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$$3 \sin{\left(x \right)} - \sin{\left(\left|{x}\right| \right)} \operatorname{sign}{\left(x \right)} + \frac{\sqrt{2} \sqrt{x}}{2 x} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = 100.495689562389$$
$$x_{2} = 40.8960186130528$$
$$x_{3} = 34.6176459701182$$
$$x_{4} = 91.1432287574617$$
$$x_{5} = 31.3527426632222$$
$$x_{6} = 3.33638279476936$$
$$x_{7} = 22.0664840127627$$
$$x_{8} = 59.7360206738079$$
$$x_{9} = 72.2982236659435$$
$$x_{10} = 6.14001432005001$$
$$x_{11} = 18.7678542451706$$
$$x_{12} = 78.5797110212012$$
$$x_{13} = 2082.88367621027$$
$$x_{14} = 53.455450939062$$
$$x_{15} = 97.4251994117561$$
$$x_{16} = 147.683951814101$$
$$x_{17} = 87.9268807753933$$
$$x_{18} = 37.6414534578481$$
$$x_{19} = 66.0169732880922$$
$$x_{20} = 25.0620593136738$$
$$x_{21} = 56.5016150332186$$
$$x_{22} = 81.6422701167997$$
$$x_{23} = 94.2113462258309$$
$$x_{24} = 9.53949957697623$$
$$x_{25} = 84.8613906302444$$
$$x_{26} = 75.357484537593$$
$$x_{27} = 28.3407951710726$$
$$x_{28} = 62.7872192998985$$
$$x_{29} = 15.7970353560729$$
$$x_{30} = 69.0724850498015$$
$$x_{31} = 47.1753876712831$$
$$x_{32} = 43.9289285415686$$
$$x_{33} = 12.4660664960616$$
$$x_{34} = 50.2155691721985$$
Signos de extremos en los puntos:
(100.4956895623887, 12.1783870621348)
(40.896018613052775, 11.0408361537307)
(34.617645970118225, 10.3171605792316)
(91.14322875746173, 15.4999782724934)
(31.352742663222234, 5.92267039291801)
(3.3363827947693583, 4.54534630193841)
(22.066484012762732, 8.6375921174486)
(59.73602067380795, 12.9282330451214)
(72.29822366594347, 14.0230965933589)
(6.140014320050014, 1.52475012173083)
(18.76785424517056, 4.13331070083742)
(78.5797110212012, 14.5347327295503)
(2082.8836762102746, 66.5426959888194)
(53.455450939061954, 12.3374330512077)
(97.42519941175608, 15.9575989872012)
(147.68395181410102, 19.1854242076374)
(87.92688077539327, 11.2624085885278)
(37.64145345784813, 6.67989587069277)
(66.01697328809216, 13.4887081836159)
(25.06205931367379, 5.08483273779398)
(56.501615033218634, 8.6325112922882)
(81.64227011679972, 10.779815587602)
(94.21134622583087, 11.7280417986598)
(9.53949957697623, 6.35480474696333)
(84.86139063024437, 15.0262960865029)
(75.35748453759302, 10.2782619540773)
(28.340795171072592, 9.52430223177848)
(62.7872192998985, 9.20799195888633)
(15.797035356072874, 7.61293172500437)
(69.07248504980153, 9.75531935462167)
(47.17538767128312, 11.7107812827346)
(43.928928541568624, 7.37609974728043)
(12.466066496061623, 3.00326117082544)
(50.215569172198514, 8.02402455099547)
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:
$$x_{1} = 100.495689562389$$
$$x_{2} = 31.3527426632222$$
$$x_{3} = 6.14001432005001$$
$$x_{4} = 18.7678542451706$$
$$x_{5} = 87.9268807753933$$
$$x_{6} = 37.6414534578481$$
$$x_{7} = 25.0620593136738$$
$$x_{8} = 56.5016150332186$$
$$x_{9} = 81.6422701167997$$
$$x_{10} = 94.2113462258309$$
$$x_{11} = 75.357484537593$$
$$x_{12} = 62.7872192998985$$
$$x_{13} = 69.0724850498015$$
$$x_{14} = 43.9289285415686$$
$$x_{15} = 12.4660664960616$$
$$x_{16} = 50.2155691721985$$
Puntos máximos de la función:
$$x_{16} = 40.8960186130528$$
$$x_{16} = 34.6176459701182$$
$$x_{16} = 91.1432287574617$$
$$x_{16} = 3.33638279476936$$
$$x_{16} = 22.0664840127627$$
$$x_{16} = 59.7360206738079$$
$$x_{16} = 72.2982236659435$$
$$x_{16} = 78.5797110212012$$
$$x_{16} = 2082.88367621027$$
$$x_{16} = 53.455450939062$$
$$x_{16} = 97.4251994117561$$
$$x_{16} = 147.683951814101$$
$$x_{16} = 66.0169732880922$$
$$x_{16} = 9.53949957697623$$
$$x_{16} = 84.8613906302444$$
$$x_{16} = 28.3407951710726$$
$$x_{16} = 15.7970353560729$$
$$x_{16} = 47.1753876712831$$
Decrece en los intervalos
$$\left[100.495689562389, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, 6.14001432005001\right]$$