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$$\frac{\cos{\left(2 x \right)}}{x \sqrt{\sin{\left(2 x \right)}}} - \frac{\sqrt{\sin{\left(2 x \right)}}}{x^{2}} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = -63.609391408151$$
$$x_{2} = 66.7513539643031$$
$$x_{3} = -99.7405539133885$$
$$x_{4} = -62.0383960813757$$
$$x_{5} = 33.7573137512912$$
$$x_{6} = 36.9001669211527$$
$$x_{7} = -46.3277006372222$$
$$x_{8} = -98.1696773784321$$
$$x_{9} = 74.6061240613667$$
$$x_{10} = 11.7384800939942$$
$$x_{11} = 58.8963735834305$$
$$x_{12} = -13.3142855057572$$
$$x_{13} = 22.7545872771683$$
$$x_{14} = 63.609391408151$$
$$x_{15} = -65.1803771434091$$
$$x_{16} = -30.6142018882607$$
$$x_{17} = -19.6094782798075$$
$$x_{18} = 76.1770585706428$$
$$x_{19} = -18.0364644916681$$
$$x_{20} = -27.4707425696429$$
$$x_{21} = 41.614089810742$$
$$x_{22} = -47.8988508196586$$
$$x_{23} = -43.1853230479384$$
$$x_{24} = -41.614089810742$$
$$x_{25} = 47.8988508196586$$
$$x_{26} = -57.3253446670262$$
$$x_{27} = 85.602559129126$$
$$x_{28} = 18.0364644916681$$
$$x_{29} = 44.7565256168206$$
$$x_{30} = -85.602559129126$$
$$x_{31} = -90.3152528534981$$
$$x_{32} = -52.6121745880344$$
$$x_{33} = 40.0428222957763$$
$$x_{34} = -5.40633666694364$$
$$x_{35} = 98.1696773784321$$
$$x_{36} = -79.3189111701245$$
$$x_{37} = 99.7405539133885$$
$$x_{38} = -40.0428222957763$$
$$x_{39} = 16.4630276170453$$
$$x_{40} = -54.1832463772731$$
$$x_{41} = 27.4707425696429$$
$$x_{42} = 3.79827300987529$$
$$x_{43} = 14.8890337004883$$
$$x_{44} = 24.3268011678533$$
$$x_{45} = 55.7543026449315$$
$$x_{46} = 88.744358541403$$
$$x_{47} = -93.4570315962354$$
$$x_{48} = 60.4673904155608$$
$$x_{49} = -69.8932832672187$$
$$x_{50} = 71.4642368192271$$
$$x_{51} = 2.13739113572906$$
$$x_{52} = 77.747987496317$$
$$x_{53} = -76.1770585706428$$
$$x_{54} = 49.4699785303278$$
$$x_{55} = 8.58137569421011$$
$$x_{56} = 19.6094782798075$$
$$x_{57} = -35.3287683589234$$
$$x_{58} = -55.7543026449315$$
$$x_{59} = -84.0316536256143$$
$$x_{60} = -32.1857948915632$$
$$x_{61} = -38.4715163039797$$
$$x_{62} = -60.4673904155608$$
$$x_{63} = -49.4699785303278$$
$$x_{64} = 32.1857948915632$$
$$x_{65} = 46.3277006372222$$
$$x_{66} = 62.0383960813757$$
$$x_{67} = 96.5987982349427$$
$$x_{68} = -71.4642368192271$$
$$x_{69} = -16.4630276170453$$
$$x_{70} = -96.5987982349427$$
$$x_{71} = 10.1611269299962$$
$$x_{72} = 82.4607439626894$$
$$x_{73} = -2.13739113572906$$
$$x_{74} = 68.322322485708$$
$$x_{75} = -87.173460698078$$
$$x_{76} = 30.6142018882607$$
$$x_{77} = -91.8861438154664$$
$$x_{78} = 84.0316536256143$$
$$x_{79} = -21.182163158836$$
$$x_{80} = -24.3268011678533$$
$$x_{81} = -10.1611269299962$$
$$x_{82} = -11.7384800939942$$
$$x_{83} = 54.1832463772731$$
$$x_{84} = -25.898843096056$$
$$x_{85} = -74.6061240613667$$
$$x_{86} = 90.3152528534981$$
$$x_{87} = -3.79827300987529$$
$$x_{88} = -33.7573137512912$$
$$x_{89} = -82.4607439626894$$
$$x_{90} = 80.8898298980315$$
$$x_{91} = 38.4715163039797$$
$$x_{92} = 69.8932832672187$$
$$x_{93} = 5.40633666694364$$
$$x_{94} = -8.58137569421011$$
$$x_{95} = 25.898843096056$$
$$x_{96} = -68.322322485708$$
$$x_{97} = 91.8861438154664$$
$$x_{98} = 52.6121745880344$$
$$x_{99} = 93.4570315962354$$
$$x_{100} = -77.747987496317$$
Signos de extremos en los puntos:
(-63.60939140815101, -0.0157199778263343*I)
(66.75135396430314, 0.0149801291078152)
(-99.74055391338845, -0.0100257601558681)
(-62.03839608137567, -0.0161180030072405)
(33.75731375129117, 0.0296167148833163*I)
(36.900166921152675, 0.0270951749854015*I)
(-46.327700637222165, -0.0215828443509178)
(-98.16967737843208, -0.0101861805750159*I)
(74.60612406136674, 0.0134031234387214*I)
(11.73848009399415, 0.0850360393335331*I)
(58.89637358343045, 0.016977750594038*I)
(-13.314285505757224, -0.0750017471309216*I)
(22.754587277168262, 0.0439259886853852)
(63.60939140815101, 0.0157199778263343)
(-65.18037714340906, -0.0153411380529724)
(-30.6142018882607, -0.0326558712102137)
(-19.60947827980746, -0.0509626466768668*I)
(76.17705857064283, 0.0131267463173753)
(-18.036464491668116, -0.0554007125582391)
(-27.470742569642862, -0.0363903155700504)
(41.614089810742044, 0.02402685466694)
(-47.89885081965864, -0.0208750533021526*I)
(-43.185323047938354, -0.0231529122413857)
(-41.614089810742044, -0.02402685466694*I)
(47.89885081965864, 0.0208750533021526)
(-57.32534466702617, -0.0174429642673855*I)
(85.60255912912602, 0.0116814952312444)
(18.036464491668116, 0.0554007125582391*I)
(44.756525616820596, 0.0223403229249365)
(-85.60255912912602, -0.0116814952312444*I)
(-90.31525285349807, -0.0110719875286708)
(-52.612174588034385, -0.0190052912967891)
(40.04282229577635, 0.0249693724726356*I)
(-5.406336666943637, -0.18341903022942)
(98.16967737843208, 0.0101861805750159)
(-79.31891117012448, -0.0126068330109199*I)
(99.74055391338845, 0.0100257601558681*I)
(-40.04282229577635, -0.0249693724726356)
(16.463027617045263, 0.0606862687517582)
(-54.18324637727305, -0.018454318078295*I)
(27.470742569642862, 0.0363903155700504*I)
(3.798273009875294, 0.258903189710353)
(14.889033700488254, 0.067087996040512*I)
(24.326801167853258, 0.0410895782745605*I)
(55.75430264493151, 0.0179343933146333*I)
(88.74435854140305, 0.0112679642422872)
(-93.45703159623535, -0.0106997982057615)
(60.46739041556085, 0.0165367089563368)
(-69.89328326721875, -0.0143067943509695*I)
(71.46423681922705, 0.0139923281824918*I)
(2.137391135729064, 0.445271188713631*I)
(77.74798749631695, 0.0128615373628376*I)
(-76.17705857064283, -0.0131267463173753*I)
(49.46997853032777, 0.0202122156003825*I)
(8.581375694210113, 0.116139144217135*I)
(19.60947827980746, 0.0509626466768668)
(-35.328768358923384, -0.028299877004799*I)
(-55.75430264493151, -0.0179343933146333)
(-84.03165362561431, -0.0118998562546094)
(-32.18579489156321, -0.0310621135424149*I)
(-38.471516303979705, -0.0259888679989479*I)
(-60.46739041556085, -0.0165367089563368*I)
(-49.46997853032777, -0.0202122156003825)
(32.18579489156321, 0.0310621135424149)
(46.327700637222165, 0.0215828443509178*I)
(62.03839608137567, 0.0161180030072405*I)
(96.59879823494268, 0.010351818331022*I)
(-71.46423681922705, -0.0139923281824918)
(-16.463027617045263, -0.0606862687517582*I)
(-96.59879823494268, -0.010351818331022)
(10.16112692999623, 0.0981774182935068)
(82.46074396268945, 0.0121265366936926)
(-2.137391135729064, -0.445271188713631)
(68.32232248570797, 0.0146357210055436*I)
(-87.17346069807797, -0.0114710038646054)
(30.6142018882607, 0.0326558712102137*I)
(-91.88614381546644, -0.0108827114792659*I)
(84.03165362561431, 0.0118998562546094*I)
(-21.18216315883595, -0.0471832636912932)
(-24.326801167853258, -0.0410895782745605)
(-10.16112692999623, -0.0981774182935068*I)
(-11.73848009399415, -0.0850360393335331)
(54.18324637727305, 0.018454318078295)
(-25.89884309605599, -0.0385973854559027*I)
(-74.60612406136674, -0.0134031234387214)
(90.31525285349807, 0.0110719875286708*I)
(-3.798273009875294, -0.258903189710353*I)
(-33.75731375129117, -0.0296167148833163)
(-82.46074396268945, -0.0121265366936926*I)
(80.88982989803151, 0.0123620212466517*I)
(38.471516303979705, 0.0259888679989479)
(69.89328326721875, 0.0143067943509695)
(5.406336666943637, 0.18341903022942*I)
(-8.581375694210113, -0.116139144217135)
(25.89884309605599, 0.0385973854559027)
(-68.32232248570797, -0.0146357210055436)
(91.88614381546644, 0.0108827114792659)
(52.612174588034385, 0.0190052912967891*I)
(93.45703159623535, 0.0106997982057615*I)
(-77.74798749631695, -0.0128615373628376)
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:
$$x_{1} = -99.7405539133885$$
$$x_{2} = -62.0383960813757$$
$$x_{3} = -46.3277006372222$$
$$x_{4} = -65.1803771434091$$
$$x_{5} = -30.6142018882607$$
$$x_{6} = -18.0364644916681$$
$$x_{7} = -27.4707425696429$$
$$x_{8} = -43.1853230479384$$
$$x_{9} = -90.3152528534981$$
$$x_{10} = -52.6121745880344$$
$$x_{11} = -5.40633666694364$$
$$x_{12} = -40.0428222957763$$
$$x_{13} = -93.4570315962354$$
$$x_{14} = -55.7543026449315$$
$$x_{15} = -84.0316536256143$$
$$x_{16} = -49.4699785303278$$
$$x_{17} = -71.4642368192271$$
$$x_{18} = -96.5987982349427$$
$$x_{19} = -2.13739113572906$$
$$x_{20} = -87.173460698078$$
$$x_{21} = -21.182163158836$$
$$x_{22} = -24.3268011678533$$
$$x_{23} = -11.7384800939942$$
$$x_{24} = -74.6061240613667$$
$$x_{25} = -33.7573137512912$$
$$x_{26} = -8.58137569421011$$
$$x_{27} = -68.322322485708$$
$$x_{28} = -77.747987496317$$
Puntos máximos de la función:
$$x_{28} = 66.7513539643031$$
$$x_{28} = 22.7545872771683$$
$$x_{28} = 63.609391408151$$
$$x_{28} = 76.1770585706428$$
$$x_{28} = 41.614089810742$$
$$x_{28} = 47.8988508196586$$
$$x_{28} = 85.602559129126$$
$$x_{28} = 44.7565256168206$$
$$x_{28} = 98.1696773784321$$
$$x_{28} = 16.4630276170453$$
$$x_{28} = 3.79827300987529$$
$$x_{28} = 88.744358541403$$
$$x_{28} = 60.4673904155608$$
$$x_{28} = 19.6094782798075$$
$$x_{28} = 32.1857948915632$$
$$x_{28} = 10.1611269299962$$
$$x_{28} = 82.4607439626894$$
$$x_{28} = 54.1832463772731$$
$$x_{28} = 38.4715163039797$$
$$x_{28} = 69.8932832672187$$
$$x_{28} = 25.898843096056$$
$$x_{28} = 91.8861438154664$$
Decrece en los intervalos
$$\left[-2.13739113572906, 3.79827300987529\right]$$
Crece en los intervalos
$$\left(-\infty, -99.7405539133885\right]$$