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$$\frac{x \cos{\left(x \right)} \left|{x}\right|}{2} + \left(\frac{x \operatorname{sign}{\left(x \right)}}{2} + \frac{\left|{x}\right|}{2}\right) \sin{\left(x \right)} + \frac{\cos{\left(x \right)}}{2} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = 92.6985527346464$$
$$x_{2} = -61.2936836399605$$
$$x_{3} = 29.9118197187497$$
$$x_{4} = -70.7141157169284$$
$$x_{5} = 39.3206953827842$$
$$x_{6} = 36.1834910972322$$
$$x_{7} = 55.0141976948257$$
$$x_{8} = 5.07497753086734$$
$$x_{9} = 98.980369804178$$
$$x_{10} = 61.2936662971292$$
$$x_{11} = 33.0471136324618$$
$$x_{12} = 64.433671639937$$
$$x_{13} = 17.3928724532687$$
$$x_{14} = -0.555968430719397$$
$$x_{15} = -89.557721665321$$
$$x_{16} = 11.1713476748979$$
$$x_{17} = 58.1538319302606$$
$$x_{18} = -55.0142216707462$$
$$x_{19} = -11.1740843244572$$
$$x_{20} = 51.8747997634236$$
$$x_{21} = -80.1355690775093$$
$$x_{22} = -2.36950073184739$$
$$x_{23} = -92.6985577527308$$
$$x_{24} = 83.2762137053534$$
$$x_{25} = 20.5172951367754$$
$$x_{26} = -45.5969490305626$$
$$x_{27} = -98.9803739265469$$
$$x_{28} = 70.7141044184235$$
$$x_{29} = -48.7357180363942$$
$$x_{30} = -73.8545059751268$$
$$x_{31} = 67.5738241989977$$
$$x_{32} = -29.9119681844492$$
$$x_{33} = 80.135561311847$$
$$x_{34} = 2.21441690507964$$
$$x_{35} = -23.6464737422827$$
$$x_{36} = -42.4585968290625$$
$$x_{37} = 42.4585447430688$$
$$x_{38} = -14.2770237477481$$
$$x_{39} = -5.09965827157433$$
$$x_{40} = -51.8748283540152$$
$$x_{41} = 73.854496056498$$
$$x_{42} = -58.1538522330151$$
$$x_{43} = 23.6461744227855$$
$$x_{44} = -8.09966225317152$$
$$x_{45} = -86.4169405511272$$
$$x_{46} = -67.5738371455493$$
$$x_{47} = -64.4336865711139$$
$$x_{48} = -20.5177517424302$$
$$x_{49} = 89.5577161008208$$
$$x_{50} = -17.3936178603206$$
$$x_{51} = -36.183575147983$$
$$x_{52} = 86.4169343579344$$
$$x_{53} = -39.3207609238378$$
$$x_{54} = 45.5969069577237$$
$$x_{55} = 14.27568833055$$
$$x_{56} = 76.9949855126353$$
$$x_{57} = -83.2762206255981$$
$$x_{58} = 95.8394388710214$$
$$x_{59} = -95.8394434119385$$
$$x_{60} = -33.0472238569584$$
$$x_{61} = 8.09275583571457$$
$$x_{62} = 26.7779839240345$$
$$x_{63} = 48.7356835678677$$
$$x_{64} = -76.9949942671742$$
$$x_{65} = -26.7781905112054$$
The values of the extrema at the points:
(92.69855273464636, -4296.01130477231)
(-61.29368363996054, -1876.95835928771)
(29.911819718749687, -446.862925291614)
(-70.71411571692843, 2498.74348069471)
(39.32069538278422, 772.561121480556)
(36.18349109723224, -654.125557565988)
(55.01419769482573, -1512.78229345467)
(5.074977530867337, -12.5079047222614)
(98.98036980417797, -4898.05721136117)
(61.29366629712923, -1877.95782736065)
(33.047113632461794, 545.559505658263)
(64.43367163993699, 2075.34998280395)
(17.39287245326871, -150.769013848249)
(-0.5559684307193966, -0.182316570648737)
(-89.557721665321, 4008.79300426882)
(11.171347674897921, -61.930323171237)
(58.15383193026061, 1690.43526511869)
(-55.01422167074623, -1511.78295361597)
(-11.17408432445722, -60.9459708969528)
(51.87479976342356, 1344.99890893021)
(-80.13556907750933, -3209.3550270832)
(-2.3695007318473897, 1.60963133183268)
(-92.69855775273078, -4295.01153743807)
(83.27621370535343, 3466.96446092473)
(20.517295136775363, 209.989090483111)
(-45.59694903056257, 1038.04184194853)
(-98.98037392654695, -4897.05741544043)
(70.71410441842355, 2499.74308097332)
(-48.73571803639418, -1186.085947952)
(-73.85450597512677, -2725.74439301833)
(67.57382419899773, -2282.61173347816)
(-29.91196818444918, -445.86515315981)
(80.13556131184697, -3210.35471578503)
(2.2144169050796365, 2.36124528340854)
(-23.64647374228269, -278.081430657965)
(-42.45859682906252, -899.867331158339)
(42.45854474306877, -900.866223572506)
(-14.277023747748137, 100.426467579168)
(-5.099658271574332, -11.5773134494116)
(-51.87482835401515, 1343.99965132192)
(73.85449605649804, -2726.74402654929)
(-58.15385223301508, 1689.43585598355)
(23.646174422785496, -279.077872857632)
(-8.09966225317152, 31.3322922959358)
(-86.41694055112717, -3732.44407488602)
(-67.57383714554935, -2281.61217118943)
(-64.4336865711139, 2074.35046418665)
(-20.517751742430185, 208.993807864045)
(89.55771610082076, 4009.79275500368)
(-17.39361786032065, -149.775560097541)
(-36.183575147983, -653.127081669827)
(86.41693435793444, -3733.44380718018)
(-39.3207609238378, 771.56241253748)
(45.59690695772369, 1039.04088137013)
(14.275688330549972, 101.416796620662)
(76.99498551263531, 2963.61457121477)
(-83.27622062559811, 3465.96474919524)
(95.83943887102137, 4592.09945680246)
(-95.83944341193853, 4591.09967447299)
(-33.047223856958354, 544.561331952917)
(8.092755835714566, 32.3031037373247)
(26.77798392403449, 358.035751231176)
(48.735683567867724, -1187.08510696841)
(-76.99499426717416, 2962.61490841307)
(-26.778190511205352, 357.038528765215)
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:
$$x_{1} = 92.6985527346464$$
$$x_{2} = -61.2936836399605$$
$$x_{3} = 29.9118197187497$$
$$x_{4} = 36.1834910972322$$
$$x_{5} = 55.0141976948257$$
$$x_{6} = 5.07497753086734$$
$$x_{7} = 98.980369804178$$
$$x_{8} = 61.2936662971292$$
$$x_{9} = 17.3928724532687$$
$$x_{10} = -0.555968430719397$$
$$x_{11} = 11.1713476748979$$
$$x_{12} = -55.0142216707462$$
$$x_{13} = -11.1740843244572$$
$$x_{14} = -80.1355690775093$$
$$x_{15} = -92.6985577527308$$
$$x_{16} = -98.9803739265469$$
$$x_{17} = -48.7357180363942$$
$$x_{18} = -73.8545059751268$$
$$x_{19} = 67.5738241989977$$
$$x_{20} = -29.9119681844492$$
$$x_{21} = 80.135561311847$$
$$x_{22} = -23.6464737422827$$
$$x_{23} = -42.4585968290625$$
$$x_{24} = 42.4585447430688$$
$$x_{25} = -5.09965827157433$$
$$x_{26} = 73.854496056498$$
$$x_{27} = 23.6461744227855$$
$$x_{28} = -86.4169405511272$$
$$x_{29} = -67.5738371455493$$
$$x_{30} = -17.3936178603206$$
$$x_{31} = -36.183575147983$$
$$x_{32} = 86.4169343579344$$
$$x_{33} = 48.7356835678677$$
Maxima of the function at points:
$$x_{33} = -70.7141157169284$$
$$x_{33} = 39.3206953827842$$
$$x_{33} = 33.0471136324618$$
$$x_{33} = 64.433671639937$$
$$x_{33} = -89.557721665321$$
$$x_{33} = 58.1538319302606$$
$$x_{33} = 51.8747997634236$$
$$x_{33} = -2.36950073184739$$
$$x_{33} = 83.2762137053534$$
$$x_{33} = 20.5172951367754$$
$$x_{33} = -45.5969490305626$$
$$x_{33} = 70.7141044184235$$
$$x_{33} = 2.21441690507964$$
$$x_{33} = -14.2770237477481$$
$$x_{33} = -51.8748283540152$$
$$x_{33} = -58.1538522330151$$
$$x_{33} = -8.09966225317152$$
$$x_{33} = -64.4336865711139$$
$$x_{33} = -20.5177517424302$$
$$x_{33} = 89.5577161008208$$
$$x_{33} = -39.3207609238378$$
$$x_{33} = 45.5969069577237$$
$$x_{33} = 14.27568833055$$
$$x_{33} = 76.9949855126353$$
$$x_{33} = -83.2762206255981$$
$$x_{33} = 95.8394388710214$$
$$x_{33} = -95.8394434119385$$
$$x_{33} = -33.0472238569584$$
$$x_{33} = 8.09275583571457$$
$$x_{33} = 26.7779839240345$$
$$x_{33} = -76.9949942671742$$
$$x_{33} = -26.7781905112054$$
Decreasing at intervals
$$\left[98.980369804178, \infty\right)$$
Increasing at intervals
$$\left(-\infty, -98.9803739265469\right]$$