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$$x \cos{\left(x \right)} + \sin{\left(x \right)} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = -20.469167402741$$
$$x_{2} = 80.1230928148503$$
$$x_{3} = 45.57503179559$$
$$x_{4} = -86.4053708116885$$
$$x_{5} = -11.085538406497$$
$$x_{6} = 95.8290108090195$$
$$x_{7} = -36.1559664195367$$
$$x_{8} = -42.4350618814099$$
$$x_{9} = 36.1559664195367$$
$$x_{10} = 7.97866571241324$$
$$x_{11} = 2.02875783811043$$
$$x_{12} = -80.1230928148503$$
$$x_{13} = 92.687771772017$$
$$x_{14} = 86.4053708116885$$
$$x_{15} = 29.8785865061074$$
$$x_{16} = 70.69997803861$$
$$x_{17} = 58.1366632448992$$
$$x_{18} = 4.91318043943488$$
$$x_{19} = 73.8409691490209$$
$$x_{20} = -48.7152107175577$$
$$x_{21} = 54.9960525574964$$
$$x_{22} = 11.085538406497$$
$$x_{23} = -4.91318043943488$$
$$x_{24} = -76.9820093304187$$
$$x_{25} = -89.5465575382492$$
$$x_{26} = -14.2074367251912$$
$$x_{27} = -33.0170010333572$$
$$x_{28} = -51.855560729152$$
$$x_{29} = 64.4181717218392$$
$$x_{30} = -67.5590428388084$$
$$x_{31} = 42.4350618814099$$
$$x_{32} = 0$$
$$x_{33} = -7.97866571241324$$
$$x_{34} = 51.855560729152$$
$$x_{35} = 26.7409160147873$$
$$x_{36} = -26.7409160147873$$
$$x_{37} = 89.5465575382492$$
$$x_{38} = -83.2642147040886$$
$$x_{39} = -2.02875783811043$$
$$x_{40} = 83.2642147040886$$
$$x_{41} = -45.57503179559$$
$$x_{42} = -98.9702722883957$$
$$x_{43} = 67.5590428388084$$
$$x_{44} = 20.469167402741$$
$$x_{45} = -54.9960525574964$$
$$x_{46} = 48.7152107175577$$
$$x_{47} = 39.295350981473$$
$$x_{48} = -17.3363779239834$$
$$x_{49} = 102.111554139654$$
$$x_{50} = -95.8290108090195$$
$$x_{51} = -64.4181717218392$$
$$x_{52} = 61.2773745335697$$
$$x_{53} = -29.8785865061074$$
$$x_{54} = 23.6042847729804$$
$$x_{55} = -39.295350981473$$
$$x_{56} = -58.1366632448992$$
$$x_{57} = 98.9702722883957$$
$$x_{58} = -23.6042847729804$$
$$x_{59} = 14.2074367251912$$
$$x_{60} = -73.8409691490209$$
$$x_{61} = -92.687771772017$$
$$x_{62} = 33.0170010333572$$
$$x_{63} = 76.9820093304187$$
$$x_{64} = -61.2773745335697$$
$$x_{65} = 17.3363779239834$$
$$x_{66} = -70.69997803861$$
The values of the extrema at the points:
(-20.46916740274095, 20.4447840582523)
(80.12309281485025, -80.1168531456592)
(45.57503179559002, 45.5640648360268)
(-86.40537081168854, -86.3995847156108)
(-11.085538406497022, -11.04070801593)
(95.82901080901948, 95.8237936084657)
(-36.15596641953672, -36.1421453722421)
(-42.43506188140989, -42.4232840772591)
(36.15596641953672, -36.1421453722421)
(7.978665712413241, 7.91672737158778)
(2.028757838110434, 1.81970574115965)
(-80.12309281485025, -80.1168531456592)
(92.687771772017, -92.6823777880592)
(86.40537081168854, -86.3995847156108)
(29.878586506107393, -29.8618661591868)
(70.69997803861, 70.6929069615931)
(58.13666324489916, 58.1280647280857)
(4.913180439434884, -4.81446988971227)
(73.8409691490209, -73.8341987715416)
(-48.715210717557724, -48.7049502253679)
(54.99605255749639, -54.9869632496976)
(11.085538406497022, -11.04070801593)
(-4.913180439434884, -4.81446988971227)
(-76.98200933041872, 76.9755151282637)
(-89.54655753824919, 89.5409743728852)
(-14.207436725191188, 14.1723741137743)
(-33.017001033357246, 33.0018677308454)
(-51.85556072915197, 51.8459212502015)
(64.41817172183916, 64.4104113393753)
(-67.5590428388084, -67.5516431209725)
(42.43506188140989, -42.4232840772591)
(0, 0)
(-7.978665712413241, 7.91672737158778)
(51.85556072915197, 51.8459212502015)
(26.74091601478731, 26.7222376646974)
(-26.74091601478731, 26.7222376646974)
(89.54655753824919, 89.5409743728852)
(-83.26421470408864, 83.2582103729533)
(-2.028757838110434, 1.81970574115965)
(83.26421470408864, 83.2582103729533)
(-45.57503179559002, 45.5640648360268)
(-98.9702722883957, -98.9652206531187)
(67.5590428388084, -67.5516431209725)
(20.46916740274095, 20.4447840582523)
(-54.99605255749639, -54.9869632496976)
(48.715210717557724, -48.7049502253679)
(39.295350981472986, 39.2826330068918)
(-17.33637792398336, -17.3076086078585)
(102.11155413965392, 102.106657886316)
(-95.82901080901948, 95.8237936084657)
(-64.41817172183916, 64.4104113393753)
(61.277374533569656, -61.2692165444766)
(-29.878586506107393, -29.8618661591868)
(23.604284772980407, -23.5831306496334)
(-39.295350981472986, 39.2826330068918)
(-58.13666324489916, 58.1280647280857)
(98.9702722883957, -98.9652206531187)
(-23.604284772980407, -23.5831306496334)
(14.207436725191188, 14.1723741137743)
(-73.8409691490209, -73.8341987715416)
(-92.687771772017, -92.6823777880592)
(33.017001033357246, 33.0018677308454)
(76.98200933041872, 76.9755151282637)
(-61.277374533569656, -61.2692165444766)
(17.33637792398336, -17.3076086078585)
(-70.69997803861, 70.6929069615931)
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:
$$x_{1} = 80.1230928148503$$
$$x_{2} = -86.4053708116885$$
$$x_{3} = -11.085538406497$$
$$x_{4} = -36.1559664195367$$
$$x_{5} = -42.4350618814099$$
$$x_{6} = 36.1559664195367$$
$$x_{7} = -80.1230928148503$$
$$x_{8} = 92.687771772017$$
$$x_{9} = 86.4053708116885$$
$$x_{10} = 29.8785865061074$$
$$x_{11} = 4.91318043943488$$
$$x_{12} = 73.8409691490209$$
$$x_{13} = -48.7152107175577$$
$$x_{14} = 54.9960525574964$$
$$x_{15} = 11.085538406497$$
$$x_{16} = -4.91318043943488$$
$$x_{17} = -67.5590428388084$$
$$x_{18} = 42.4350618814099$$
$$x_{19} = 0$$
$$x_{20} = -98.9702722883957$$
$$x_{21} = 67.5590428388084$$
$$x_{22} = -54.9960525574964$$
$$x_{23} = 48.7152107175577$$
$$x_{24} = -17.3363779239834$$
$$x_{25} = 61.2773745335697$$
$$x_{26} = -29.8785865061074$$
$$x_{27} = 23.6042847729804$$
$$x_{28} = 98.9702722883957$$
$$x_{29} = -23.6042847729804$$
$$x_{30} = -73.8409691490209$$
$$x_{31} = -92.687771772017$$
$$x_{32} = -61.2773745335697$$
$$x_{33} = 17.3363779239834$$
Maxima of the function at points:
$$x_{33} = -20.469167402741$$
$$x_{33} = 45.57503179559$$
$$x_{33} = 95.8290108090195$$
$$x_{33} = 7.97866571241324$$
$$x_{33} = 2.02875783811043$$
$$x_{33} = 70.69997803861$$
$$x_{33} = 58.1366632448992$$
$$x_{33} = -76.9820093304187$$
$$x_{33} = -89.5465575382492$$
$$x_{33} = -14.2074367251912$$
$$x_{33} = -33.0170010333572$$
$$x_{33} = -51.855560729152$$
$$x_{33} = 64.4181717218392$$
$$x_{33} = -7.97866571241324$$
$$x_{33} = 51.855560729152$$
$$x_{33} = 26.7409160147873$$
$$x_{33} = -26.7409160147873$$
$$x_{33} = 89.5465575382492$$
$$x_{33} = -83.2642147040886$$
$$x_{33} = -2.02875783811043$$
$$x_{33} = 83.2642147040886$$
$$x_{33} = -45.57503179559$$
$$x_{33} = 20.469167402741$$
$$x_{33} = 39.295350981473$$
$$x_{33} = 102.111554139654$$
$$x_{33} = -95.8290108090195$$
$$x_{33} = -64.4181717218392$$
$$x_{33} = -39.295350981473$$
$$x_{33} = -58.1366632448992$$
$$x_{33} = 14.2074367251912$$
$$x_{33} = 33.0170010333572$$
$$x_{33} = 76.9820093304187$$
$$x_{33} = -70.69997803861$$
Decreasing at intervals
$$\left[98.9702722883957, \infty\right)$$
Increasing at intervals
$$\left(-\infty, -98.9702722883957\right]$$