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$$x^{2} \cos{\left(x \right)} + x \sin{\left(x \right)} + \cos{\left(x \right)} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = -89.5465561460837$$
$$x_{2} = -54.9960465516394$$
$$x_{3} = -11.0848217516286$$
$$x_{4} = 73.8409666666368$$
$$x_{5} = 76.9820071395694$$
$$x_{6} = 45.5750212470478$$
$$x_{7} = 29.8785491411967$$
$$x_{8} = 33.0169733260496$$
$$x_{9} = -76.9820071395694$$
$$x_{10} = -33.0169733260496$$
$$x_{11} = -7.97678462750978$$
$$x_{12} = -58.1366581602094$$
$$x_{13} = -4.90565966567415$$
$$x_{14} = 1.95588396372027$$
$$x_{15} = 70.6999752105956$$
$$x_{16} = 11.0848217516286$$
$$x_{17} = -64.4181679836523$$
$$x_{18} = -1.95588396372027$$
$$x_{19} = -61.2773701909402$$
$$x_{20} = -23.6042091422871$$
$$x_{21} = -48.7152020786718$$
$$x_{22} = 36.1559453106627$$
$$x_{23} = -36.1559453106627$$
$$x_{24} = -92.6877705166201$$
$$x_{25} = -73.8409666666368$$
$$x_{26} = -70.6999752105956$$
$$x_{27} = -14.2070931146973$$
$$x_{28} = -26.7408639367464$$
$$x_{29} = 83.264212972531$$
$$x_{30} = -42.4350488166082$$
$$x_{31} = -29.8785491411967$$
$$x_{32} = 17.3361878945599$$
$$x_{33} = 98.9702712571724$$
$$x_{34} = 14.2070931146973$$
$$x_{35} = 4.90565966567415$$
$$x_{36} = -39.2953345326852$$
$$x_{37} = 7.97678462750978$$
$$x_{38} = 92.6877705166201$$
$$x_{39} = -86.4053692621466$$
$$x_{40} = -17.3361878945599$$
$$x_{41} = -98.9702712571724$$
$$x_{42} = 61.2773701909402$$
$$x_{43} = 42.4350488166082$$
$$x_{44} = 86.4053692621466$$
$$x_{45} = -95.8290096730487$$
$$x_{46} = 67.559039597919$$
$$x_{47} = 64.4181679836523$$
$$x_{48} = 26.7408639367464$$
$$x_{49} = 39.2953345326852$$
$$x_{50} = 20.4690516301297$$
$$x_{51} = 58.1366581602094$$
$$x_{52} = 95.8290096730487$$
$$x_{53} = 23.6042091422871$$
$$x_{54} = -80.1230908716212$$
$$x_{55} = -67.559039597919$$
$$x_{56} = -83.264212972531$$
$$x_{57} = 48.7152020786718$$
$$x_{58} = -45.5750212470478$$
$$x_{59} = -20.4690516301297$$
$$x_{60} = -51.8555535655761$$
$$x_{61} = 80.1230908716212$$
$$x_{62} = 54.9960465516394$$
$$x_{63} = 51.8555535655761$$
$$x_{64} = 89.5465561460837$$
The values of the extrema at the points:
(-89.5465561460837, -8017.08607606954)
(-54.9960465516394, 3023.06608618409)
(-11.0848217516286, 121.396268979983)
(73.8409666666368, -5450.98888533752)
(76.9820071395694, 5924.72990819201)
(45.5750212470478, 2075.5839443892)
(29.8785491411967, -891.230911505695)
(33.0169733260496, 1088.62315974023)
(-76.9820071395694, -5924.72990819201)
(-33.0169733260496, -1088.62315974023)
(-7.97678462750978, -62.1728011783095)
(-58.1366581602094, -3378.37187211994)
(-4.90565966567415, 22.6751995394057)
(1.95588396372027, 2.81061592389021)
(70.6999752105956, 4996.9870697054)
(11.0848217516286, -121.396268979983)
(-64.4181679836523, -4148.20105883203)
(-1.95588396372027, -2.81061592389021)
(-61.2773701909402, 3753.41686274186)
(-23.6042091422871, 555.66382949141)
(-48.7152020786718, 2371.67212392755)
(36.1559453106627, -1305.75457696247)
(-36.1559453106627, 1305.75457696247)
(-92.6877705166201, 8589.52313790947)
(-73.8409666666368, 5450.98888533752)
(-70.6999752105956, -4996.9870697054)
(-14.2070931146973, -200.35558843529)
(-26.7408639367464, -713.577812585208)
(83.264212972531, 6931.42957649394)
(-42.4350488166082, 1799.2349627339)
(-29.8785491411967, 891.230911505695)
(17.3361878945599, -299.052908734495)
(98.9702712571724, -9793.61488616748)
(14.2070931146973, 200.35558843529)
(4.90565966567415, -22.6751995394057)
(-39.2953345326852, -1542.62517534231)
(7.97678462750978, 62.1728011783095)
(92.6877705166201, -8589.52313790947)
(-86.4053692621466, 7464.38822230175)
(-17.3361878945599, 299.052908734495)
(-98.9702712571724, 9793.61488616748)
(61.2773701909402, -3753.41686274186)
(42.4350488166082, -1799.2349627339)
(86.4053692621466, -7464.38822230175)
(-95.8290096730487, -9181.69940791564)
(67.559039597919, -4562.72446099531)
(64.4181679836523, 4148.20105883203)
(26.7408639367464, 713.577812585208)
(39.2953345326852, 1542.62517534231)
(20.4690516301297, 417.488901448924)
(58.1366581602094, 3378.37187211994)
(95.8290096730487, 9181.69940791564)
(23.6042091422871, -555.66382949141)
(-80.1230908716212, 6418.21013851154)
(-67.559039597919, 4562.72446099531)
(-83.264212972531, -6931.42957649394)
(48.7152020786718, -2371.67212392755)
(-45.5750212470478, -2075.5839443892)
(-20.4690516301297, -417.488901448924)
(-51.8555535655761, -2687.49950390865)
(80.1230908716212, -6418.21013851154)
(54.9960465516394, -3023.06608618409)
(51.8555535655761, 2687.49950390865)
(89.5465561460837, 8017.08607606954)
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:
$$x_{1} = -89.5465561460837$$
$$x_{2} = 73.8409666666368$$
$$x_{3} = 29.8785491411967$$
$$x_{4} = -76.9820071395694$$
$$x_{5} = -33.0169733260496$$
$$x_{6} = -7.97678462750978$$
$$x_{7} = -58.1366581602094$$
$$x_{8} = 11.0848217516286$$
$$x_{9} = -64.4181679836523$$
$$x_{10} = -1.95588396372027$$
$$x_{11} = 36.1559453106627$$
$$x_{12} = -70.6999752105956$$
$$x_{13} = -14.2070931146973$$
$$x_{14} = -26.7408639367464$$
$$x_{15} = 17.3361878945599$$
$$x_{16} = 98.9702712571724$$
$$x_{17} = 4.90565966567415$$
$$x_{18} = -39.2953345326852$$
$$x_{19} = 92.6877705166201$$
$$x_{20} = 61.2773701909402$$
$$x_{21} = 42.4350488166082$$
$$x_{22} = 86.4053692621466$$
$$x_{23} = -95.8290096730487$$
$$x_{24} = 67.559039597919$$
$$x_{25} = 23.6042091422871$$
$$x_{26} = -83.264212972531$$
$$x_{27} = 48.7152020786718$$
$$x_{28} = -45.5750212470478$$
$$x_{29} = -20.4690516301297$$
$$x_{30} = -51.8555535655761$$
$$x_{31} = 80.1230908716212$$
$$x_{32} = 54.9960465516394$$
Maxima of the function at points:
$$x_{32} = -54.9960465516394$$
$$x_{32} = -11.0848217516286$$
$$x_{32} = 76.9820071395694$$
$$x_{32} = 45.5750212470478$$
$$x_{32} = 33.0169733260496$$
$$x_{32} = -4.90565966567415$$
$$x_{32} = 1.95588396372027$$
$$x_{32} = 70.6999752105956$$
$$x_{32} = -61.2773701909402$$
$$x_{32} = -23.6042091422871$$
$$x_{32} = -48.7152020786718$$
$$x_{32} = -36.1559453106627$$
$$x_{32} = -92.6877705166201$$
$$x_{32} = -73.8409666666368$$
$$x_{32} = 83.264212972531$$
$$x_{32} = -42.4350488166082$$
$$x_{32} = -29.8785491411967$$
$$x_{32} = 14.2070931146973$$
$$x_{32} = 7.97678462750978$$
$$x_{32} = -86.4053692621466$$
$$x_{32} = -17.3361878945599$$
$$x_{32} = -98.9702712571724$$
$$x_{32} = 64.4181679836523$$
$$x_{32} = 26.7408639367464$$
$$x_{32} = 39.2953345326852$$
$$x_{32} = 20.4690516301297$$
$$x_{32} = 58.1366581602094$$
$$x_{32} = 95.8290096730487$$
$$x_{32} = -80.1230908716212$$
$$x_{32} = -67.559039597919$$
$$x_{32} = 51.8555535655761$$
$$x_{32} = 89.5465561460837$$
Decreasing at intervals
$$\left[98.9702712571724, \infty\right)$$
Increasing at intervals
$$\left(-\infty, -95.8290096730487\right]$$