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$$2 \sin{\left(x \right)} + \frac{1}{x} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = 31.4000022971774$$
$$x_{2} = -97.3945060436357$$
$$x_{3} = 9.4775586655226$$
$$x_{4} = -50.2555331400595$$
$$x_{5} = 22.0138634883979$$
$$x_{6} = -43.9709257550159$$
$$x_{7} = 81.6752871523161$$
$$x_{8} = -6.20248488876007$$
$$x_{9} = -94.2424741193738$$
$$x_{10} = 78.5461820645165$$
$$x_{11} = -103.677380239879$$
$$x_{12} = 87.9589097974439$$
$$x_{13} = -100.525991056308$$
$$x_{14} = -18.8229895348165$$
$$x_{15} = 50.2555331400595$$
$$x_{16} = -59.6986359168773$$
$$x_{17} = -56.5398243250897$$
$$x_{18} = -3.29397401984354$$
$$x_{19} = -69.1078032428717$$
$$x_{20} = 53.4164356626158$$
$$x_{21} = -37.6858438724328$$
$$x_{22} = 40.8529438223281$$
$$x_{23} = 84.8288958989938$$
$$x_{24} = -34.5719822719983$$
$$x_{25} = 37.6858438724328$$
$$x_{26} = -15.7397353480342$$
$$x_{27} = -62.8238942324921$$
$$x_{28} = -40.8529438223281$$
$$x_{29} = 43.9709257550159$$
$$x_{30} = 56.5398243250897$$
$$x_{31} = 62.8238942324921$$
$$x_{32} = -2921.68099670415$$
$$x_{33} = 25.1128297713109$$
$$x_{34} = 59.6986359168773$$
$$x_{35} = -75.3915915982152$$
$$x_{36} = 100.525991056308$$
$$x_{37} = 28.2920076380021$$
$$x_{38} = 94.2424741193738$$
$$x_{39} = -87.9589097974439$$
$$x_{40} = -9471.90190336093$$
$$x_{41} = 47.1344979443735$$
$$x_{42} = -9.4775586655226$$
$$x_{43} = 97.3945060436357$$
$$x_{44} = 75.3915915982152$$
$$x_{45} = -91.111674752578$$
$$x_{46} = 3.29397401984354$$
$$x_{47} = 12.5264444510967$$
$$x_{48} = 6.20248488876007$$
$$x_{49} = -28.2920076380021$$
$$x_{50} = -53.4164356626158$$
$$x_{51} = 69.1078032428717$$
$$x_{52} = -31.4000022971774$$
$$x_{53} = 15.7397353480342$$
$$x_{54} = -72.2635502053457$$
$$x_{55} = 91.111674752578$$
$$x_{56} = -81.6752871523161$$
$$x_{57} = 18.8229895348165$$
$$x_{58} = 65.9810237342926$$
$$x_{59} = -12.5264444510967$$
$$x_{60} = -84.8288958989938$$
$$x_{61} = 72.2635502053457$$
$$x_{62} = -78.5461820645165$$
$$x_{63} = -65.9810237342926$$
$$x_{64} = -25.1128297713109$$
$$x_{65} = 34.5719822719983$$
$$x_{66} = -175.926346498733$$
$$x_{67} = -47.1344979443735$$
$$x_{68} = -22.0138634883979$$
The values of the extrema at the points:
(31.40000229717736, 1.44706154209297)
(-97.39450604363567, 6.57874344727141 + pi*I)
(9.477558665522597, 4.24614160223831)
(-50.25553314005948, 1.91721964121698 + pi*I)
(22.0138634883979, 5.09115646835529)
(-43.970925755015905, 1.78365794444932 + pi*I)
(81.67528715231606, 2.40278895009342)
(-6.20248488876007, -0.168540975883722 + pi*I)
(-94.24247411937377, 2.54589912108789 + pi*I)
(78.54618206451649, 6.36364623604322)
(-103.67738023987934, 6.64126070642214 + pi*I)
(87.95890979744387, 2.47690208479871)
(-100.52599105630847, 2.61044105075712 + pi*I)
(-18.822989534816458, 0.935784701758213 + pi*I)
(50.25553314005948, 1.91721964121698)
(-59.6986359168773, 6.08923902260607 + pi*I)
(-56.53982432508972, 2.03502345108285 + pi*I)
(-3.2939740198435428, 3.16891956007841 + pi*I)
(-69.10780324287171, 2.23571999817831 + pi*I)
(53.41643566261576, 5.97803086326822)
(-37.68584387243278, 1.62946056630104 + pi*I)
(40.85294382232808, 5.70982908368098)
(84.82889589899379, 6.44060149617356)
(-34.57198227199833, 5.54283441599595 + pi*I)
(37.68584387243278, 1.62946056630104)
(-15.739735348034166, 4.75517904871546 + pi*I)
(-62.82389423249211, 2.14039882530727 + pi*I)
(-40.85294382232808, 5.70982908368098 + pi*I)
(43.970925755015905, 1.78365794444932)
(56.53982432508972, 2.03502345108285)
(62.82389423249211, 2.14039882530727)
(-2921.6809967041495, 5.97991444270974 + pi*I)
(25.11282977131086, 1.22377531485183)
(59.6986359168773, 6.08923902260607)
(-75.39159159821523, 2.32273973595957 + pi*I)
(100.52599105630847, 2.61044105075712)
(28.292007638002087, 5.34226699563449)
(94.24247411937377, 2.54589912108789)
(-87.95890979744387, 2.47690208479871 + pi*I)
(-9471.901903360935, 11.1560849977972 + pi*I)
(47.13449794437348, 5.85289264176121)
(-9.477558665522597, 4.24614160223831 + pi*I)
(97.39450604363567, 6.57874344727141)
(75.39159159821523, 2.32273973595957)
(-91.111674752578, 6.51205583335522 + pi*I)
(3.2939740198435428, 3.16891956007841)
(12.526444451096722, 0.529435852519932)
(6.20248488876007, -0.168540975883722)
(-28.292007638002087, 5.34226699563449 + pi*I)
(-53.41643566261576, 5.97803086326822 + pi*I)
(69.10780324287171, 2.23571999817831)
(-31.40000229717736, 1.44706154209297 + pi*I)
(15.739735348034166, 4.75517904871546)
(-72.26355020534572, 6.28027198074176 + pi*I)
(91.111674752578, 6.51205583335522)
(-81.67528715231606, 2.40278895009342 + pi*I)
(18.822989534816458, 0.935784701758213)
(65.98102373429263, 6.1893097555633)
(-12.526444451096722, 0.529435852519932 + pi*I)
(-84.82889589899379, 6.44060149617356 + pi*I)
(72.26355020534572, 6.28027198074176)
(-78.54618206451649, 6.36364623604322 + pi*I)
(-65.98102373429263, 6.1893097555633 + pi*I)
(-25.11282977131086, 1.22377531485183 + pi*I)
(34.57198227199833, 5.54283441599595)
(-175.92634649873347, 3.1700734991859 + pi*I)
(-47.13449794437348, 5.85289264176121 + pi*I)
(-22.0138634883979, 5.09115646835529 + pi*I)
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:
$$x_{1} = 31.4000022971774$$
$$x_{2} = 81.6752871523161$$
$$x_{3} = 87.9589097974439$$
$$x_{4} = 50.2555331400595$$
$$x_{5} = 37.6858438724328$$
$$x_{6} = 43.9709257550159$$
$$x_{7} = 56.5398243250897$$
$$x_{8} = 62.8238942324921$$
$$x_{9} = 25.1128297713109$$
$$x_{10} = 100.525991056308$$
$$x_{11} = 94.2424741193738$$
$$x_{12} = 75.3915915982152$$
$$x_{13} = 12.5264444510967$$
$$x_{14} = 6.20248488876007$$
$$x_{15} = 69.1078032428717$$
$$x_{16} = 18.8229895348165$$
Maxima of the function at points:
$$x_{16} = 9.4775586655226$$
$$x_{16} = 22.0138634883979$$
$$x_{16} = 78.5461820645165$$
$$x_{16} = 53.4164356626158$$
$$x_{16} = 40.8529438223281$$
$$x_{16} = 84.8288958989938$$
$$x_{16} = 59.6986359168773$$
$$x_{16} = 28.2920076380021$$
$$x_{16} = 47.1344979443735$$
$$x_{16} = 97.3945060436357$$
$$x_{16} = 3.29397401984354$$
$$x_{16} = 15.7397353480342$$
$$x_{16} = 91.111674752578$$
$$x_{16} = 65.9810237342926$$
$$x_{16} = 72.2635502053457$$
$$x_{16} = 34.5719822719983$$
Decreasing at intervals
$$\left[100.525991056308, \infty\right)$$
Increasing at intervals
$$\left(-\infty, 6.20248488876007\right]$$