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$$\frac{\sqrt{x \sin{\left(x \right)}} \left(\frac{x \cos{\left(x \right)}}{2} + \frac{\sin{\left(x \right)}}{2}\right)}{x \left(\sqrt{x \sin{\left(x \right)}} + 1\right) \sin{\left(x \right)}} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = -26.7409160147873$$
$$x_{2} = -89.5465575382492$$
$$x_{3} = 73.8409691490209$$
$$x_{4} = -39.295350981473$$
$$x_{5} = 70.69997803861$$
$$x_{6} = -2.02875783811043$$
$$x_{7} = -64.4181717218392$$
$$x_{8} = -20.469167402741$$
$$x_{9} = -61.2773745335697$$
$$x_{10} = 26.7409160147873$$
$$x_{11} = 61.2773745335697$$
$$x_{12} = -98.9702722883957$$
$$x_{13} = -83.2642147040886$$
$$x_{14} = -51.855560729152$$
$$x_{15} = -86.4053708116885$$
$$x_{16} = 54.9960525574964$$
$$x_{17} = -92.687771772017$$
$$x_{18} = -58.1366632448992$$
$$x_{19} = -4.91318043943488$$
$$x_{20} = 23.6042847729804$$
$$x_{21} = 48.7152107175577$$
$$x_{22} = 17.3363779239834$$
$$x_{23} = 95.8290108090195$$
$$x_{24} = -70.69997803861$$
$$x_{25} = 33.0170010333572$$
$$x_{26} = -36.1559664195367$$
$$x_{27} = -54.9960525574964$$
$$x_{28} = 76.9820093304187$$
$$x_{29} = 80.1230928148503$$
$$x_{30} = -76.9820093304187$$
$$x_{31} = 64.4181717218392$$
$$x_{32} = -23.6042847729804$$
$$x_{33} = -80.1230928148503$$
$$x_{34} = 98.9702722883957$$
$$x_{35} = 14.2074367251912$$
$$x_{36} = 42.4350618814099$$
$$x_{37} = 51.855560729152$$
$$x_{38} = 7.97866571241324$$
$$x_{39} = -33.0170010333572$$
$$x_{40} = -67.5590428388084$$
$$x_{41} = 58.1366632448992$$
$$x_{42} = 11.085538406497$$
$$x_{43} = 29.8785865061074$$
$$x_{44} = 92.687771772017$$
$$x_{45} = 67.5590428388084$$
$$x_{46} = -45.57503179559$$
$$x_{47} = -73.8409691490209$$
$$x_{48} = -11.085538406497$$
$$x_{49} = -14.2074367251912$$
$$x_{50} = -48.7152107175577$$
$$x_{51} = 86.4053708116885$$
$$x_{52} = -17.3363779239834$$
$$x_{53} = 20.469167402741$$
$$x_{54} = 45.57503179559$$
$$x_{55} = -29.8785865061074$$
$$x_{56} = 4.91318043943488$$
$$x_{57} = -7.97866571241324$$
$$x_{58} = -42.4350618814099$$
$$x_{59} = 2.02875783811043$$
$$x_{60} = 83.2642147040886$$
$$x_{61} = 89.5465575382492$$
$$x_{62} = -95.8290108090195$$
$$x_{63} = 39.295350981473$$
The values of the extrema at the points:
(-26.74091601478731, 1.81959439703613)
(-89.54655753824919, 2.34780788061745)
(73.8409691490209, 2.15763749166158 + 1.45493941316353*I)
(-39.295350981472986, 1.98342417304979)
(70.69997803861, 2.24155054914526)
(-2.028757838110434, 0.853974674628296)
(-64.41817172183916, 2.20006606067476)
(-20.46916740274095, 1.70866602110739)
(-61.277374533569656, 2.0657335936083 + 1.44372937374428*I)
(26.74091601478731, 1.81959439703613)
(61.277374533569656, 2.0657335936083 + 1.44372937374428*I)
(-98.9702722883957, 2.30241116601255 + 1.47061142377839*I)
(-83.26421470408864, 2.31496748478214)
(-51.85556072915197, 2.10418429767025)
(-86.40537081168854, 2.23524526555628 + 1.46362539911914*I)
(54.99605255749639, 2.01255943226034 + 1.43674906731317*I)
(-92.687771772017, 2.26995505049594 + 1.46729480728072*I)
(-58.13666324489916, 2.15456946051663)
(-4.913180439434884, 0.88017480962742 + 1.14317179398954*I)
(23.604284772980407, 1.60103023087858 + 1.36771465614255*I)
(48.715210717557724, 1.95305226513135 + 1.42847588502425*I)
(17.33637792398336, 1.45365837217991 + 1.3349008968913*I)
(95.82901080901948, 2.37852365290896)
(-70.69997803861, 2.24155054914526)
(33.017001033357246, 1.90876074892496)
(-36.15596641953672, 1.80737615936079 + 1.40596686862376*I)
(-54.99605255749639, 2.01255943226034 + 1.43674906731317*I)
(76.98200933041872, 2.2796817137158)
(80.12309281485025, 2.19794537325366 + 1.45953587048814*I)
(-76.98200933041872, 2.2796817137158)
(64.41817172183916, 2.20006606067476)
(-23.604284772980407, 1.60103023087858 + 1.36771465614255*I)
(-80.12309281485025, 2.19794537325366 + 1.45953587048814*I)
(98.9702722883957, 2.30241116601255 + 1.47061142377839*I)
(14.207436725191188, 1.56121807481243)
(42.43506188140989, 1.88549789835899 + 1.41845426567079*I)
(51.85556072915197, 2.10418429767025)
(7.978665712413241, 1.3385914443644)
(-33.017001033357246, 1.90876074892496)
(-67.5590428388084, 2.11379368769621 + 1.4497218389579*I)
(58.13666324489916, 2.15456946051663)
(11.085538406497022, 1.24414662173076 + 1.27846363206011*I)
(29.878586506107393, 1.71476065844424 + 1.3898029639899*I)
(92.687771772017, 2.26995505049594 + 1.46729480728072*I)
(67.5590428388084, 2.11379368769621 + 1.4497218389579*I)
(-45.57503179559002, 2.04770779974173)
(-73.8409691490209, 2.15763749166158 + 1.45493941316353*I)
(-11.085538406497022, 1.24414662173076 + 1.27846363206011*I)
(-14.207436725191188, 1.56121807481243)
(-48.715210717557724, 1.95305226513135 + 1.42847588502425*I)
(86.40537081168854, 2.23524526555628 + 1.46362539911914*I)
(-17.33637792398336, 1.45365837217991 + 1.3349008968913*I)
(20.46916740274095, 1.70866602110739)
(45.57503179559002, 2.04770779974173)
(-29.878586506107393, 1.71476065844424 + 1.3898029639899*I)
(4.913180439434884, 0.88017480962742 + 1.14317179398954*I)
(-7.978665712413241, 1.3385914443644)
(-42.43506188140989, 1.88549789835899 + 1.41845426567079*I)
(2.028757838110434, 0.853974674628296)
(83.26421470408864, 2.31496748478214)
(89.54655753824919, 2.34780788061745)
(-95.82901080901948, 2.37852365290896)
(39.295350981472986, 1.98342417304979)
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:
The function has no minima
Maxima of the function at points:
$$x_{63} = -26.7409160147873$$
$$x_{63} = -89.5465575382492$$
$$x_{63} = -39.295350981473$$
$$x_{63} = 70.69997803861$$
$$x_{63} = -2.02875783811043$$
$$x_{63} = -64.4181717218392$$
$$x_{63} = -20.469167402741$$
$$x_{63} = 26.7409160147873$$
$$x_{63} = -83.2642147040886$$
$$x_{63} = -51.855560729152$$
$$x_{63} = -58.1366632448992$$
$$x_{63} = 95.8290108090195$$
$$x_{63} = -70.69997803861$$
$$x_{63} = 33.0170010333572$$
$$x_{63} = 76.9820093304187$$
$$x_{63} = -76.9820093304187$$
$$x_{63} = 64.4181717218392$$
$$x_{63} = 14.2074367251912$$
$$x_{63} = 51.855560729152$$
$$x_{63} = 7.97866571241324$$
$$x_{63} = -33.0170010333572$$
$$x_{63} = 58.1366632448992$$
$$x_{63} = -45.57503179559$$
$$x_{63} = -14.2074367251912$$
$$x_{63} = 20.469167402741$$
$$x_{63} = 45.57503179559$$
$$x_{63} = -7.97866571241324$$
$$x_{63} = 2.02875783811043$$
$$x_{63} = 83.2642147040886$$
$$x_{63} = 89.5465575382492$$
$$x_{63} = -95.8290108090195$$
$$x_{63} = 39.295350981473$$
Decreasing at intervals
$$\left(-\infty, -95.8290108090195\right]$$
Increasing at intervals
$$\left[95.8290108090195, \infty\right)$$