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$$2 x^{2} \cos{\left(2 x \right)} + 2 x \sin{\left(2 x \right)} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = 16.5235843473527$$
$$x_{2} = 60.4839244878466$$
$$x_{3} = 40.0677825970372$$
$$x_{4} = -11.8231619098018$$
$$x_{5} = 25.9374070267134$$
$$x_{6} = 82.4728694594266$$
$$x_{7} = 38.4974949445838$$
$$x_{8} = -41.6381085824888$$
$$x_{9} = -63.6251091208926$$
$$x_{10} = 27.5071048394191$$
$$x_{11} = 77.760847792972$$
$$x_{12} = -98.1798629425939$$
$$x_{13} = -62.0545116429054$$
$$x_{14} = -99.7505790857949$$
$$x_{15} = -27.5071048394191$$
$$x_{16} = 74.6195257807054$$
$$x_{17} = 10.2587614549708$$
$$x_{18} = 0$$
$$x_{19} = 33.7869153354295$$
$$x_{20} = 8.69662198229738$$
$$x_{21} = -21.2292853858495$$
$$x_{22} = 85.6142396947314$$
$$x_{23} = 54.2016970313842$$
$$x_{24} = -77.760847792972$$
$$x_{25} = 11.8231619098018$$
$$x_{26} = -18.0917665453763$$
$$x_{27} = 52.6311758774383$$
$$x_{28} = 63.6251091208926$$
$$x_{29} = -69.9075883539626$$
$$x_{30} = -55.7722336752062$$
$$x_{31} = -71.4782275499213$$
$$x_{32} = 18.0917665453763$$
$$x_{33} = -4.04808180161146$$
$$x_{34} = 35.3570550332742$$
$$x_{35} = -76.1901839979235$$
$$x_{36} = -49.4901859325761$$
$$x_{37} = -79.3315168346756$$
$$x_{38} = -5.58635293416499$$
$$x_{39} = 32.2168395518658$$
$$x_{40} = 24.3678503974527$$
$$x_{41} = -10.2587614549708$$
$$x_{42} = 84.0435524991391$$
$$x_{43} = -46.3492776216985$$
$$x_{44} = -33.7869153354295$$
$$x_{45} = 69.9075883539626$$
$$x_{46} = -38.4974949445838$$
$$x_{47} = -40.0677825970372$$
$$x_{48} = 62.0545116429054$$
$$x_{49} = 68.3369563786298$$
$$x_{50} = 88.7556256712795$$
$$x_{51} = -25.9374070267134$$
$$x_{52} = -93.4677306800165$$
$$x_{53} = -57.3427845371101$$
$$x_{54} = -3.42962943093331 \cdot 10^{-7}$$
$$x_{55} = -54.2016970313842$$
$$x_{56} = 71.4782275499213$$
$$x_{57} = 30.6468374831214$$
$$x_{58} = -24.3678503974527$$
$$x_{59} = 41.6381085824888$$
$$x_{60} = -84.0435524991391$$
$$x_{61} = 4.04808180161146$$
$$x_{62} = -60.4839244878466$$
$$x_{63} = -68.3369563786298$$
$$x_{64} = 46.3492776216985$$
$$x_{65} = -47.9197205706165$$
$$x_{66} = -32.2168395518658$$
$$x_{67} = 65.1957161761796$$
$$x_{68} = 47.9197205706165$$
$$x_{69} = 98.1798629425939$$
$$x_{70} = -91.8970257752571$$
$$x_{71} = 55.7722336752062$$
$$x_{72} = 91.8970257752571$$
$$x_{73} = -44.7788594413622$$
$$x_{74} = -2.54349254705114$$
$$x_{75} = -3.16473361148914 \cdot 10^{-7}$$
$$x_{76} = -13.3890435377793$$
$$x_{77} = -82.4728694594266$$
$$x_{78} = -85.6142396947314$$
$$x_{79} = 19.6603640661261$$
$$x_{80} = 2.54349254705114$$
$$x_{81} = 76.1901839979235$$
$$x_{82} = 49.4901859325761$$
$$x_{83} = -19.6603640661261$$
$$x_{84} = 90.3263240494369$$
$$x_{85} = -35.3570550332742$$
$$x_{86} = -90.3263240494369$$
$$x_{87} = 7.13817645916824$$
$$x_{88} = 96.6091494063022$$
$$x_{89} = 95.0384386061415$$
$$x_{90} = 5.58635293416499$$
$$x_{91} = 99.7505790857949$$
The values of the extrema at the points:
(16.5235843473527, 272.530208986636)
(60.4839244878466, 3657.80522393468)
(40.0677825970372, -1604.92743570495)
(-11.8231619098018, 139.289824302256)
(25.9374070267134, 672.24963999419)
(82.4728694594266, 6801.27425199754)
(38.4974949445838, 1481.55736989275)
(-41.6381085824888, -1733.23230251961)
(-63.6251091208926, -4047.65460326123)
(27.5071048394191, -756.141311713221)
(77.760847792972, -6046.24951149001)
(-98.1798629425939, -9638.78552632646)
(-62.0545116429054, 3850.26251260173)
(-99.7505790857949, 9949.67806563604)
(-27.5071048394191, 756.141311713221)
(74.6195257807054, -5567.57369507552)
(10.2587614549708, 104.745721818108)
(0, 0)
(33.7869153354295, -1141.05597614296)
(8.69662198229738, -75.1361381644989)
(-21.2292853858495, 450.183388529538)
(85.6142396947314, 7329.29808966213)
(54.2016970313842, 2937.32408869126)
(-77.760847792972, 6046.24951149001)
(11.8231619098018, -139.289824302256)
(-18.0917665453763, 326.813159519034)
(52.6311758774383, -2769.54080957821)
(63.6251091208926, 4047.65460326123)
(-69.9075883539626, -4886.57098618708)
(-55.7722336752062, 3110.04216964728)
(-71.4782275499213, 5108.63708706427)
(18.0917665453763, -326.813159519034)
(-4.04808180161146, -15.9087454878886)
(35.3570550332742, 1249.62164039704)
(-76.1901839979235, -5804.44420222827)
(-49.4901859325761, 2448.7786566952)
(-79.3315168346756, -6292.98962286791)
(-5.58635293416499, 30.719043378479)
(32.2168395518658, 1037.4251117187)
(24.3678503974527, -593.292763641772)
(-10.2587614549708, -104.745721818108)
(84.0435524991391, -7062.81876976048)
(-46.3492776216985, 2147.75571054583)
(-33.7869153354295, 1141.05597614296)
(69.9075883539626, 4886.57098618708)
(-38.4974949445838, -1481.55736989275)
(-40.0677825970372, 1604.92743570495)
(62.0545116429054, -3850.26251260173)
(68.3369563786298, -4669.43968738125)
(88.7556256712795, 7877.06113589882)
(-25.9374070267134, -672.24963999419)
(-93.4677306800165, 8735.71672139277)
(-57.3427845371101, -3287.69505248487)
(-3.42962943093331e-7, -8.06810585778905e-20)
(-54.2016970313842, -2937.32408869126)
(71.4782275499213, -5108.63708706427)
(30.6468374831214, -938.729046626741)
(-24.3678503974527, 593.292763641772)
(41.6381085824888, 1733.23230251961)
(-84.0435524991391, 7062.81876976048)
(4.04808180161146, 15.9087454878886)
(-60.4839244878466, -3657.80522393468)
(-68.3369563786298, 4669.43968738125)
(46.3492776216985, -2147.75571054583)
(-47.9197205706165, -2295.79978281294)
(-32.2168395518658, -1037.4251117187)
(65.1957161761796, -4249.98149593298)
(47.9197205706165, 2295.79978281294)
(98.1798629425939, 9638.78552632646)
(-91.8970257752571, -8444.56339073853)
(55.7722336752062, -3110.04216964728)
(91.8970257752571, 8444.56339073853)
(-44.7788594413622, -2004.64643981036)
(-2.54349254705114, 6.02074005576708)
(-3.16473361148914e-7, -6.33930247556382e-20)
(-13.3890435377793, -178.768569037428)
(-82.4728694594266, -6801.27425199754)
(-85.6142396947314, -7329.29808966213)
(19.6603640661261, 386.030883296424)
(2.54349254705114, -6.02074005576708)
(76.1901839979235, 5804.44420222827)
(49.4901859325761, -2448.7786566952)
(-19.6603640661261, -386.030883296424)
(90.3263240494369, -8158.34486224158)
(-35.3570550332742, -1249.62164039704)
(-90.3263240494369, 8158.34486224158)
(7.13817645916824, 50.4608044704652)
(96.6091494063022, -9332.82778918424)
(95.0384386061415, 9031.80485420714)
(5.58635293416499, -30.719043378479)
(99.7505790857949, -9949.67806563604)
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:
$$x_{1} = 40.0677825970372$$
$$x_{2} = -41.6381085824888$$
$$x_{3} = -63.6251091208926$$
$$x_{4} = 27.5071048394191$$
$$x_{5} = 77.760847792972$$
$$x_{6} = -98.1798629425939$$
$$x_{7} = 74.6195257807054$$
$$x_{8} = 33.7869153354295$$
$$x_{9} = 8.69662198229738$$
$$x_{10} = 11.8231619098018$$
$$x_{11} = 52.6311758774383$$
$$x_{12} = -69.9075883539626$$
$$x_{13} = 18.0917665453763$$
$$x_{14} = -4.04808180161146$$
$$x_{15} = -76.1901839979235$$
$$x_{16} = -79.3315168346756$$
$$x_{17} = 24.3678503974527$$
$$x_{18} = -10.2587614549708$$
$$x_{19} = 84.0435524991391$$
$$x_{20} = -38.4974949445838$$
$$x_{21} = 62.0545116429054$$
$$x_{22} = 68.3369563786298$$
$$x_{23} = -25.9374070267134$$
$$x_{24} = -57.3427845371101$$
$$x_{25} = -54.2016970313842$$
$$x_{26} = 71.4782275499213$$
$$x_{27} = 30.6468374831214$$
$$x_{28} = -60.4839244878466$$
$$x_{29} = 46.3492776216985$$
$$x_{30} = -47.9197205706165$$
$$x_{31} = -32.2168395518658$$
$$x_{32} = 65.1957161761796$$
$$x_{33} = -91.8970257752571$$
$$x_{34} = 55.7722336752062$$
$$x_{35} = -44.7788594413622$$
$$x_{36} = -13.3890435377793$$
$$x_{37} = -82.4728694594266$$
$$x_{38} = -85.6142396947314$$
$$x_{39} = 2.54349254705114$$
$$x_{40} = 49.4901859325761$$
$$x_{41} = -19.6603640661261$$
$$x_{42} = 90.3263240494369$$
$$x_{43} = -35.3570550332742$$
$$x_{44} = 96.6091494063022$$
$$x_{45} = 5.58635293416499$$
$$x_{46} = 99.7505790857949$$
Maxima of the function at points:
$$x_{46} = 16.5235843473527$$
$$x_{46} = 60.4839244878466$$
$$x_{46} = -11.8231619098018$$
$$x_{46} = 25.9374070267134$$
$$x_{46} = 82.4728694594266$$
$$x_{46} = 38.4974949445838$$
$$x_{46} = -62.0545116429054$$
$$x_{46} = -99.7505790857949$$
$$x_{46} = -27.5071048394191$$
$$x_{46} = 10.2587614549708$$
$$x_{46} = -21.2292853858495$$
$$x_{46} = 85.6142396947314$$
$$x_{46} = 54.2016970313842$$
$$x_{46} = -77.760847792972$$
$$x_{46} = -18.0917665453763$$
$$x_{46} = 63.6251091208926$$
$$x_{46} = -55.7722336752062$$
$$x_{46} = -71.4782275499213$$
$$x_{46} = 35.3570550332742$$
$$x_{46} = -49.4901859325761$$
$$x_{46} = -5.58635293416499$$
$$x_{46} = 32.2168395518658$$
$$x_{46} = -46.3492776216985$$
$$x_{46} = -33.7869153354295$$
$$x_{46} = 69.9075883539626$$
$$x_{46} = -40.0677825970372$$
$$x_{46} = 88.7556256712795$$
$$x_{46} = -93.4677306800165$$
$$x_{46} = -24.3678503974527$$
$$x_{46} = 41.6381085824888$$
$$x_{46} = -84.0435524991391$$
$$x_{46} = 4.04808180161146$$
$$x_{46} = -68.3369563786298$$
$$x_{46} = 47.9197205706165$$
$$x_{46} = 98.1798629425939$$
$$x_{46} = 91.8970257752571$$
$$x_{46} = -2.54349254705114$$
$$x_{46} = 19.6603640661261$$
$$x_{46} = 76.1901839979235$$
$$x_{46} = -90.3263240494369$$
$$x_{46} = 7.13817645916824$$
$$x_{46} = 95.0384386061415$$
Decreasing at intervals
$$\left[99.7505790857949, \infty\right)$$
Increasing at intervals
$$\left(-\infty, -98.1798629425939\right]$$