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$$- 2 x \cos{\left(2 x \right)} - \sin{\left(2 x \right)} + \frac{1}{x} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = -5.55078614385824$$
$$x_{2} = 69.9039616572184$$
$$x_{3} = 52.6265175248244$$
$$x_{4} = -35.3497890165078$$
$$x_{5} = 88.7527775101222$$
$$x_{6} = -60.4797237593826$$
$$x_{7} = 60.4797237593826$$
$$x_{8} = 54.1970008368134$$
$$x_{9} = -79.3283261948234$$
$$x_{10} = 84.0406135966029$$
$$x_{11} = 11.8039318424589$$
$$x_{12} = 55.7678327334044$$
$$x_{13} = -33.779740439879$$
$$x_{14} = -22.7870347804882$$
$$x_{15} = -62.050548722829$$
$$x_{16} = 91.8942760038802$$
$$x_{17} = -25.9274086756167$$
$$x_{18} = 98.1772908798262$$
$$x_{19} = -93.465084832123$$
$$x_{20} = 71.4747794848494$$
$$x_{21} = -19.6470284601706$$
$$x_{22} = -91.8942760038802$$
$$x_{23} = -3.97386307124954$$
$$x_{24} = 5.55078614385824$$
$$x_{25} = -68.3333522239591$$
$$x_{26} = -54.1970008368134$$
$$x_{27} = -85.6112858437368$$
$$x_{28} = 33.779740439879$$
$$x_{29} = -46.3440022656785$$
$$x_{30} = -49.4852382198545$$
$$x_{31} = 35.3497890165078$$
$$x_{32} = 76.1868601386139$$
$$x_{33} = -57.3383498626967$$
$$x_{34} = -10.2322043845486$$
$$x_{35} = 25.9274086756167$$
$$x_{36} = -40.0617021401075$$
$$x_{37} = -77.757674595807$$
$$x_{38} = -11.8039318424589$$
$$x_{39} = 32.2088449631489$$
$$x_{40} = -71.4747794848494$$
$$x_{41} = -32.2088449631489$$
$$x_{42} = 18.0787472314028$$
$$x_{43} = 16.5075843463746$$
$$x_{44} = -27.4983567329359$$
$$x_{45} = -84.0406135966029$$
$$x_{46} = 63.6211189047024$$
$$x_{47} = 41.6319631430707$$
$$x_{48} = 19.6470284601706$$
$$x_{49} = -24.3580264551541$$
$$x_{50} = 46.3440022656785$$
$$x_{51} = -98.1772908798262$$
$$x_{52} = 40.0617021401075$$
$$x_{53} = 3.97386307124954$$
$$x_{54} = -16.5075843463746$$
$$x_{55} = -13.3690621886223$$
$$x_{56} = 24.3580264551541$$
$$x_{57} = 62.050548722829$$
$$x_{58} = 99.7480981700469$$
$$x_{59} = 2.49453567490861$$
$$x_{60} = 74.6162208523254$$
$$x_{61} = 8.6714972018692$$
$$x_{62} = 38.490835965076$$
$$x_{63} = -76.1868601386139$$
$$x_{64} = 49.4852382198545$$
$$x_{65} = -69.9039616572184$$
$$x_{66} = 82.469801775061$$
$$x_{67} = -82.469801775061$$
$$x_{68} = -90.3235872317126$$
$$x_{69} = 90.3235872317126$$
$$x_{70} = 66.7625323470719$$
$$x_{71} = -38.490835965076$$
$$x_{72} = 96.6065886767859$$
$$x_{73} = -99.7480981700469$$
$$x_{74} = -63.6211189047024$$
$$x_{75} = -18.0787472314028$$
$$x_{76} = 10.2322043845486$$
$$x_{77} = 68.3333522239591$$
$$x_{78} = 77.757674595807$$
$$x_{79} = 85.6112858437368$$
$$x_{80} = 30.6389534733369$$
$$x_{81} = -47.9143965272863$$
$$x_{82} = 27.4983567329359$$
$$x_{83} = -55.7678327334044$$
$$x_{84} = -41.6319631430707$$
$$x_{85} = 47.9143965272863$$
The values of the extrema at the points:
(-5.550786143858235, 7.23357175501643 + pi*I)
(69.90396165721837, -65.655102026686)
(52.62651752482435, 56.5872714686413)
(-35.34978901650776, -31.7811582279713 + pi*I)
(88.75277751012221, -84.2655459712149)
(-60.479723759382644, -56.375416670045 + pi*I)
(60.479723759382644, -56.375416670045)
(54.19700083681336, -50.202153330937)
(-79.3283261948234, -74.9531947190737 + pi*I)
(84.04061359660294, 88.470390831253)
(11.803931842458944, 14.259922212362)
(55.767832733404376, 59.7867075631906)
(-33.779740439879006, 37.2956795461122 + pi*I)
(-22.78703478048822, -19.655830136878 + pi*I)
(-62.050548722828985, 66.1764182397145 + pi*I)
(91.89427600388024, -87.3723064757993)
(-25.927408675616707, -22.6676527980116 + pi*I)
(98.17729087982616, -93.5892685670194)
(-93.46508483212298, 98.001306634918 + pi*I)
(71.47477948484938, 75.7423260568176)
(-19.647028460170628, -16.663373933412 + pi*I)
(-91.89427600388024, -87.3723064757993 + pi*I)
(-3.9738630712495366, -2.57667594793945 + pi*I)
(5.550786143858235, 7.23357175501643)
(-68.33335222395908, 72.5558670681559 + pi*I)
(-54.19700083681336, -50.202153330937 + pi*I)
(-85.61128584373682, -81.1600425841317 + pi*I)
(33.779740439879006, 37.2956795461122)
(-46.344002265678526, 50.1772795211067 + pi*I)
(-49.48523821985451, 53.3842837025034 + pi*I)
(35.34978901650776, -31.7811582279713)
(76.18686013861385, -71.8520732660392)
(-57.33834986269668, -53.2872756358037 + pi*I)
(-10.232204384548577, -7.89673571473403 + pi*I)
(25.927408675616707, -22.6676527980116)
(-40.06170214010753, 43.7488454482748 + pi*I)
(-77.75767459580699, 82.1096227300581 + pi*I)
(-11.803931842458944, 14.259922212362 + pi*I)
(32.2088449631489, -28.7329608269299)
(-71.47477948484938, 75.7423260568176 + pi*I)
(-32.2088449631489, -28.7329608269299 + pi*I)
(18.07874723140276, 20.965788618298)
(16.507584346374557, -13.697086182269)
(-27.498356732935857, 30.8076018639768 + pi*I)
(-84.04061359660294, 88.470390831253 + pi*I)
(63.62111890470238, -59.4662700409492)
(41.6319631430707, -37.9002352374976)
(19.647028460170628, -16.663373933412)
(-24.358026455154075, 27.5453278926012 + pi*I)
(46.344002265678526, 50.1772795211067)
(-98.17729087982616, -93.5892685670194 + pi*I)
(40.06170214010753, 43.7488454482748)
(3.9738630712495366, -2.57667594793945)
(-16.507584346374557, -13.697086182269 + pi*I)
(-13.3690621886223, -10.7681233814929 + pi*I)
(24.358026455154075, 27.5453278926012)
(62.050548722828985, 66.1764182397145)
(99.7480981700469, 104.349467774819)
(2.4945356749086143, 3.31376359318726)
(74.6162208523254, 78.9268583893914)
(8.6714972018692, 10.8136550371811)
(38.490835965076016, -34.8373351866165)
(-76.18686013861385, -71.8520732660392 + pi*I)
(49.48523821985451, 53.3842837025034)
(-69.90396165721837, -65.655102026686 + pi*I)
(82.46980177506099, -78.0558904569359)
(-82.46980177506099, -78.0558904569359 + pi*I)
(-90.32358723171255, 94.8255711742018 + pi*I)
(90.32358723171255, 94.8255711742018)
(66.76253234707185, -62.5595737534091)
(-38.490835965076016, -34.8373351866165 + pi*I)
(96.60658867678588, 101.175914814762)
(-99.7480981700469, 104.349467774819 + pi*I)
(-63.62111890470238, -59.4662700409492 + pi*I)
(-18.07874723140276, 20.965788618298 + pi*I)
(10.232204384548577, -7.89673571473403)
(68.33335222395908, 72.5558670681559)
(77.75767459580699, 82.1096227300581)
(85.61128584373682, -81.1600425841317)
(30.638953473336933, 34.0568761167198)
(-47.91439652728628, -44.0424796550028 + pi*I)
(27.498356732935857, 30.8076018639768)
(-55.767832733404376, 59.7867075631906 + pi*I)
(-41.6319631430707, -37.9002352374976 + pi*I)
(47.91439652728628, -44.0424796550028)
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:
$$x_{1} = 69.9039616572184$$
$$x_{2} = 88.7527775101222$$
$$x_{3} = 60.4797237593826$$
$$x_{4} = 54.1970008368134$$
$$x_{5} = 91.8942760038802$$
$$x_{6} = 98.1772908798262$$
$$x_{7} = 35.3497890165078$$
$$x_{8} = 76.1868601386139$$
$$x_{9} = 25.9274086756167$$
$$x_{10} = 32.2088449631489$$
$$x_{11} = 16.5075843463746$$
$$x_{12} = 63.6211189047024$$
$$x_{13} = 41.6319631430707$$
$$x_{14} = 19.6470284601706$$
$$x_{15} = 3.97386307124954$$
$$x_{16} = 38.490835965076$$
$$x_{17} = 82.469801775061$$
$$x_{18} = 66.7625323470719$$
$$x_{19} = 10.2322043845486$$
$$x_{20} = 85.6112858437368$$
$$x_{21} = 47.9143965272863$$
Maxima of the function at points:
$$x_{21} = 52.6265175248244$$
$$x_{21} = 84.0406135966029$$
$$x_{21} = 11.8039318424589$$
$$x_{21} = 55.7678327334044$$
$$x_{21} = 71.4747794848494$$
$$x_{21} = 5.55078614385824$$
$$x_{21} = 33.779740439879$$
$$x_{21} = 18.0787472314028$$
$$x_{21} = 46.3440022656785$$
$$x_{21} = 40.0617021401075$$
$$x_{21} = 24.3580264551541$$
$$x_{21} = 62.050548722829$$
$$x_{21} = 99.7480981700469$$
$$x_{21} = 2.49453567490861$$
$$x_{21} = 74.6162208523254$$
$$x_{21} = 8.6714972018692$$
$$x_{21} = 49.4852382198545$$
$$x_{21} = 90.3235872317126$$
$$x_{21} = 96.6065886767859$$
$$x_{21} = 68.3333522239591$$
$$x_{21} = 77.757674595807$$
$$x_{21} = 30.6389534733369$$
$$x_{21} = 27.4983567329359$$
Decreasing at intervals
$$\left[98.1772908798262, \infty\right)$$
Increasing at intervals
$$\left(-\infty, 3.97386307124954\right]$$