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$$\sqrt{x} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{2 \sqrt{x}} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = -86.3995849739529$$
$$x_{2} = 17.3076405374146$$
$$x_{3} = -1.83659720315213$$
$$x_{4} = 11.0408298179713$$
$$x_{5} = 51.8459224452234$$
$$x_{6} = -92.682377997352$$
$$x_{7} = -76.9755154935569$$
$$x_{8} = 26.7222463741877$$
$$x_{9} = -95.8237937978449$$
$$x_{10} = -36.1421488970061$$
$$x_{11} = 86.3995849739529$$
$$x_{12} = 67.5516436614121$$
$$x_{13} = -58.1280655761511$$
$$x_{14} = -51.8459224452234$$
$$x_{15} = -61.2692172687226$$
$$x_{16} = -26.7222463741877$$
$$x_{17} = 39.2826357527234$$
$$x_{18} = 45.5640665961997$$
$$x_{19} = 98.9652208250325$$
$$x_{20} = 7.91705268466621$$
$$x_{21} = -29.861872403816$$
$$x_{22} = 29.861872403816$$
$$x_{23} = -48.7049516666752$$
$$x_{24} = -89.5409746049841$$
$$x_{25} = 61.2692172687226$$
$$x_{26} = 1.83659720315213$$
$$x_{27} = -54.9869642514883$$
$$x_{28} = -42.4232862577008$$
$$x_{29} = -64.410411962776$$
$$x_{30} = -11.0408298179713$$
$$x_{31} = 83.2582106616487$$
$$x_{32} = 76.9755154935569$$
$$x_{33} = 20.4448034666183$$
$$x_{34} = -73.8341991854591$$
$$x_{35} = -39.2826357527234$$
$$x_{36} = -67.5516436614121$$
$$x_{37} = -80.1168534696549$$
$$x_{38} = 33.0018723591446$$
$$x_{39} = 54.9869642514883$$
$$x_{40} = -14.1724320747999$$
$$x_{41} = 73.8341991854591$$
$$x_{42} = -4.81584231784594$$
$$x_{43} = -7.91705268466621$$
$$x_{44} = 64.410411962776$$
$$x_{45} = -83.2582106616487$$
$$x_{46} = -98.9652208250325$$
$$x_{47} = 36.1421488970061$$
$$x_{48} = 48.7049516666752$$
$$x_{49} = -20.4448034666183$$
$$x_{50} = 14.1724320747999$$
$$x_{51} = 58.1280655761511$$
$$x_{52} = -23.5831433102848$$
$$x_{53} = 4.81584231784594$$
$$x_{54} = 80.1168534696549$$
$$x_{55} = -70.692907433161$$
$$x_{56} = 23.5831433102848$$
$$x_{57} = -45.5640665961997$$
$$x_{58} = 42.4232862577008$$
$$x_{59} = -17.3076405374146$$
$$x_{60} = 95.8237937978449$$
$$x_{61} = 92.682377997352$$
$$x_{62} = 70.692907433161$$
$$x_{63} = -33.0018723591446$$
$$x_{64} = 89.5409746049841$$
The values of the extrema at the points:
(-86.3995849739529, 9.29498206229774*I)
(17.307640537414635, -4.15851032158028)
(-1.8365972031521258, -1.30761941299144*I)
(11.040829817971295, -3.31937237072132)
(51.84592244522343, 7.20007645193272)
(-92.68237799735202, 9.6270286533*I)
(-76.97551549355693, -8.77338405887965*I)
(26.72224637418772, 5.16845181340769)
(-95.82379379784489, -9.78882959875799*I)
(-36.142148897006074, 6.01125886058877*I)
(86.3995849739529, -9.29498206229774)
(67.5516436614121, -8.21875556224649)
(-58.12806557615112, -7.6238943490782*I)
(-51.84592244522343, -7.20007645193272*I)
(-61.269217268722585, 7.82720494097395*I)
(-26.72224637418772, -5.16845181340769*I)
(39.282635752723394, 6.26707847792961)
(45.56406659619972, 6.74970965872142)
(98.96522082503246, -9.94799953505937)
(7.917052684666207, 2.808131180007)
(-29.861872403816044, 5.46383591176171*I)
(29.861872403816044, -5.46383591176171)
(-48.70495166667517, 6.97852557917854*I)
(-89.54097460498406, -9.46246176606193*I)
(61.269217268722585, -7.82720494097395)
(1.8365972031521258, 1.30761941299144)
(-54.98696425148828, 7.4150130205716*I)
(-42.423286257700816, 6.51286373926386*I)
(-64.41041196277601, -8.02536795646149*I)
(-11.040829817971295, 3.31937237072132*I)
(83.25821066164869, 9.12442919108264)
(76.97551549355693, 8.77338405887965)
(20.4448034666183, 4.52024144595309)
(-73.83419918545908, 8.59248586707723*I)
(-39.282635752723394, -6.26707847792961*I)
(-67.5516436614121, 8.21875556224649*I)
(-80.11685346965491, 8.95062752823053*I)
(33.00187235914463, 5.74406639671223)
(54.98696425148828, -7.4150130205716)
(-14.172432074799941, -3.76228841574689*I)
(73.83419918545908, -8.59248586707723)
(-4.815842317845935, 2.18276978467772*I)
(-7.917052684666207, -2.808131180007*I)
(64.41041196277601, 8.02536795646149)
(-83.25821066164869, -9.12442919108264*I)
(-98.96522082503246, 9.94799953505937*I)
(36.142148897006074, -6.01125886058877)
(48.70495166667517, -6.97852557917854)
(-20.4448034666183, -4.52024144595309*I)
(14.172432074799941, 3.76228841574689)
(58.12806557615112, 7.6238943490782)
(-23.583143310284843, 4.85515677204621*I)
(4.815842317845935, -2.18276978467772)
(80.11685346965491, -8.95062752823053)
(-70.692907433161, -8.40769713937167*I)
(23.583143310284843, -4.85515677204621)
(-45.56406659619972, -6.74970965872142*I)
(42.423286257700816, -6.51286373926386)
(-17.307640537414635, 4.15851032158028*I)
(95.82379379784489, 9.78882959875799)
(92.68237799735202, -9.6270286533)
(70.692907433161, 8.40769713937167)
(-33.00187235914463, -5.74406639671223*I)
(89.54097460498406, 9.46246176606193)
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:
$$x_{1} = 17.3076405374146$$
$$x_{2} = 11.0408298179713$$
$$x_{3} = 86.3995849739529$$
$$x_{4} = 67.5516436614121$$
$$x_{5} = 98.9652208250325$$
$$x_{6} = 29.861872403816$$
$$x_{7} = 61.2692172687226$$
$$x_{8} = 54.9869642514883$$
$$x_{9} = 73.8341991854591$$
$$x_{10} = 36.1421488970061$$
$$x_{11} = 48.7049516666752$$
$$x_{12} = 4.81584231784594$$
$$x_{13} = 80.1168534696549$$
$$x_{14} = 23.5831433102848$$
$$x_{15} = 42.4232862577008$$
$$x_{16} = 92.682377997352$$
Maxima of the function at points:
$$x_{16} = 51.8459224452234$$
$$x_{16} = 26.7222463741877$$
$$x_{16} = 39.2826357527234$$
$$x_{16} = 45.5640665961997$$
$$x_{16} = 7.91705268466621$$
$$x_{16} = 1.83659720315213$$
$$x_{16} = 83.2582106616487$$
$$x_{16} = 76.9755154935569$$
$$x_{16} = 20.4448034666183$$
$$x_{16} = 33.0018723591446$$
$$x_{16} = 64.410411962776$$
$$x_{16} = 14.1724320747999$$
$$x_{16} = 58.1280655761511$$
$$x_{16} = 95.8237937978449$$
$$x_{16} = 70.692907433161$$
$$x_{16} = 89.5409746049841$$
Decreasing at intervals
$$\left[98.9652208250325, \infty\right)$$
Increasing at intervals
$$\left(-\infty, 4.81584231784594\right]$$