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$$2 \log{\left(x \right)} \sin{\left(x \right)} \cos{\left(x \right)} + \frac{\sin^{2}{\left(x \right)}}{x} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = 95.8197196421744$$
$$x_{2} = 80.1120364968984$$
$$x_{3} = 70.6874957987652$$
$$x_{4} = -6.28318530717959$$
$$x_{5} = -31.4159265358979$$
$$x_{6} = 67.5459991590417$$
$$x_{7} = 21.9911485751286$$
$$x_{8} = 43.9822971502571$$
$$x_{9} = 15.707963267949$$
$$x_{10} = 42.4146464836393$$
$$x_{11} = 7.88468215226503$$
$$x_{12} = -53.4070751110265$$
$$x_{13} = 92.6781744605177$$
$$x_{14} = 58.1215816459086$$
$$x_{15} = -9.42477796076938$$
$$x_{16} = 78.5398163397448$$
$$x_{17} = -81.6814089933346$$
$$x_{18} = -94.2477796076938$$
$$x_{19} = 36.1321731425561$$
$$x_{20} = 26.7092361429331$$
$$x_{21} = 94.2477796076938$$
$$x_{22} = 12.5663706143592$$
$$x_{23} = 81.6814089933346$$
$$x_{24} = 29.8500622839131$$
$$x_{25} = -50.2654824574367$$
$$x_{26} = 23.5686584612553$$
$$x_{27} = -15.707963267949$$
$$x_{28} = 64.4045132832651$$
$$x_{29} = -59.6902604182061$$
$$x_{30} = 270.176968208722$$
$$x_{31} = 51.8387217793455$$
$$x_{32} = 6.28318530717959$$
$$x_{33} = 48.697328558041$$
$$x_{34} = 73.829001694965$$
$$x_{35} = -87.9645943005142$$
$$x_{36} = 34.5575191894877$$
$$x_{37} = 65.9734457253857$$
$$x_{38} = -40.8407044966673$$
$$x_{39} = -37.6991118430775$$
$$x_{40} = 72.2566310325652$$
$$x_{41} = -43.9822971502571$$
$$x_{42} = 20.4284648400094$$
$$x_{43} = 14.1505011586627$$
$$x_{44} = -100.530964914873$$
$$x_{45} = 45.5559674357001$$
$$x_{46} = -75.398223686155$$
$$x_{47} = 1.94119311490964$$
$$x_{48} = -97.3893722612836$$
$$x_{49} = 87.9645943005142$$
$$x_{50} = 59.6902604182061$$
$$x_{51} = 100.530964914873$$
$$x_{52} = 28.2743338823081$$
$$x_{53} = -72.2566310325652$$
$$x_{54} = -21.9911485751286$$
$$x_{55} = 56.5486677646163$$
$$x_{56} = 89.5366330613754$$
$$x_{57} = -28.2743338823081$$
$$x_{58} = 50.2654824574367$$
$$x_{59} = -65.9734457253857$$
$$x_{60} = 37.6991118430775$$
$$x_{61} = 86.3950958997003$$
Signos de extremos en los puntos:
(95.81971964217442, 4.56246253755753)
(80.11203649689836, 4.38341722467833)
(70.68749579876517, 4.25825694491077)
(-6.283185307179586, 1.10254964387092e-31 + 5.99903913064743e-32*pi*I)
(-31.41592653589793, 5.17014436343721e-30 + 1.49975978266186e-30*pi*I)
(67.54599915904168, 4.21279582809083)
(21.991148575128552, 2.27125663724702e-30)
(43.982297150257104, 1.11225529081318e-29)
(15.707963267948966, 1.03264752464199e-30)
(42.414646483639295, 3.74745665652845)
(7.884682152265031, 2.06297628658956)
(-53.40707511102649, 8.60546142301337e-30 + 2.16329417632678e-30*pi*I)
(92.67817446051765, 4.52912657571395)
(58.12158164590861, 4.06251883524572)
(-9.42477796076938, 3.0280269348815e-31 + 1.34978380439567e-31*pi*I)
(78.53981633974483, 1.05239675280611e-30)
(-81.68140899333463, 6.77020503227825e-29 + 1.53769519406264e-29*pi*I)
(-94.2477796076938, 5.35287607041647e-29 + 1.17751027577409e-29*pi*I)
(36.132173142556084, 3.5871303095993)
(26.709236142933094, 3.28490275263666)
(94.2477796076938, 5.35287607041647e-29)
(12.566370614359172, 6.07348539927449e-31)
(81.68140899333463, 6.77020503227825e-29)
(29.85006228391305, 3.39610431359338)
(-50.26548245743669, 1.50400944749698e-29 + 3.83938504361436e-30*pi*I)
(23.568658461255307, 3.15977537682115)
(-15.707963267948966, 1.03264752464199e-30 + 3.74939945665464e-31*pi*I)
(64.40451328326512, 4.16516924258626)
(-59.69026041820607, 6.14517616924322e-30 + 1.5027934458313e-30*pi*I)
(270.1769682087222, 1.7417470442363e-27)
(51.838721779345526, 3.94811383137987)
(6.283185307179586, 1.10254964387092e-31)
(48.69732855804098, 3.88559704250618)
(73.82900169496496, 4.30174096928494)
(-87.96459430051421, 5.26403170691023e-29 + 1.1758116696069e-29*pi*I)
(34.55751918948773, 1.72337415243203e-29)
(65.97344572538566, 4.03120648719441e-30)
(-40.840704496667314, 1.42608898598829e-29 + 3.84423798515659e-30*pi*I)
(-37.69911184307752, 7.83875937863388e-30 + 2.15965408703307e-30*pi*I)
(72.25663103256524, 1.73645580663874e-28)
(-43.982297150257104, 1.11225529081318e-29 + 2.93952917401724e-30*pi*I)
(20.428464840009365, 3.01673071130889)
(14.150501158662737, 2.64927893980534)
(-100.53096491487338, 7.08054135721405e-29 + 1.53575401744574e-29*pi*I)
(45.55596743570007, 3.81891008060303)
(-75.39822368615503, 3.73428700801825e-29 + 8.6386163481323e-30*pi*I)
(1.9411931149096389, 0.576387980778498)
(-97.3893722612836, 2.15581661527067e-28 + 4.70834203716472e-29*pi*I)
(87.96459430051421, 5.26403170691023e-29)
(59.69026041820607, 6.14517616924322e-30)
(100.53096491487338, 7.08054135721405e-29)
(28.274333882308138, 4.0598244084922e-30)
(-72.25663103256524, 1.73645580663874e-28 + 4.05692731349557e-29*pi*I)
(-21.991148575128552, 2.27125663724702e-30 + 7.3488229350431e-31*pi*I)
(56.548667764616276, 1.96074534521452e-29)
(89.53663306137544, 4.49464091136003)
(-28.274333882308138, 4.0598244084922e-30 + 1.2148054239561e-30*pi*I)
(50.26548245743669, 1.50400944749698e-29)
(-65.97344572538566, 4.03120648719441e-30 + 9.62273497948765e-31*pi*I)
(37.69911184307752, 7.83875937863388e-30)
(86.39509589970032, 4.45892340221529)
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:
$$x_{1} = -6.28318530717959$$
$$x_{2} = -31.4159265358979$$
$$x_{3} = 21.9911485751286$$
$$x_{4} = 43.9822971502571$$
$$x_{5} = 15.707963267949$$
$$x_{6} = -53.4070751110265$$
$$x_{7} = -9.42477796076938$$
$$x_{8} = 78.5398163397448$$
$$x_{9} = -81.6814089933346$$
$$x_{10} = -94.2477796076938$$
$$x_{11} = 94.2477796076938$$
$$x_{12} = 12.5663706143592$$
$$x_{13} = 81.6814089933346$$
$$x_{14} = -50.2654824574367$$
$$x_{15} = -15.707963267949$$
$$x_{16} = -59.6902604182061$$
$$x_{17} = 270.176968208722$$
$$x_{18} = 6.28318530717959$$
$$x_{19} = -87.9645943005142$$
$$x_{20} = 34.5575191894877$$
$$x_{21} = 65.9734457253857$$
$$x_{22} = -40.8407044966673$$
$$x_{23} = -37.6991118430775$$
$$x_{24} = 72.2566310325652$$
$$x_{25} = -43.9822971502571$$
$$x_{26} = -100.530964914873$$
$$x_{27} = -75.398223686155$$
$$x_{28} = -97.3893722612836$$
$$x_{29} = 87.9645943005142$$
$$x_{30} = 59.6902604182061$$
$$x_{31} = 100.530964914873$$
$$x_{32} = 28.2743338823081$$
$$x_{33} = -72.2566310325652$$
$$x_{34} = -21.9911485751286$$
$$x_{35} = 56.5486677646163$$
$$x_{36} = -28.2743338823081$$
$$x_{37} = 50.2654824574367$$
$$x_{38} = -65.9734457253857$$
$$x_{39} = 37.6991118430775$$
Puntos máximos de la función:
$$x_{39} = 95.8197196421744$$
$$x_{39} = 80.1120364968984$$
$$x_{39} = 70.6874957987652$$
$$x_{39} = 67.5459991590417$$
$$x_{39} = 42.4146464836393$$
$$x_{39} = 7.88468215226503$$
$$x_{39} = 92.6781744605177$$
$$x_{39} = 58.1215816459086$$
$$x_{39} = 36.1321731425561$$
$$x_{39} = 26.7092361429331$$
$$x_{39} = 29.8500622839131$$
$$x_{39} = 23.5686584612553$$
$$x_{39} = 64.4045132832651$$
$$x_{39} = 51.8387217793455$$
$$x_{39} = 48.697328558041$$
$$x_{39} = 73.829001694965$$
$$x_{39} = 20.4284648400094$$
$$x_{39} = 14.1505011586627$$
$$x_{39} = 45.5559674357001$$
$$x_{39} = 1.94119311490964$$
$$x_{39} = 89.5366330613754$$
$$x_{39} = 86.3950958997003$$
Decrece en los intervalos
$$\left[270.176968208722, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -100.530964914873\right]$$