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$$2 x \sin{\left(x \right)} + \left(x^{2} + 3\right) \cos{\left(x \right)} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = -51.8747712151257$$
$$x_{2} = -70.7140931289331$$
$$x_{3} = 17.3921364947438$$
$$x_{4} = -80.135553551012$$
$$x_{5} = -64.4336567231008$$
$$x_{6} = -39.3206300098787$$
$$x_{7} = 61.2936489726992$$
$$x_{8} = 92.6985477188943$$
$$x_{9} = 98.9803656834899$$
$$x_{10} = -67.573811263755$$
$$x_{11} = -73.8544861451257$$
$$x_{12} = 67.573811263755$$
$$x_{13} = -29.9116719072751$$
$$x_{14} = -8.08620023474585$$
$$x_{15} = 89.5577105390911$$
$$x_{16} = -14.2743775435397$$
$$x_{17} = -55.014173750457$$
$$x_{18} = 45.5968649653267$$
$$x_{19} = 55.014173750457$$
$$x_{20} = -48.7356491570733$$
$$x_{21} = 73.8544861451257$$
$$x_{22} = -95.8394343320789$$
$$x_{23} = -76.9949767639905$$
$$x_{24} = -11.1686903887487$$
$$x_{25} = 33.0470038069223$$
$$x_{26} = 95.8394343320789$$
$$x_{27} = 5.05280036176264$$
$$x_{28} = 70.7140931289331$$
$$x_{29} = -139.815174513663$$
$$x_{30} = 20.5168427394902$$
$$x_{31} = -42.4584927718185$$
$$x_{32} = -61.2936489726992$$
$$x_{33} = -33.0470038069223$$
$$x_{34} = -20.5168427394902$$
$$x_{35} = 51.8747712151257$$
$$x_{36} = -5.05280036176264$$
$$x_{37} = -36.1834073007459$$
$$x_{38} = 26.7777784687197$$
$$x_{39} = -98.9803656834899$$
$$x_{40} = -58.1538116514278$$
$$x_{41} = 14.2743775435397$$
$$x_{42} = 48.7356491570733$$
$$x_{43} = -2.08694849327138$$
$$x_{44} = 23.6458771958085$$
$$x_{45} = -83.2762067890928$$
$$x_{46} = 86.4169281680532$$
$$x_{47} = -45.5968649653267$$
$$x_{48} = 8.08620023474585$$
$$x_{49} = 64.4336567231008$$
$$x_{50} = 42.4584927718185$$
$$x_{51} = 83.2762067890928$$
$$x_{52} = -23.6458771958085$$
$$x_{53} = 58.1538116514278$$
$$x_{54} = 2.08694849327138$$
$$x_{55} = -17.3921364947438$$
$$x_{56} = -92.6985477188943$$
$$x_{57} = 36.1834073007459$$
$$x_{58} = -86.4169281680532$$
$$x_{59} = -89.5577105390911$$
$$x_{60} = 11.1686903887487$$
$$x_{61} = 29.9116719072751$$
$$x_{62} = -26.7777784687197$$
$$x_{63} = 39.3206300098787$$
$$x_{64} = 76.9949767639905$$
$$x_{65} = 80.135553551012$$
The values of the extrema at the points:
(-51.87477121512573, -2691.99633527749)
(-70.71409312893314, -5001.48536314231)
(17.39213649474379, -303.52510387424)
(-80.13555355101198, 6422.70880936104)
(-64.43365672310077, -4152.69900376829)
(-39.32063000987867, -1547.11966749316)
(61.2936489726992, -3757.91459199749)
(92.69854771889428, -8594.0221444295)
(98.98036568348985, -9798.11401473032)
(-67.57381126375503, 4567.22259229932)
(-73.85448614512573, 5455.48732069721)
(67.57381126375503, -4567.22259229932)
(-29.911671907275117, 895.721414590771)
(-8.086200234745847, -66.5510144264642)
(89.55771053909113, 8021.58501172545)
(-14.274377543539684, -204.814616709738)
(-55.01417375045699, 3027.56326832712)
(45.59686496532671, 2080.07984526547)
(55.01417375045699, -3027.56326832712)
(-48.7356491570733, 2376.16853479274)
(73.85448614512573, -5455.48732069721)
(-95.83943433207891, -9186.19847845327)
(-76.99497676399054, -5929.22846848739)
(-11.16869038874869, 125.830294133831)
(33.0470038069223, 1093.11537200813)
(95.83943433207891, 9186.19847845327)
(5.052800361762636, -26.8936203679094)
(70.71409312893314, 5001.48536314231)
(-139.8151745136635, -19549.2836379102)
(20.516842739490215, 421.968834187781)
(-42.458492771818506, 1803.7302368668)
(-61.2936489726992, 3757.91459199749)
(-33.0470038069223, -1093.11537200813)
(-20.516842739490215, -421.968834187781)
(51.87477121512573, 2691.99633527749)
(-5.052800361762636, 26.8936203679094)
(-36.18340730074592, 1310.24807618067)
(26.77777846871971, 718.065978041015)
(-98.98036568348985, 9798.11401473032)
(-58.1538116514278, -3382.8693499021)
(14.274377543539684, 204.814616709738)
(48.7356491570733, -2376.16853479274)
(-2.0869484932713838, -6.39713278797974)
(23.645877195808538, -560.148680302034)
(-83.27620678909281, -6935.92834564062)
(86.41692816805318, -7468.88707923516)
(-45.59686496532671, -2080.07984526547)
(8.086200234745847, 66.5510144264642)
(64.43365672310077, 4152.69900376829)
(42.458492771818506, -1803.7302368668)
(83.27620678909281, 6935.92834564062)
(-23.645877195808538, 560.148680302034)
(58.1538116514278, 3382.8693499021)
(2.0869484932713838, 6.39713278797974)
(-17.39213649474379, 303.52510387424)
(-92.69854771889428, 8594.0221444295)
(36.18340730074592, -1310.24807618067)
(-86.41692816805318, 7468.88707923516)
(-89.55771053909113, -8021.58501172545)
(11.16869038874869, -125.830294133831)
(29.911671907275117, -895.721414590771)
(-26.77777846871971, -718.065978041015)
(39.32063000987867, 1547.11966749316)
(76.99497676399054, 5929.22846848739)
(80.13555355101198, -6422.70880936104)
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:
$$x_{1} = -51.8747712151257$$
$$x_{2} = -70.7140931289331$$
$$x_{3} = 17.3921364947438$$
$$x_{4} = -64.4336567231008$$
$$x_{5} = -39.3206300098787$$
$$x_{6} = 61.2936489726992$$
$$x_{7} = 92.6985477188943$$
$$x_{8} = 98.9803656834899$$
$$x_{9} = 67.573811263755$$
$$x_{10} = -8.08620023474585$$
$$x_{11} = -14.2743775435397$$
$$x_{12} = 55.014173750457$$
$$x_{13} = 73.8544861451257$$
$$x_{14} = -95.8394343320789$$
$$x_{15} = -76.9949767639905$$
$$x_{16} = 5.05280036176264$$
$$x_{17} = -139.815174513663$$
$$x_{18} = -33.0470038069223$$
$$x_{19} = -20.5168427394902$$
$$x_{20} = -58.1538116514278$$
$$x_{21} = 48.7356491570733$$
$$x_{22} = -2.08694849327138$$
$$x_{23} = 23.6458771958085$$
$$x_{24} = -83.2762067890928$$
$$x_{25} = 86.4169281680532$$
$$x_{26} = -45.5968649653267$$
$$x_{27} = 42.4584927718185$$
$$x_{28} = 36.1834073007459$$
$$x_{29} = -89.5577105390911$$
$$x_{30} = 11.1686903887487$$
$$x_{31} = 29.9116719072751$$
$$x_{32} = -26.7777784687197$$
$$x_{33} = 80.135553551012$$
Maxima of the function at points:
$$x_{33} = -80.135553551012$$
$$x_{33} = -67.573811263755$$
$$x_{33} = -73.8544861451257$$
$$x_{33} = -29.9116719072751$$
$$x_{33} = 89.5577105390911$$
$$x_{33} = -55.014173750457$$
$$x_{33} = 45.5968649653267$$
$$x_{33} = -48.7356491570733$$
$$x_{33} = -11.1686903887487$$
$$x_{33} = 33.0470038069223$$
$$x_{33} = 95.8394343320789$$
$$x_{33} = 70.7140931289331$$
$$x_{33} = 20.5168427394902$$
$$x_{33} = -42.4584927718185$$
$$x_{33} = -61.2936489726992$$
$$x_{33} = 51.8747712151257$$
$$x_{33} = -5.05280036176264$$
$$x_{33} = -36.1834073007459$$
$$x_{33} = 26.7777784687197$$
$$x_{33} = -98.9803656834899$$
$$x_{33} = 14.2743775435397$$
$$x_{33} = 8.08620023474585$$
$$x_{33} = 64.4336567231008$$
$$x_{33} = 83.2762067890928$$
$$x_{33} = -23.6458771958085$$
$$x_{33} = 58.1538116514278$$
$$x_{33} = 2.08694849327138$$
$$x_{33} = -17.3921364947438$$
$$x_{33} = -92.6985477188943$$
$$x_{33} = -86.4169281680532$$
$$x_{33} = 39.3206300098787$$
$$x_{33} = 76.9949767639905$$
Decreasing at intervals
$$\left[98.9803656834899, \infty\right)$$
Increasing at intervals
$$\left(-\infty, -139.815174513663\right]$$