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$$- 8 \cdot \frac{1}{3 x} x - 3 \cos{\left(3 x \right)} - \frac{- 4 x^{2} + 2}{3 x^{2}} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = -9.79399613491955$$
$$x_{2} = 71.8865662557331$$
$$x_{3} = 73.9809604649365$$
$$x_{4} = -95.6650489018407$$
$$x_{5} = -29.9987400979142$$
$$x_{6} = -82.3585379571436$$
$$x_{7} = 78.1697490939037$$
$$x_{8} = -48.8482387828323$$
$$x_{9} = -3.50490760181752$$
$$x_{10} = 64.2491113693648$$
$$x_{11} = -36.2818963327114$$
$$x_{12} = 23.7156099376309$$
$$x_{13} = -38.3762847652043$$
$$x_{14} = -13.9832259109932$$
$$x_{15} = 57.9659214840189$$
$$x_{16} = -78.1697490939037$$
$$x_{17} = 40.470674206219$$
$$x_{18} = 11.8886685462031$$
$$x_{19} = -76.0753547468905$$
$$x_{20} = 29.9987400979142$$
$$x_{21} = 88.6417215972974$$
$$x_{22} = 95.6650489018407$$
$$x_{23} = -53.7771272959775$$
$$x_{24} = 16.0777240784174$$
$$x_{25} = 13.9832259109932$$
$$x_{26} = 38.3762847652043$$
$$x_{27} = 62.1547148941877$$
$$x_{28} = -42.5650644622225$$
$$x_{29} = 60.0603182729038$$
$$x_{30} = -45.3995353597057$$
$$x_{31} = -67.6977780924208$$
$$x_{32} = -57.9659214840189$$
$$x_{33} = -97.7594443872384$$
$$x_{34} = 53.7771272959775$$
$$x_{35} = 7.69906717612363$$
$$x_{36} = 5.60342975884779$$
$$x_{37} = -25.8099821429868$$
$$x_{38} = 42.5650644622225$$
$$x_{39} = 20.2666329000236$$
$$x_{40} = -89.3818622796945$$
$$x_{41} = 34.1875091633272$$
$$x_{42} = -21.6212447049234$$
$$x_{43} = -93.5706533904375$$
$$x_{44} = -65.6033841604836$$
$$x_{45} = 9.79399613491955$$
$$x_{46} = -51.6827298291268$$
$$x_{47} = -73.9809604649365$$
$$x_{48} = 87.2874666746391$$
$$x_{49} = -16.0777240784174$$
$$x_{50} = -99.8538398487895$$
$$x_{51} = -1.37163640284868$$
$$x_{52} = 99.8538398487895$$
$$x_{53} = -55.8715245019358$$
$$x_{54} = -91.4762578506202$$
$$x_{55} = 86.5473270104187$$
$$x_{56} = 84.4529324623676$$
$$x_{57} = 68.4379039512863$$
$$x_{58} = 36.2818963327114$$
$$x_{59} = -47.4939339226263$$
$$x_{60} = -11.8886685462031$$
$$x_{61} = -463.538434786562$$
$$x_{62} = 80.2641434992748$$
$$x_{63} = -34.1875091633272$$
$$x_{64} = 44.6594553839724$$
$$x_{65} = -60.0603182729038$$
$$x_{66} = -49.5883320563873$$
$$x_{67} = 18.1721887001287$$
$$x_{68} = 66.3435077165907$$
$$x_{69} = -23.7156099376309$$
$$x_{70} = 27.9043593082823$$
$$x_{71} = -69.792172128143$$
$$x_{72} = 82.3585379571436$$
$$x_{73} = 25.8099821429868$$
$$x_{74} = 92.8305108737602$$
$$x_{75} = 76.0753547468905$$
$$x_{76} = -80.2641434992748$$
$$x_{77} = 32.093123597352$$
$$x_{78} = 55.8715245019358$$
$$x_{79} = -32.093123597352$$
$$x_{80} = -86.5473270104187$$
$$x_{81} = -27.9043593082823$$
$$x_{82} = -71.8865662557331$$
$$x_{83} = -18.1721887001287$$
$$x_{84} = 22.3610639591715$$
$$x_{85} = 51.6827298291268$$
$$x_{86} = -0.819394360115899$$
$$x_{87} = 48.8482387828323$$
$$x_{88} = 97.7594443872384$$
$$x_{89} = -7.69906717612363$$
$$x_{90} = -5.60342975884779$$
$$x_{91} = -62.1547148941877$$
The values of the extrema at the points:
(-9.79399613491955, 12.0959393213563)
(71.8865662557331, -96.7352662177534)
(73.9809604649365, -99.5280555634645)
(-95.6650489018407, 126.650635407907)
(-29.9987400979142, 40.871780835358)
(-82.3585379571436, 110.699079416119)
(78.1697490939037, -105.113592049234)
(-48.8482387828323, 66.0130975373671)
(-3.50490760181752, 3.59639924710933)
(64.2491113693648, -84.7593258418893)
(-36.2818963327114, 49.253209781807)
(23.7156099376309, -32.4883126481304)
(-38.3762847652043, 52.0467393852872)
(-13.9832259109932, 17.7013833925527)
(57.9659214840189, -76.3806207011282)
(-78.1697490939037, 105.113592049234)
(40.470674206219, -54.8401651973731)
(11.8886685462031, -14.9004576138098)
(-76.0753547468905, 102.32083045331)
(29.9987400979142, -40.871780835358)
(88.6417215972974, -119.077233600926)
(95.6650489018407, -126.650635407907)
(-53.7771272959775, 70.7946712618512)
(16.0777240784174, -20.5001208192217)
(13.9832259109932, -17.7013833925527)
(38.3762847652043, -52.0467393852872)
(62.1547148941877, -81.9664493946416)
(-42.5650644622225, 57.633502539013)
(60.0603182729038, -79.1735485621318)
(-45.3995353597057, 59.6222764583461)
(-67.6977780924208, 91.1496387920123)
(-57.9659214840189, 76.3806207011282)
(-97.7594443872384, 129.44331150956)
(53.7771272959775, -70.7946712618512)
(7.69906717612363, -9.28489571149604)
(5.60342975884779, -6.46000490959713)
(-25.8099821429868, 35.2831205568066)
(42.5650644622225, -57.633502539013)
(20.2666329000236, -26.0937446234305)
(-89.3818622796945, 118.272565122617)
(34.1875091633272, -46.4595573116555)
(-21.6212447049234, 29.6930628151633)
(-93.5706533904375, 123.857952622749)
(-65.6033841604836, 88.3567976002888)
(9.79399613491955, -12.0959393213563)
(-51.6827298291268, 68.0016420879503)
(-73.9809604649365, 99.5280555634645)
(87.2874666746391, -115.479859355266)
(-16.0777240784174, 20.5001208192217)
(-99.8538398487895, 132.235981348257)
(-1.37163640284868, 0.516055061769062)
(99.8538398487895, -132.235981348257)
(-55.8715245019358, 73.5876627720626)
(-91.4762578506202, 121.065262695017)
(86.5473270104187, -116.284524796006)
(84.4529324623676, -113.491806963966)
(68.4379039512863, -90.3450145357121)
(36.2818963327114, -49.253209781807)
(-47.4939339226263, 62.4154508182383)
(-11.8886685462031, 14.9004576138098)
(-463.538434786562, 618.945614073165)
(80.2641434992748, -107.906341391901)
(-34.1875091633272, 46.4595573116555)
(44.6594553839724, -60.4267638571503)
(-60.0603182729038, 79.1735485621318)
(-49.5883320563873, 65.2085704780983)
(18.1721887001287, -23.2974266048808)
(66.3435077165907, -87.5521802133293)
(-23.7156099376309, 32.4883126481304)
(27.9043593082823, -38.077586046588)
(-69.792172128143, 93.9424611147587)
(82.3585379571436, -110.699079416119)
(25.8099821429868, -35.2831205568066)
(92.8305108737602, -124.662626573384)
(76.0753547468905, -102.32083045331)
(-80.2641434992748, 107.906341391901)
(32.093123597352, -43.6657579202123)
(55.8715245019358, -73.5876627720626)
(-32.093123597352, 43.6657579202123)
(-86.5473270104187, 116.284524796006)
(-27.9043593082823, 38.077586046588)
(-71.8865662557331, 96.7352662177534)
(-18.1721887001287, 23.2974266048808)
(22.3610639591715, -28.8893524376558)
(51.6827298291268, -68.0016420879503)
(-0.819394360115899, 0.910357318885714)
(48.8482387828323, -66.0130975373671)
(97.7594443872384, -129.44331150956)
(-7.69906717612363, 9.28489571149604)
(-5.60342975884779, 6.46000490959713)
(-62.1547148941877, 81.9664493946416)
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:
$$x_{1} = -9.79399613491955$$
$$x_{2} = 71.8865662557331$$
$$x_{3} = 73.9809604649365$$
$$x_{4} = -95.6650489018407$$
$$x_{5} = 78.1697490939037$$
$$x_{6} = -3.50490760181752$$
$$x_{7} = 23.7156099376309$$
$$x_{8} = -13.9832259109932$$
$$x_{9} = 40.470674206219$$
$$x_{10} = 29.9987400979142$$
$$x_{11} = 88.6417215972974$$
$$x_{12} = -53.7771272959775$$
$$x_{13} = 38.3762847652043$$
$$x_{14} = -45.3995353597057$$
$$x_{15} = -57.9659214840189$$
$$x_{16} = -97.7594443872384$$
$$x_{17} = 42.5650644622225$$
$$x_{18} = -89.3818622796945$$
$$x_{19} = 34.1875091633272$$
$$x_{20} = -93.5706533904375$$
$$x_{21} = -51.6827298291268$$
$$x_{22} = -16.0777240784174$$
$$x_{23} = -99.8538398487895$$
$$x_{24} = -1.37163640284868$$
$$x_{25} = -55.8715245019358$$
$$x_{26} = -91.4762578506202$$
$$x_{27} = 86.5473270104187$$
$$x_{28} = 84.4529324623676$$
$$x_{29} = 36.2818963327114$$
$$x_{30} = -47.4939339226263$$
$$x_{31} = -11.8886685462031$$
$$x_{32} = 80.2641434992748$$
$$x_{33} = 44.6594553839724$$
$$x_{34} = -60.0603182729038$$
$$x_{35} = -49.5883320563873$$
$$x_{36} = 27.9043593082823$$
$$x_{37} = 82.3585379571436$$
$$x_{38} = 25.8099821429868$$
$$x_{39} = 92.8305108737602$$
$$x_{40} = 76.0753547468905$$
$$x_{41} = 32.093123597352$$
$$x_{42} = -18.1721887001287$$
$$x_{43} = 48.8482387828323$$
$$x_{44} = -7.69906717612363$$
$$x_{45} = -5.60342975884779$$
$$x_{46} = -62.1547148941877$$
Maxima of the function at points:
$$x_{46} = -29.9987400979142$$
$$x_{46} = -82.3585379571436$$
$$x_{46} = -48.8482387828323$$
$$x_{46} = 64.2491113693648$$
$$x_{46} = -36.2818963327114$$
$$x_{46} = -38.3762847652043$$
$$x_{46} = 57.9659214840189$$
$$x_{46} = -78.1697490939037$$
$$x_{46} = 11.8886685462031$$
$$x_{46} = -76.0753547468905$$
$$x_{46} = 95.6650489018407$$
$$x_{46} = 16.0777240784174$$
$$x_{46} = 13.9832259109932$$
$$x_{46} = 62.1547148941877$$
$$x_{46} = -42.5650644622225$$
$$x_{46} = 60.0603182729038$$
$$x_{46} = -67.6977780924208$$
$$x_{46} = 53.7771272959775$$
$$x_{46} = 7.69906717612363$$
$$x_{46} = 5.60342975884779$$
$$x_{46} = -25.8099821429868$$
$$x_{46} = 20.2666329000236$$
$$x_{46} = -21.6212447049234$$
$$x_{46} = -65.6033841604836$$
$$x_{46} = 9.79399613491955$$
$$x_{46} = -73.9809604649365$$
$$x_{46} = 87.2874666746391$$
$$x_{46} = 99.8538398487895$$
$$x_{46} = 68.4379039512863$$
$$x_{46} = -463.538434786562$$
$$x_{46} = -34.1875091633272$$
$$x_{46} = 18.1721887001287$$
$$x_{46} = 66.3435077165907$$
$$x_{46} = -23.7156099376309$$
$$x_{46} = -69.792172128143$$
$$x_{46} = -80.2641434992748$$
$$x_{46} = 55.8715245019358$$
$$x_{46} = -32.093123597352$$
$$x_{46} = -86.5473270104187$$
$$x_{46} = -27.9043593082823$$
$$x_{46} = -71.8865662557331$$
$$x_{46} = 22.3610639591715$$
$$x_{46} = 51.6827298291268$$
$$x_{46} = -0.819394360115899$$
$$x_{46} = 97.7594443872384$$
Decreasing at intervals
$$\left[92.8305108737602, \infty\right)$$
Increasing at intervals
$$\left(-\infty, -99.8538398487895\right]$$