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$$2 x \cos{\left(x^{2} + 4 \right)} - \frac{2}{2 x - 1} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = -90.3643272246607$$
$$x_{2} = 80.0989679461124$$
$$x_{3} = -85.5973511564295$$
$$x_{4} = 56.3078965806725$$
$$x_{5} = -31.8167520296521$$
$$x_{6} = -93.2389360956885$$
$$x_{7} = -9.74500778543442$$
$$x_{8} = -81.7104442738241$$
$$x_{9} = -8.72449571618325$$
$$x_{10} = 72.9774748965956$$
$$x_{11} = 4.04790440787845$$
$$x_{12} = -95.5026481717499$$
$$x_{13} = -71.9368847913903$$
$$x_{14} = 1.91532575171434$$
$$x_{15} = 85.6340452801487$$
$$x_{16} = 13.7553876875434$$
$$x_{17} = -48.0275089011207$$
$$x_{18} = -24.1227317710406$$
$$x_{19} = -13.7553806952644$$
$$x_{20} = -1.93591150791451$$
$$x_{21} = -77.5684358359589$$
$$x_{22} = -80.0008534123524$$
$$x_{23} = 86.1096459130926$$
$$x_{24} = 20.0708126777615$$
$$x_{25} = -26.7767846943908$$
$$x_{26} = -34.288022115607$$
$$x_{27} = 70.1906672715433$$
$$x_{28} = -73.8972293923511$$
$$x_{29} = -33.780321053161$$
$$x_{30} = -40.2712614994376$$
$$x_{31} = -64.8879546806254$$
$$x_{32} = -45.7494626479764$$
$$x_{33} = -43.7845197020382$$
$$x_{34} = 1.17827400007491$$
$$x_{35} = 30.2475667697536$$
$$x_{36} = -1.02013689250565$$
$$x_{37} = -14.424279072548$$
$$x_{38} = -3.64853385453414$$
$$x_{39} = -54.1462664068119$$
$$x_{40} = 84.47052287509$$
$$x_{41} = 28.4823440214636$$
$$x_{42} = 60.8916402816016$$
$$x_{43} = 33.780320861126$$
$$x_{44} = -17.2963021103696$$
$$x_{45} = -4.42548135143114$$
$$x_{46} = -7.14052072632612$$
$$x_{47} = -96.7770635854397$$
$$x_{48} = 55.0667697735665$$
$$x_{49} = 2.66131755760155$$
$$x_{50} = -15.875822010098$$
$$x_{51} = 27.4144552296651$$
$$x_{52} = 48.2233467305809$$
$$x_{53} = -39.8398841355098$$
$$x_{54} = 27.8690710088216$$
$$x_{55} = 8.35673637282801$$
$$x_{56} = -43.8203747459463$$
$$x_{57} = 78.6743287092362$$
$$x_{58} = -29.8292232093822$$
$$x_{59} = -4.04885181338452$$
$$x_{60} = 42.2508851938536$$
$$x_{61} = 8.16565041898796$$
$$x_{62} = 6.19884304394702$$
$$x_{63} = -99.7188702681725$$
$$x_{64} = 40.736637258373$$
$$x_{65} = 10.3697359939432$$
$$x_{66} = 82.1705234682079$$
$$x_{67} = -85.908750805183$$
$$x_{68} = 24.1877964109212$$
$$x_{69} = -195.213296909077$$
$$x_{70} = -29.7765355927321$$
$$x_{71} = -57.4676092926258$$
$$x_{72} = 1.17827400007491$$
$$x_{73} = 18.9434786293785$$
$$x_{74} = -76.8155158585219$$
$$x_{75} = 35.635883886335$$
$$x_{76} = 16.1698053757139$$
$$x_{77} = 87.8791624710794$$
The values of the extrema at the points:
(-90.36432722466066, -4.20251466680011 - pi*I)
(80.09896794611235, -6.07014830475016)
(-85.5973511564295, -6.1486258244852 - pi*I)
(56.30789658067252, -3.7150625683759)
(-31.816752029652086, -5.16873279679676 - pi*I)
(-93.23893609568849, -4.23366082510317 - pi*I)
(-9.745007785434423, -4.0199251814183 - pi*I)
(-81.71044427382408, -4.10242953661313 - pi*I)
(-8.724495716183249, -3.91499040448169 - pi*I)
(72.97747489659561, -3.9764230075328)
(4.047904407878453, -0.960110532015081)
(-95.50264817174993, -4.25752295825672 - pi*I)
(-71.93688479139031, -3.97586281326983 - pi*I)
(1.9153257517143445, -0.057664385200122)
(85.63404528014873, -6.13737419583034)
(13.755387687543417, -4.27754750833872)
(-48.02750890112069, -5.57527798853532 - pi*I)
(-24.122731771040648, -2.89681760686596 - pi*I)
(-13.755380695264357, -4.35027835772283 - pi*I)
(-1.9359115079145115, -0.589105121571768 - pi*I)
(-77.56843583595888, -4.05073300849548 - pi*I)
(-80.00085341235238, -4.08141496933779 - pi*I)
(86.1096459130926, -6.1429451409595)
(20.070812677761484, -4.66718567767349)
(-26.776784694390788, -4.99918290916131 - pi*I)
(-34.28802211560699, -5.24242022815759 - pi*I)
(70.19066727154333, -5.93721358559409)
(-73.89722939235108, -6.00256587821985 - pi*I)
(-33.780321053160975, -3.22771873307362 - pi*I)
(-40.27126149943764, -5.4011245924004 - pi*I)
(-64.8879546806254, -5.87348523592107 - pi*I)
(-45.74946264797638, -5.52719699810816 - pi*I)
(-43.784519702038246, -5.48378232090194 - pi*I)
(1.1782740000749066, -1.085062040913)
(30.247566769753636, -3.0858946743472)
(-1.0201368925056489, -2.05854254947143 - pi*I)
(-14.424279072547954, -4.3961338379854 - pi*I)
(-3.6485338545341377, -3.11535640331258 - pi*I)
(-54.14626640681191, -3.69402808886108 - pi*I)
(84.47052287509003, -4.12361300214793)
(28.482344021463557, -5.02472072439555)
(60.89164028160158, -5.79399786074438)
(33.78032086112602, -3.19811353662882)
(-17.296302110369563, -2.57213918896926 - pi*I)
(-4.425481351431137, -3.28730606982724 - pi*I)
(-7.140520726326124, -3.72657094277794 - pi*I)
(-96.77706358543969, -4.27071041455006 - pi*I)
(55.066769773566534, -3.69257227966988)
(2.6613175576015475, -2.46007988013207)
(-15.875822010098048, -4.48895131050745 - pi*I)
(27.41445522966508, -2.98581092157328)
(48.223346730580914, -5.55856788439027)
(-39.83988413550981, -3.39048788927878 - pi*I)
(27.869071008821564, -3.00256097594099)
(8.356736372828012, -3.75448948366589)
(-43.82037474594632, -3.48459171156219 - pi*I)
(78.67432870923625, -6.05208849353633)
(-29.829223209382196, -3.10525904335882 - pi*I)
(-4.048851813384522, -1.20839060528751 - pi*I)
(42.250885193853605, -5.42486779370956)
(8.165650418987957, -1.72992844841816)
(6.198843043947024, -3.43331019018606)
(-99.71887026817252, -4.30050367876083 - pi*I)
(40.73663725837304, -5.38792508919559)
(10.369735993943243, -3.9826083518604)
(82.17052346820789, -4.0958403302383)
(-85.90875080518299, -4.15223613597703 - pi*I)
(24.18779641092117, -4.85810679593934)
(-195.21329690907734, -6.96979799793902 - pi*I)
(-29.77653559273207, -5.10352003639281 - pi*I)
(-57.46760929262584, -3.75303158590127 - pi*I)
(1.1782740000749052, -1.085062040913)
(18.94347862937845, -4.6078570027297)
(-76.8155158585219, -6.0410418350873 - pi*I)
(35.635883886334966, -5.25237004201725)
(16.169805375713867, -2.44488476370106)
(87.87916247107941, -6.16340401701326)
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:
$$x_{1} = 80.0989679461124$$
$$x_{2} = -85.5973511564295$$
$$x_{3} = -31.8167520296521$$
$$x_{4} = -9.74500778543442$$
$$x_{5} = -8.72449571618325$$
$$x_{6} = 85.6340452801487$$
$$x_{7} = 13.7553876875434$$
$$x_{8} = -48.0275089011207$$
$$x_{9} = -13.7553806952644$$
$$x_{10} = 86.1096459130926$$
$$x_{11} = 20.0708126777615$$
$$x_{12} = -26.7767846943908$$
$$x_{13} = -34.288022115607$$
$$x_{14} = 70.1906672715433$$
$$x_{15} = -73.8972293923511$$
$$x_{16} = -40.2712614994376$$
$$x_{17} = -64.8879546806254$$
$$x_{18} = -45.7494626479764$$
$$x_{19} = -43.7845197020382$$
$$x_{20} = 1.17827400007491$$
$$x_{21} = -1.02013689250565$$
$$x_{22} = -14.424279072548$$
$$x_{23} = -3.64853385453414$$
$$x_{24} = 28.4823440214636$$
$$x_{25} = 60.8916402816016$$
$$x_{26} = -4.42548135143114$$
$$x_{27} = -7.14052072632612$$
$$x_{28} = 2.66131755760155$$
$$x_{29} = -15.875822010098$$
$$x_{30} = 48.2233467305809$$
$$x_{31} = 8.35673637282801$$
$$x_{32} = 78.6743287092362$$
$$x_{33} = 42.2508851938536$$
$$x_{34} = 6.19884304394702$$
$$x_{35} = 40.736637258373$$
$$x_{36} = 10.3697359939432$$
$$x_{37} = 24.1877964109212$$
$$x_{38} = -195.213296909077$$
$$x_{39} = -29.7765355927321$$
$$x_{40} = 1.17827400007491$$
$$x_{41} = 18.9434786293785$$
$$x_{42} = -76.8155158585219$$
$$x_{43} = 35.635883886335$$
$$x_{44} = 87.8791624710794$$
Maxima of the function at points:
$$x_{44} = -90.3643272246607$$
$$x_{44} = 56.3078965806725$$
$$x_{44} = -93.2389360956885$$
$$x_{44} = -81.7104442738241$$
$$x_{44} = 72.9774748965956$$
$$x_{44} = 4.04790440787845$$
$$x_{44} = -95.5026481717499$$
$$x_{44} = -71.9368847913903$$
$$x_{44} = 1.91532575171434$$
$$x_{44} = -24.1227317710406$$
$$x_{44} = -1.93591150791451$$
$$x_{44} = -77.5684358359589$$
$$x_{44} = -80.0008534123524$$
$$x_{44} = -33.780321053161$$
$$x_{44} = 30.2475667697536$$
$$x_{44} = -54.1462664068119$$
$$x_{44} = 84.47052287509$$
$$x_{44} = 33.780320861126$$
$$x_{44} = -17.2963021103696$$
$$x_{44} = -96.7770635854397$$
$$x_{44} = 55.0667697735665$$
$$x_{44} = 27.4144552296651$$
$$x_{44} = -39.8398841355098$$
$$x_{44} = 27.8690710088216$$
$$x_{44} = -43.8203747459463$$
$$x_{44} = -29.8292232093822$$
$$x_{44} = -4.04885181338452$$
$$x_{44} = 8.16565041898796$$
$$x_{44} = -99.7188702681725$$
$$x_{44} = 82.1705234682079$$
$$x_{44} = -85.908750805183$$
$$x_{44} = -57.4676092926258$$
$$x_{44} = 16.1698053757139$$
Decreasing at intervals
$$\left[87.8791624710794, \infty\right)$$
Increasing at intervals
$$\left(-\infty, -195.213296909077\right]$$