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$$- x^{2} \sin{\left(x \right)} + 2 x \cos{\left(x \right)} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = -87.9873209346887$$
$$x_{2} = 69.1439554764926$$
$$x_{3} = -44.0276918992479$$
$$x_{4} = -100.550852725424$$
$$x_{5} = -9.62956034329743$$
$$x_{6} = 31.479374920314$$
$$x_{7} = -40.8895777660408$$
$$x_{8} = -47.1662676027767$$
$$x_{9} = 97.4099011706723$$
$$x_{10} = -6.57833373272234$$
$$x_{11} = 28.3447768697864$$
$$x_{12} = 100.550852725424$$
$$x_{13} = -59.7237354324305$$
$$x_{14} = 18.954681766529$$
$$x_{15} = 59.7237354324305$$
$$x_{16} = -37.7520396346102$$
$$x_{17} = -1.0768739863118$$
$$x_{18} = 50.3052188363296$$
$$x_{19} = -56.5839987378634$$
$$x_{20} = -34.6152330552306$$
$$x_{21} = 40.8895777660408$$
$$x_{22} = 84.8465692433091$$
$$x_{23} = -15.8336114149477$$
$$x_{24} = -31.479374920314$$
$$x_{25} = 62.863657228703$$
$$x_{26} = 9.62956034329743$$
$$x_{27} = 47.1662676027767$$
$$x_{28} = 53.4444796697636$$
$$x_{29} = -50.3052188363296$$
$$x_{30} = -78.5652673845995$$
$$x_{31} = 34.6152330552306$$
$$x_{32} = 12.7222987717666$$
$$x_{33} = 75.4247339745236$$
$$x_{34} = -3.6435971674254$$
$$x_{35} = -69.1439554764926$$
$$x_{36} = 66.0037377708277$$
$$x_{37} = 25.2119030642106$$
$$x_{38} = 6.57833373272234$$
$$x_{39} = 91.1281305511393$$
$$x_{40} = 81.7058821480364$$
$$x_{41} = 44.0276918992479$$
$$x_{42} = 56.5839987378634$$
$$x_{43} = 37.7520396346102$$
$$x_{44} = 22.0814757672807$$
$$x_{45} = -18.954681766529$$
$$x_{46} = -81.7058821480364$$
$$x_{47} = 1.0768739863118$$
$$x_{48} = 15.8336114149477$$
$$x_{49} = 87.9873209346887$$
$$x_{50} = -72.2842925036825$$
$$x_{51} = -75.4247339745236$$
$$x_{52} = -12.7222987717666$$
$$x_{53} = -94.2689923093066$$
$$x_{54} = -66.0037377708277$$
$$x_{55} = -62.863657228703$$
$$x_{56} = -28.3447768697864$$
$$x_{57} = -97.4099011706723$$
$$x_{58} = 72.2842925036825$$
$$x_{59} = -22.0814757672807$$
$$x_{60} = 0$$
$$x_{61} = -53.4444796697636$$
$$x_{62} = -25.2119030642106$$
$$x_{63} = -84.8465692433091$$
$$x_{64} = 78.5652673845995$$
$$x_{65} = -91.1281305511393$$
$$x_{66} = 94.2689923093066$$
$$x_{67} = 3.6435971674254$$
The values of the extrema at the points:
(-87.9873209346887, 7739.76941994707)
(69.1439554764926, 4778.88783305817)
(-44.02769189924788, 1936.44074393829)
(-100.55085272542402, 10108.4745770583)
(-9.62956034329743, -90.7908960221418)
(31.479374920314047, 988.957079867956)
(-40.889577766040844, -1669.96115135336)
(-47.1662676027767, -2222.65949258718)
(97.40990117067226, -9486.68947818992)
(-6.578333732722339, 41.4032422108864)
(28.344776869786372, -801.433812963046)
(100.55085272542402, 10108.4745770583)
(-59.72373543243046, -3564.92625455388)
(18.954681766529042, 357.296507493256)
(59.72373543243046, -3564.92625455388)
(-37.75203963461023, 1423.22069663783)
(-1.0768739863118038, 0.549774025605498)
(50.30521883632959, 2528.6174100174)
(-56.58399873786344, 3199.75078519348)
(-34.61523305523058, -1196.21935302944)
(40.889577766040844, -1669.96115135336)
(84.84656924330915, -7196.94114542993)
(-15.833611414947718, -248.726869289185)
(-31.479374920314047, 988.957079867956)
(62.863657228703005, 3949.84091716867)
(9.62956034329743, -90.7908960221418)
(47.1662676027767, -2222.65949258718)
(53.44447966976355, -2854.3145053339)
(-50.30521883632959, 2528.6174100174)
(-78.56526738459954, -6170.50221074225)
(34.61523305523058, -1196.21935302944)
(12.722298771766635, 159.893208545431)
(75.4247339745236, 5686.89154919726)
(-3.643597167425401, -11.6378292117556)
(-69.1439554764926, 4778.88783305817)
(66.00373777082767, -4354.4947759217)
(25.21190306421058, 633.649446194351)
(6.578333732722339, 41.4032422108864)
(91.1281305511393, -8302.3368999697)
(81.70588214803641, 6673.85207590208)
(44.02769189924788, 1936.44074393829)
(56.58399873786344, 3199.75078519348)
(37.75203963461023, 1423.22069663783)
(22.081475767280747, -485.603793917741)
(-18.954681766529042, 357.296507493256)
(-81.70588214803641, 6673.85207590208)
(1.0768739863118038, 0.549774025605498)
(15.833611414947718, -248.726869289185)
(87.9873209346887, 7739.76941994707)
(-72.2842925036825, -5223.02009034704)
(-75.4247339745236, 5686.89154919726)
(-12.722298771766635, 159.893208545431)
(-94.26899230930657, 8884.64358592955)
(-66.00373777082767, -4354.4947759217)
(-62.863657228703005, 3949.84091716867)
(-28.344776869786372, -801.433812963046)
(-97.40990117067226, -9486.68947818992)
(72.2842925036825, -5223.02009034704)
(-22.081475767280747, -485.603793917741)
(0, 0)
(-53.44447966976355, -2854.3145053339)
(-25.21190306421058, 633.649446194351)
(-84.84656924330915, -7196.94114542993)
(78.56526738459954, -6170.50221074225)
(-91.1281305511393, -8302.3368999697)
(94.26899230930657, 8884.64358592955)
(3.643597167425401, -11.6378292117556)
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:
$$x_{1} = -9.62956034329743$$
$$x_{2} = -40.8895777660408$$
$$x_{3} = -47.1662676027767$$
$$x_{4} = 97.4099011706723$$
$$x_{5} = 28.3447768697864$$
$$x_{6} = -59.7237354324305$$
$$x_{7} = 59.7237354324305$$
$$x_{8} = -34.6152330552306$$
$$x_{9} = 40.8895777660408$$
$$x_{10} = 84.8465692433091$$
$$x_{11} = -15.8336114149477$$
$$x_{12} = 9.62956034329743$$
$$x_{13} = 47.1662676027767$$
$$x_{14} = 53.4444796697636$$
$$x_{15} = -78.5652673845995$$
$$x_{16} = 34.6152330552306$$
$$x_{17} = -3.6435971674254$$
$$x_{18} = 66.0037377708277$$
$$x_{19} = 91.1281305511393$$
$$x_{20} = 22.0814757672807$$
$$x_{21} = 15.8336114149477$$
$$x_{22} = -72.2842925036825$$
$$x_{23} = -66.0037377708277$$
$$x_{24} = -28.3447768697864$$
$$x_{25} = -97.4099011706723$$
$$x_{26} = 72.2842925036825$$
$$x_{27} = -22.0814757672807$$
$$x_{28} = 0$$
$$x_{29} = -53.4444796697636$$
$$x_{30} = -84.8465692433091$$
$$x_{31} = 78.5652673845995$$
$$x_{32} = -91.1281305511393$$
$$x_{33} = 3.6435971674254$$
Maxima of the function at points:
$$x_{33} = -87.9873209346887$$
$$x_{33} = 69.1439554764926$$
$$x_{33} = -44.0276918992479$$
$$x_{33} = -100.550852725424$$
$$x_{33} = 31.479374920314$$
$$x_{33} = -6.57833373272234$$
$$x_{33} = 100.550852725424$$
$$x_{33} = 18.954681766529$$
$$x_{33} = -37.7520396346102$$
$$x_{33} = -1.0768739863118$$
$$x_{33} = 50.3052188363296$$
$$x_{33} = -56.5839987378634$$
$$x_{33} = -31.479374920314$$
$$x_{33} = 62.863657228703$$
$$x_{33} = -50.3052188363296$$
$$x_{33} = 12.7222987717666$$
$$x_{33} = 75.4247339745236$$
$$x_{33} = -69.1439554764926$$
$$x_{33} = 25.2119030642106$$
$$x_{33} = 6.57833373272234$$
$$x_{33} = 81.7058821480364$$
$$x_{33} = 44.0276918992479$$
$$x_{33} = 56.5839987378634$$
$$x_{33} = 37.7520396346102$$
$$x_{33} = -18.954681766529$$
$$x_{33} = -81.7058821480364$$
$$x_{33} = 1.0768739863118$$
$$x_{33} = 87.9873209346887$$
$$x_{33} = -75.4247339745236$$
$$x_{33} = -12.7222987717666$$
$$x_{33} = -94.2689923093066$$
$$x_{33} = -62.863657228703$$
$$x_{33} = -25.2119030642106$$
$$x_{33} = 94.2689923093066$$
Decreasing at intervals
$$\left[97.4099011706723, \infty\right)$$
Increasing at intervals
$$\left(-\infty, -97.4099011706723\right]$$