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$$6 \log{\left(x \right)}^{2} \sin^{2}{\left(2 x \right)} \cos{\left(2 x \right)} + \frac{2 \log{\left(x \right)} \sin^{3}{\left(2 x \right)}}{x} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = 28.2743338655222$$
$$x_{2} = 87.9645943337434$$
$$x_{3} = 20.4203521848575$$
$$x_{4} = 14.1371669972345$$
$$x_{5} = 21.9911485850751$$
$$x_{6} = 50.2654824464643$$
$$x_{7} = 40.0564338300669$$
$$x_{8} = 7.85398170926124$$
$$x_{9} = 94.2477796093544$$
$$x_{10} = 32.2028153200555$$
$$x_{11} = 14.9266956256452$$
$$x_{12} = 6.28318528515891$$
$$x_{13} = 76.184126720141$$
$$x_{14} = 46.3394292450536$$
$$x_{15} = 90.3211985427413$$
$$x_{16} = 54.1932435608028$$
$$x_{17} = 68.3302176140681$$
$$x_{18} = 10.217194538067$$
$$x_{19} = 98.1751405440235$$
$$x_{20} = 84.0380510365345$$
$$x_{21} = 65.973445751761$$
$$x_{22} = 43.9822971689231$$
$$x_{23} = 3.95756657756918$$
$$x_{24} = 69.900997945409$$
$$x_{25} = 72.2566310277385$$
$$x_{26} = 18.0673451397774$$
$$x_{27} = 11.7867040144534$$
$$x_{28} = 91.8919863292189$$
$$x_{29} = 80.8964797955064$$
$$x_{30} = 24.3494870609614$$
$$x_{31} = 62.0471056348408$$
$$x_{32} = 25.9201147964456$$
$$x_{33} = 47.9101870193282$$
Signos de extremos en los puntos:
(28.274333865522188, -4.22599457735672e-22)
(87.96459433374336, 5.88314100551173e-21)
(20.420352184857492, 1.86182086894047e-20)
(14.13716699723455, -9.89978771373222e-21)
(21.99114858507506, 7.5196665969877e-23)
(50.26548244646428, -1.62171201484649e-22)
(40.0564338300669, -13.6181312860135)
(7.853981709261238, -1.45014157207409e-20)
(94.24777960935442, 7.57096593840344e-25)
(32.20281532005548, 12.0549974378264)
(14.92669562564515, -7.3062787781205)
(6.283185285158907, -2.88546004559296e-22)
(76.184126720141, 18.7761873343664)
(46.339429245053644, -14.7147662327291)
(90.32119854274134, -20.2803406490493)
(54.193243560802784, 15.9404486056561)
(68.33021761406813, -17.8451149139526)
(10.217194538066972, 5.39971467987353)
(98.17514054402346, 21.0382860848242)
(84.03805103653451, -19.6361274203226)
(65.97344575176098, 2.57606124495216e-21)
(43.982297168923104, 7.44898472812582e-22)
(3.9575665775691755, 1.8817644765114)
(69.90099794540905, 18.0376537869824)
(72.25663102773849, -1.64811628124724e-23)
(18.067345139777416, -8.3753399905399)
(11.786704014453388, -6.08475195069731)
(91.89198632921891, 20.4359296232023)
(80.89647979550644, -19.2999199057376)
(24.34948706096143, -10.1918440073735)
(62.04710563484077, -17.0394644679274)
(25.92011479644556, 10.5949025801098)
(47.910187019328184, 14.9716277576272)
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:
$$x_{1} = 40.0564338300669$$
$$x_{2} = 14.9266956256452$$
$$x_{3} = 46.3394292450536$$
$$x_{4} = 90.3211985427413$$
$$x_{5} = 68.3302176140681$$
$$x_{6} = 84.0380510365345$$
$$x_{7} = 18.0673451397774$$
$$x_{8} = 11.7867040144534$$
$$x_{9} = 80.8964797955064$$
$$x_{10} = 24.3494870609614$$
$$x_{11} = 62.0471056348408$$
Puntos máximos de la función:
$$x_{11} = 32.2028153200555$$
$$x_{11} = 76.184126720141$$
$$x_{11} = 54.1932435608028$$
$$x_{11} = 10.217194538067$$
$$x_{11} = 98.1751405440235$$
$$x_{11} = 3.95756657756918$$
$$x_{11} = 69.900997945409$$
$$x_{11} = 91.8919863292189$$
$$x_{11} = 25.9201147964456$$
$$x_{11} = 47.9101870193282$$
Decrece en los intervalos
$$\left[90.3211985427413, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, 11.7867040144534\right]$$