 6.43729817917195$$
$$x_{60} = 84.8347887180423$$
$$x_{61} = -50.2853663377737$$
$$x_{62} = 72.270467060309$$
$$x_{63} = 59.7070073053355$$
$$x_{64} = -28.309642854452$$
$$x_{65} = 0.86033358901938$$
$$x_{66} = -87.9759605524932$$
$$x_{67} = 44.0050179208308$$
$$x_{68} = 12.6452872238566$$
The values of the extrema at the points:
(53.42579047739466, -8.90273902664936)
(-0.8603335890193797, -0.0935160563651742)
(-84.83478871804229, 14.1381492539428)
(-75.41148348884815, -12.567475678867)
(56.56634427982152, 9.42625119547937)
(-40.86517033048807, 6.8088234107529)
(34.58642421528892, -5.76199612226474)
(-44.005017920830845, -7.33227666318441)
(-53.42579047739466, 8.90273902664936)
(-12.645287223856643, -2.10098854964878)
(-78.55254598424293, 13.091030265289)
(75.41148348884815, 12.567475678867)
(47.14509773676103, -7.85574929292366)
(-3.4256184594817283, 0.548061899265149)
(-100.54091078684232, -16.755989675971)
(-147.66162685535437, 24.6097068086237)
(97.39963887907376, -16.2324176326039)
(-94.25838834503986, -15.7088473708514)
(-15.771284874815882, 2.62327949368896)
(87.97596055249322, 14.6617129554041)
(91.11716139446474, -15.1852790749412)
(-69.12950297389526, -11.5203785511536)
(15.771284874815882, -2.62327949368896)
(69.12950297389526, 11.5203785511536)
(-31.447714637546234, -5.23863787975577)
(-18.902409956860023, -3.14600228299484)
(22.036496727938566, -3.66897367985974)
(18.902409956860023, 3.14600228299484)
(-72.27046706030896, 12.0439249330416)
(-22.036496727938566, 3.66897367985974)
(-9.529334405361963, 1.57954904324663)
(28.30964285445201, -4.71533292318239)
(81.69364923560168, 13.6145882494208)
(-47.14509773676103, 7.85574929292366)
(25.172446326646664, 4.19210113631192)
(3.4256184594817283, -0.548061899265149)
(78.55254598424293, -13.091030265289)
(37.7256128277765, 6.28539436880166)
(-81.69364923560168, -13.6145882494208)
(100.54091078684232, 16.755989675971)
(62.84776319445445, 10.4733014953591)
(-91.11716139446474, 15.1852790749412)
(31.447714637546234, 5.23863787975577)
(-97.39963887907376, 16.2324176326039)
(-65.98859869849039, 10.9968371561319)
(40.86517033048807, -6.8088234107529)
(65.98859869849039, -10.9968371561319)
(-56.56634427982152, -9.42625119547937)
(-116.2475303039321, 19.3738715626645)
(-59.70700730533546, 9.94977247337763)
(-34.58642421528892, 5.76199612226474)
(50.28536633777365, 8.3792376725662)
(94.25838834503986, 15.7088473708514)
(-25.172446326646664, -4.19210113631192)
(-6.437298179171947, -1.06016732413898)
(-62.84776319445445, -10.4733014953591)
(9.529334405361963, -1.57954904324663)
(-37.7256128277765, -6.28539436880166)
(6.437298179171947, 1.06016732413898)
(84.83478871804229, -14.1381492539428)
(-50.28536633777365, -8.3792376725662)
(72.27046706030896, -12.0439249330416)
(59.70700730533546, -9.94977247337763)
(-28.30964285445201, 4.71533292318239)
(0.8603335890193797, 0.0935160563651742)
(-87.97596055249322, -14.6617129554041)
(44.005017920830845, 7.33227666318441)
(12.645287223856643, 2.10098854964878)
Intervals of increase and decrease of the function:Let's find intervals where the function increases and decreases, as well as minima and maxima of the function, for this let's look how the function behaves itself in the extremas and at the slightest deviation from:
Minima of the function at points:
$$x_{1} = 53.4257904773947$$
$$x_{2} = -0.86033358901938$$
$$x_{3} = -75.4114834888481$$
$$x_{4} = 34.5864242152889$$
$$x_{5} = -44.0050179208308$$
$$x_{6} = -12.6452872238566$$
$$x_{7} = 47.145097736761$$
$$x_{8} = -100.540910786842$$
$$x_{9} = 97.3996388790738$$
$$x_{10} = -94.2583883450399$$
$$x_{11} = 91.1171613944647$$
$$x_{12} = -69.1295029738953$$
$$x_{13} = 15.7712848748159$$
$$x_{14} = -31.4477146375462$$
$$x_{15} = -18.90240995686$$
$$x_{16} = 22.0364967279386$$
$$x_{17} = 28.309642854452$$
$$x_{18} = 3.42561845948173$$
$$x_{19} = 78.5525459842429$$
$$x_{20} = -81.6936492356017$$
$$x_{21} = 40.8651703304881$$
$$x_{22} = 65.9885986984904$$
$$x_{23} = -56.5663442798215$$
$$x_{24} = -25.1724463266467$$
$$x_{25} = -6.43729817917195$$
$$x_{26} = -62.8477631944545$$
$$x_{27} = 9.52933440536196$$
$$x_{28} = -37.7256128277765$$
$$x_{29} = 84.8347887180423$$
$$x_{30} = -50.2853663377737$$
$$x_{31} = 72.270467060309$$
$$x_{32} = 59.7070073053355$$
$$x_{33} = -87.9759605524932$$
Maxima of the function at points:
$$x_{33} = -84.8347887180423$$
$$x_{33} = 56.5663442798215$$
$$x_{33} = -40.8651703304881$$
$$x_{33} = -53.4257904773947$$
$$x_{33} = -78.5525459842429$$
$$x_{33} = 75.4114834888481$$
$$x_{33} = -3.42561845948173$$
$$x_{33} = -147.661626855354$$
$$x_{33} = -15.7712848748159$$
$$x_{33} = 87.9759605524932$$
$$x_{33} = 69.1295029738953$$
$$x_{33} = 18.90240995686$$
$$x_{33} = -72.270467060309$$
$$x_{33} = -22.0364967279386$$
$$x_{33} = -9.52933440536196$$
$$x_{33} = 81.6936492356017$$
$$x_{33} = -47.145097736761$$
$$x_{33} = 25.1724463266467$$
$$x_{33} = 37.7256128277765$$
$$x_{33} = 100.540910786842$$
$$x_{33} = 62.8477631944545$$
$$x_{33} = -91.1171613944647$$
$$x_{33} = 31.4477146375462$$
$$x_{33} = -97.3996388790738$$
$$x_{33} = -65.9885986984904$$
$$x_{33} = -116.247530303932$$
$$x_{33} = -59.7070073053355$$
$$x_{33} = -34.5864242152889$$
$$x_{33} = 50.2853663377737$$
$$x_{33} = 94.2583883450399$$
$$x_{33} = 6.43729817917195$$
$$x_{33} = -28.309642854452$$
$$x_{33} = 0.86033358901938$$
$$x_{33} = 44.0050179208308$$
$$x_{33} = 12.6452872238566$$
Decreasing at intervals
$$\left[97.3996388790738, \infty\right)$$
Increasing at intervals
$$\left(-\infty, -100.540910786842\right]$$