Para hallar los extremos hay que resolver la ecuación
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
$$\frac{d}{d x} f{\left(x \right)} = $$
primera derivada$$2 \cos{\left(2 x \right)} + \frac{1}{\sqrt{x}} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = 54.2264489965236$$
$$x_{2} = 91.9176729238735$$
$$x_{3} = 4.05249228238638$$
$$x_{4} = 68.2993712933574$$
$$x_{5} = 99.7205212709303$$
$$x_{6} = 24.2965370158447$$
$$x_{7} = 55.7297559553562$$
$$x_{8} = 96.5785241280874$$
$$x_{9} = 11.7076455899589$$
$$x_{10} = 19.6914120013524$$
$$x_{11} = 88.7765382108335$$
$$x_{12} = 90.2944673262361$$
$$x_{13} = 98.2000092239488$$
$$x_{14} = 33.7290211418778$$
$$x_{15} = 46.3017183027706$$
$$x_{16} = 66.7894534688458$$
$$x_{17} = 10.2884361592707$$
$$x_{18} = 62.014687268041$$
$$x_{19} = 25.9672783709104$$
$$x_{20} = 77.7260461856895$$
$$x_{21} = 30.5852618290448$$
$$x_{22} = 40.015744372293$$
$$x_{23} = 82.4943461155287$$
$$x_{24} = 47.9454243410335$$
$$x_{25} = 69.9303499758061$$
$$x_{26} = 63.6486079292757$$
$$x_{27} = 74.583861392211$$
$$x_{28} = 18.0051033417267$$
$$x_{29} = 52.5871749058489$$
$$x_{30} = 8.55347700287888$$
$$x_{31} = 41.6648721648553$$
$$x_{32} = 38.5248318580907$$
$$x_{33} = 51.0858867746697$$
$$x_{34} = 84.0103143757507$$
$$x_{35} = 76.212274530907$$
$$x_{36} = 60.5078198947171$$
$$x_{37} = 32.2454074676654$$
$$x_{38} = 2.18360612490942$$
$$x_{39} = 16.5549606350518$$
$$x_{40} = 85.6354284885093$$
Signos de extremos en los puntos:
(54.22644899652355, 25.7254142118051)
(91.91767292387351, 30.1733801426603)
(4.0524922823863765, 14.9948243995096)
(68.29937129335744, 25.5305185904362)
(99.7205212709303, 28.9732868625648)
(24.296537015844724, 18.8634615773202)
(55.72975595535619, 23.9327184819399)
(96.57852412808738, 28.6561697469637)
(11.7076455899589, 15.8540217915937)
(19.69141200135244, 19.868633073971)
(88.77653821083346, 29.8428514524883)
(90.29446732623609, 28.0060655137699)
(98.20000922394883, 30.8179098342604)
(33.72902114187779, 20.6190512321855)
(46.30171830277059, 22.6117765322634)
(66.78945346884576, 27.3430896435674)
(10.288436159270653, 17.4028941637693)
(62.01468726804098, 24.751898609548)
(25.96727837091043, 21.1867943746821)
(77.72604618568948, 26.634084794659)
(30.585261829044782, 20.0648840509687)
(40.01574437229298, 21.6547284643237)
(82.49434611552867, 29.1637632366255)
(47.94542434103352, 24.8459163639033)
(69.93034997580608, 27.7230845954385)
(63.6486079292757, 26.9540496937683)
(74.58386139221105, 26.2740670615819)
(18.005103341726727, 17.493450902053)
(52.58717490584895, 23.5057828731936)
(8.553477002878882, 14.8639879091996)
(41.66487216485534, 23.9066618381896)
(38.52483185809073, 23.4104250756333)
(51.0858867746697, 25.2924284913512)
(84.01031437575068, 27.3329171570655)
(76.21227453090701, 28.4582868508966)
(60.50781989471712, 26.5552865303941)
(32.2454074676654, 22.3531238989554)
(2.1836061249094185, 12.0143903105987)
(16.554960635051764, 19.1299781991271)
(85.63542848850933, 29.5064217385136)
Intervalos de crecimiento y decrecimiento de la función:Hallemos los intervalos donde la función crece y decrece y también los puntos mínimos y máximos de la función, para lo cual miramos cómo se comporta la función en los extremos con desviación mínima del extremo:
Puntos mínimos de la función:
$$x_{1} = 68.2993712933574$$
$$x_{2} = 99.7205212709303$$
$$x_{3} = 24.2965370158447$$
$$x_{4} = 55.7297559553562$$
$$x_{5} = 96.5785241280874$$
$$x_{6} = 11.7076455899589$$
$$x_{7} = 90.2944673262361$$
$$x_{8} = 33.7290211418778$$
$$x_{9} = 46.3017183027706$$
$$x_{10} = 62.014687268041$$
$$x_{11} = 77.7260461856895$$
$$x_{12} = 30.5852618290448$$
$$x_{13} = 40.015744372293$$
$$x_{14} = 74.583861392211$$
$$x_{15} = 18.0051033417267$$
$$x_{16} = 52.5871749058489$$
$$x_{17} = 8.55347700287888$$
$$x_{18} = 84.0103143757507$$
$$x_{19} = 2.18360612490942$$
Puntos máximos de la función:
$$x_{19} = 54.2264489965236$$
$$x_{19} = 91.9176729238735$$
$$x_{19} = 4.05249228238638$$
$$x_{19} = 19.6914120013524$$
$$x_{19} = 88.7765382108335$$
$$x_{19} = 98.2000092239488$$
$$x_{19} = 66.7894534688458$$
$$x_{19} = 10.2884361592707$$
$$x_{19} = 25.9672783709104$$
$$x_{19} = 82.4943461155287$$
$$x_{19} = 47.9454243410335$$
$$x_{19} = 69.9303499758061$$
$$x_{19} = 63.6486079292757$$
$$x_{19} = 41.6648721648553$$
$$x_{19} = 38.5248318580907$$
$$x_{19} = 51.0858867746697$$
$$x_{19} = 76.212274530907$$
$$x_{19} = 60.5078198947171$$
$$x_{19} = 32.2454074676654$$
$$x_{19} = 16.5549606350518$$
$$x_{19} = 85.6354284885093$$
Decrece en los intervalos
$$\left[99.7205212709303, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, 2.18360612490942\right]$$