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$$x^{2} \cos{\left(x \right)} + 2 x \sin{\left(x \right)} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = 29.9118938695518$$
$$x_{2} = 48.7357007949054$$
$$x_{3} = 86.4169374541167$$
$$x_{4} = 95.839441141233$$
$$x_{5} = 83.2762171649775$$
$$x_{6} = -51.8748140534268$$
$$x_{7} = 36.1835330907526$$
$$x_{8} = 0$$
$$x_{9} = 51.8748140534268$$
$$x_{10} = 67.573830670859$$
$$x_{11} = 33.0471686947054$$
$$x_{12} = 8.09616360322292$$
$$x_{13} = -26.7780870755585$$
$$x_{14} = 5.08698509410227$$
$$x_{15} = -55.0142096788381$$
$$x_{16} = -92.6985552433969$$
$$x_{17} = 61.2936749662429$$
$$x_{18} = 11.17270586833$$
$$x_{19} = 20.5175229099417$$
$$x_{20} = -36.1835330907526$$
$$x_{21} = -23.6463238196036$$
$$x_{22} = -58.153842078645$$
$$x_{23} = -20.5175229099417$$
$$x_{24} = 70.7141100665485$$
$$x_{25} = 45.5969279840735$$
$$x_{26} = 14.2763529183365$$
$$x_{27} = 42.458570771699$$
$$x_{28} = -5.08698509410227$$
$$x_{29} = -29.9118938695518$$
$$x_{30} = 98.9803718651523$$
$$x_{31} = -42.458570771699$$
$$x_{32} = -80.1355651940744$$
$$x_{33} = -89.5577188827244$$
$$x_{34} = -11.17270586833$$
$$x_{35} = 2.2889297281034$$
$$x_{36} = -48.7357007949054$$
$$x_{37} = -17.3932439645948$$
$$x_{38} = -120.967848975693$$
$$x_{39} = 92.6985552433969$$
$$x_{40} = 39.3207281322521$$
$$x_{41} = -39.3207281322521$$
$$x_{42} = -83.2762171649775$$
$$x_{43} = 73.8545010149048$$
$$x_{44} = 58.153842078645$$
$$x_{45} = -8.09616360322292$$
$$x_{46} = -76.9949898891676$$
$$x_{47} = 64.4336791037316$$
$$x_{48} = -64.4336791037316$$
$$x_{49} = 89.5577188827244$$
$$x_{50} = 55.0142096788381$$
$$x_{51} = -33.0471686947054$$
$$x_{52} = -67.573830670859$$
$$x_{53} = 80.1355651940744$$
$$x_{54} = 76.9949898891676$$
$$x_{55} = -70.7141100665485$$
$$x_{56} = -61.2936749662429$$
$$x_{57} = 17.3932439645948$$
$$x_{58} = 26.7780870755585$$
$$x_{59} = -14.2763529183365$$
$$x_{60} = -98.9803718651523$$
$$x_{61} = 23.6463238196036$$
$$x_{62} = -86.4169374541167$$
$$x_{63} = -73.8545010149048$$
$$x_{64} = -45.5969279840735$$
$$x_{65} = 3.95930141892882 \cdot 10^{-7}$$
$$x_{66} = -2.2889297281034$$
$$x_{67} = -95.839441141233$$
The values of the extrema at the points:
(29.911893869551772, -892.728075975236)
(48.73570079490539, -2373.17105456709)
(86.4169374541167, -7465.88788203037)
(95.83944114123304, 9183.19913125177)
(83.27621716497754, 6932.92921007843)
(-51.874814053426775, -2688.99855997676)
(36.18353309075258, -1307.25263807613)
(0, 0)
(51.874814053426775, 2688.99855997676)
(67.573830670859, -4564.22390457183)
(33.04716869470536, 1090.12083594654)
(8.096163603222921, 63.6349819515545)
(-26.778087075558506, -715.074276149712)
(5.08698509410227, -24.0829602230683)
(-55.01420967883812, 3024.56524685288)
(-92.69855524339692, 8591.02284218332)
(61.2936749662429, -3754.91618650696)
(11.172705868329984, -122.876173513916)
(20.51752290994169, 418.982887272434)
(-36.18353309075258, 1307.25263807613)
(-23.64632381960362, 557.159297209023)
(-58.153842078645, -3379.87112092779)
(-20.51752290994169, -418.982887272434)
(70.7141100665485, 4998.48656158818)
(45.59692798407349, 2077.08272285774)
(14.276352918336478, 201.843217881861)
(42.458570771699044, -1800.73355411815)
(-5.08698509410227, 24.0829602230683)
(-29.911893869551772, 892.728075975236)
(98.98037186515228, -9795.11462678079)
(-42.458570771699044, 1800.73355411815)
(-80.13556519407445, 6419.70974281978)
(-89.55771888272442, -8018.58575924144)
(-11.172705868329984, 122.876173513916)
(2.2889297281034042, 3.94530162528433)
(-48.73570079490539, 2373.17105456709)
(-17.393243964594753, 300.544552657996)
(-120.96784897569329, -14631.2208957387)
(92.69855524339692, -8591.02284218332)
(39.32072813225213, 1544.1235331857)
(-39.32072813225213, -1544.1235331857)
(-83.27621716497754, -6932.92921007843)
(73.85450101490484, -5452.4884195005)
(58.153842078645, 3379.87112092779)
(-8.096163603222921, -63.6349819515545)
(-76.9949898891676, -5926.22947957101)
(64.43367910373156, 4149.70044687478)
(-64.43367910373156, -4149.70044687478)
(89.55771888272442, 8018.58575924144)
(55.01420967883812, -3024.56524685288)
(-33.04716869470536, -1090.12083594654)
(-67.573830670859, 4564.22390457183)
(80.13556519407445, -6419.70974281978)
(76.9949898891676, 5926.22947957101)
(-70.7141100665485, -4998.48656158818)
(-61.2936749662429, 3754.91618650696)
(17.393243964594753, -300.544552657996)
(26.778087075558506, 715.074276149712)
(-14.276352918336478, -201.843217881861)
(-98.98037186515228, 9795.11462678079)
(23.64632381960362, -557.159297209023)
(-86.4169374541167, 7465.88788203037)
(-73.85450101490484, 5452.4884195005)
(-45.59692798407349, -2077.08272285774)
(3.9593014189288195e-07, 6.20662771905043e-20)
(-2.2889297281034042, -3.94530162528433)
(-95.83944114123304, -9183.19913125177)
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:
$$x_{1} = 29.9118938695518$$
$$x_{2} = 48.7357007949054$$
$$x_{3} = 86.4169374541167$$
$$x_{4} = -51.8748140534268$$
$$x_{5} = 36.1835330907526$$
$$x_{6} = 67.573830670859$$
$$x_{7} = -26.7780870755585$$
$$x_{8} = 5.08698509410227$$
$$x_{9} = 61.2936749662429$$
$$x_{10} = 11.17270586833$$
$$x_{11} = -58.153842078645$$
$$x_{12} = -20.5175229099417$$
$$x_{13} = 42.458570771699$$
$$x_{14} = 98.9803718651523$$
$$x_{15} = -89.5577188827244$$
$$x_{16} = -120.967848975693$$
$$x_{17} = 92.6985552433969$$
$$x_{18} = -39.3207281322521$$
$$x_{19} = -83.2762171649775$$
$$x_{20} = 73.8545010149048$$
$$x_{21} = -8.09616360322292$$
$$x_{22} = -76.9949898891676$$
$$x_{23} = -64.4336791037316$$
$$x_{24} = 55.0142096788381$$
$$x_{25} = -33.0471686947054$$
$$x_{26} = 80.1355651940744$$
$$x_{27} = -70.7141100665485$$
$$x_{28} = 17.3932439645948$$
$$x_{29} = -14.2763529183365$$
$$x_{30} = 23.6463238196036$$
$$x_{31} = -45.5969279840735$$
$$x_{32} = -2.2889297281034$$
$$x_{33} = -95.839441141233$$
Maxima of the function at points:
$$x_{33} = 95.839441141233$$
$$x_{33} = 83.2762171649775$$
$$x_{33} = 51.8748140534268$$
$$x_{33} = 33.0471686947054$$
$$x_{33} = 8.09616360322292$$
$$x_{33} = -55.0142096788381$$
$$x_{33} = -92.6985552433969$$
$$x_{33} = 20.5175229099417$$
$$x_{33} = -36.1835330907526$$
$$x_{33} = -23.6463238196036$$
$$x_{33} = 70.7141100665485$$
$$x_{33} = 45.5969279840735$$
$$x_{33} = 14.2763529183365$$
$$x_{33} = -5.08698509410227$$
$$x_{33} = -29.9118938695518$$
$$x_{33} = -42.458570771699$$
$$x_{33} = -80.1355651940744$$
$$x_{33} = -11.17270586833$$
$$x_{33} = 2.2889297281034$$
$$x_{33} = -48.7357007949054$$
$$x_{33} = -17.3932439645948$$
$$x_{33} = 39.3207281322521$$
$$x_{33} = 58.153842078645$$
$$x_{33} = 64.4336791037316$$
$$x_{33} = 89.5577188827244$$
$$x_{33} = -67.573830670859$$
$$x_{33} = 76.9949898891676$$
$$x_{33} = -61.2936749662429$$
$$x_{33} = 26.7780870755585$$
$$x_{33} = -98.9803718651523$$
$$x_{33} = -86.4169374541167$$
$$x_{33} = -73.8545010149048$$
Decreasing at intervals
$$\left[98.9803718651523, \infty\right)$$
Increasing at intervals
$$\left(-\infty, -120.967848975693\right]$$