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$$\frac{\cos{\left(x \right)}}{\sqrt{x}} - \frac{\sin{\left(x \right)}}{2 x^{\frac{3}{2}}} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = -39.2571723324086$$
$$x_{2} = -26.6848024909251$$
$$x_{3} = -58.1108600600615$$
$$x_{4} = -1.16556118520721$$
$$x_{5} = -7.78988375114457$$
$$x_{6} = 7.78988375114457$$
$$x_{7} = -51.8266315338985$$
$$x_{8} = 67.5368388204916$$
$$x_{9} = -80.1043708909521$$
$$x_{10} = -67.5368388204916$$
$$x_{11} = 20.3958423573092$$
$$x_{12} = 64.3948849627586$$
$$x_{13} = 117.80548025038$$
$$x_{14} = 70.6787605627689$$
$$x_{15} = -45.5421150692309$$
$$x_{16} = 32.9715594404485$$
$$x_{17} = 4.60421677720058$$
$$x_{18} = -23.5407082923052$$
$$x_{19} = -64.3948849627586$$
$$x_{20} = -70.6787605627689$$
$$x_{21} = -76.9625234358705$$
$$x_{22} = -73.8206542907788$$
$$x_{23} = 73.8206542907788$$
$$x_{24} = 48.6844162648433$$
$$x_{25} = 80.1043708909521$$
$$x_{26} = -20.3958423573092$$
$$x_{27} = -92.6715879363332$$
$$x_{28} = 26.6848024909251$$
$$x_{29} = -36.1144715353049$$
$$x_{30} = 158.64727737108$$
$$x_{31} = 92.6715879363332$$
$$x_{32} = 10.9499436485412$$
$$x_{33} = 83.2461991121237$$
$$x_{34} = -98.9551158352145$$
$$x_{35} = 54.9687756155963$$
$$x_{36} = -29.8283692130955$$
$$x_{37} = 14.1017251335659$$
$$x_{38} = -4.60421677720058$$
$$x_{39} = 215.19677332017$$
$$x_{40} = -61.2528940466862$$
$$x_{41} = -83.2461991121237$$
$$x_{42} = -246.612995841404$$
$$x_{43} = -54.9687756155963$$
$$x_{44} = 76.9625234358705$$
$$x_{45} = 36.1144715353049$$
$$x_{46} = -48.6844162648433$$
$$x_{47} = 29.8283692130955$$
$$x_{48} = 58.1108600600615$$
$$x_{49} = -42.3997088362447$$
$$x_{50} = 17.2497818346079$$
$$x_{51} = -32.9715594404485$$
$$x_{52} = 23.5407082923052$$
$$x_{53} = 1.16556118520721$$
$$x_{54} = -17.2497818346079$$
$$x_{55} = -10.9499436485412$$
$$x_{56} = 51.8266315338985$$
$$x_{57} = 45.5421150692309$$
$$x_{58} = -86.3880101981266$$
$$x_{59} = 89.5298059530594$$
$$x_{60} = -95.8133575027966$$
$$x_{61} = 39.2571723324086$$
$$x_{62} = -89.5298059530594$$
$$x_{63} = 61.2528940466862$$
$$x_{64} = 95.8133575027966$$
$$x_{65} = -14.1017251335659$$
$$x_{66} = 98.9551158352145$$
$$x_{67} = 86.3880101981266$$
$$x_{68} = 42.3997088362447$$
The values of the extrema at the points:
(-39.25717233240859, 0.159589851348603*I)
(-26.68480249092507, 0.19354937797769*I)
(-58.110860060061505, 0.131176268600912*I)
(-1.1655611852072114, 0.851241066782324*I)
(-7.789883751144573, 0.357554083426262*I)
(7.789883751144573, 0.357554083426262)
(-51.82663153389846, 0.138900336703391*I)
(67.53683882049161, -0.121679588990783)
(-80.1043708909521, -0.111728362291416*I)
(-67.53683882049161, -0.121679588990783*I)
(20.395842357309167, 0.221359780635401)
(64.39488496275855, 0.124612389237314)
(117.80548025038037, -0.0921326029924126)
(70.67876056276886, 0.118944583684481)
(-45.5421150692309, 0.148172370731446*I)
(32.97155944044848, 0.17413269656851)
(4.604216777200577, -0.463314891176637)
(-23.54070829230515, -0.206059336815155*I)
(-64.39488496275855, 0.124612389237314*I)
(-70.67876056276886, 0.118944583684481*I)
(-76.96252343587051, 0.113985913925499*I)
(-73.82065429077876, -0.116386094038002*I)
(73.82065429077876, -0.116386094038002)
(48.68441626484328, -0.143311853691665)
(80.1043708909521, -0.111728362291416)
(-20.395842357309167, 0.221359780635401*I)
(-92.67158793633321, -0.103877233902111*I)
(26.68480249092507, 0.19354937797769)
(-36.11447153530485, -0.166386370791913*I)
(158.6472773710796, 0.0793928754394215)
(92.67158793633321, -0.103877233902111)
(10.94994364854116, -0.301885161430297)
(83.24619911212368, 0.109599849994829)
(-98.95511583521451, -0.100525289012326*I)
(54.96877561559635, -0.134872684738376)
(-29.828369213095506, -0.183072974858657*I)
(14.101725133565873, 0.266128298234218)
(-4.604216777200577, -0.463314891176637*I)
(215.1967733201699, 0.0681680624478802)
(-61.252894046686194, -0.127768037744087*I)
(-83.24619911212368, 0.109599849994829*I)
(-246.61299584140428, 0.0636782512070729*I)
(-54.96877561559635, -0.134872684738376*I)
(76.96252343587051, 0.113985913925499)
(36.11447153530485, -0.166386370791913)
(-48.68441626484328, -0.143311853691665*I)
(29.828369213095506, -0.183072974858657)
(58.110860060061505, 0.131176268600912)
(-42.39970883624466, -0.15356362930828*I)
(17.249781834607894, -0.240672145897842)
(-32.97155944044848, 0.17413269656851*I)
(23.54070829230515, -0.206059336815155)
(1.1655611852072114, 0.851241066782324)
(-17.249781834607894, -0.240672145897842*I)
(-10.94994364854116, -0.301885161430297*I)
(51.82663153389846, 0.138900336703391)
(45.5421150692309, 0.148172370731446)
(-86.38801019812658, -0.107588534144322*I)
(89.52980595305935, 0.105684039776562)
(-95.81335750279658, 0.102160040658152*I)
(39.25717233240859, 0.159589851348603)
(-89.52980595305935, 0.105684039776562*I)
(61.252894046686194, -0.127768037744087)
(95.81335750279658, 0.102160040658152)
(-14.101725133565873, 0.266128298234218*I)
(98.95511583521451, -0.100525289012326)
(86.38801019812658, -0.107588534144322)
(42.39970883624466, -0.15356362930828)
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:
$$x_{1} = 67.5368388204916$$
$$x_{2} = 117.80548025038$$
$$x_{3} = 4.60421677720058$$
$$x_{4} = 73.8206542907788$$
$$x_{5} = 48.6844162648433$$
$$x_{6} = 80.1043708909521$$
$$x_{7} = 92.6715879363332$$
$$x_{8} = 10.9499436485412$$
$$x_{9} = 54.9687756155963$$
$$x_{10} = 36.1144715353049$$
$$x_{11} = 29.8283692130955$$
$$x_{12} = 17.2497818346079$$
$$x_{13} = 23.5407082923052$$
$$x_{14} = 61.2528940466862$$
$$x_{15} = 98.9551158352145$$
$$x_{16} = 86.3880101981266$$
$$x_{17} = 42.3997088362447$$
Maxima of the function at points:
$$x_{17} = 7.78988375114457$$
$$x_{17} = 20.3958423573092$$
$$x_{17} = 64.3948849627586$$
$$x_{17} = 70.6787605627689$$
$$x_{17} = 32.9715594404485$$
$$x_{17} = 26.6848024909251$$
$$x_{17} = 158.64727737108$$
$$x_{17} = 83.2461991121237$$
$$x_{17} = 14.1017251335659$$
$$x_{17} = 215.19677332017$$
$$x_{17} = 76.9625234358705$$
$$x_{17} = 58.1108600600615$$
$$x_{17} = 1.16556118520721$$
$$x_{17} = 51.8266315338985$$
$$x_{17} = 45.5421150692309$$
$$x_{17} = 89.5298059530594$$
$$x_{17} = 39.2571723324086$$
$$x_{17} = 95.8133575027966$$
Decreasing at intervals
$$\left[117.80548025038, \infty\right)$$
Increasing at intervals
$$\left(-\infty, 4.60421677720058\right]$$