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$$- \log{\left(x \right)} \sin{\left(x \right)} + \frac{\cos{\left(x \right)}}{x} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = 62.8356966541501$$
$$x_{2} = 75.4012916642681$$
$$x_{3} = 28.2849113047725$$
$$x_{4} = 47.1293968198114$$
$$x_{5} = 59.6943570030875$$
$$x_{6} = 31.4251563350128$$
$$x_{7} = 50.27056033759$$
$$x_{8} = 9.47170218677955$$
$$x_{9} = 100.533122377741$$
$$x_{10} = 43.9883049460921$$
$$x_{11} = 84.8256564376189$$
$$x_{12} = 15.7310277752208$$
$$x_{13} = 37.7064180281721$$
$$x_{14} = 97.3916147574604$$
$$x_{15} = 12.5976921976804$$
$$x_{16} = 1.27285069827148$$
$$x_{17} = 91.1086195251935$$
$$x_{18} = 81.6841895128946$$
$$x_{19} = 40.8473034495909$$
$$x_{20} = 72.2598642156451$$
$$x_{21} = 65.9770636783598$$
$$x_{22} = 3.37991614208723$$
$$x_{23} = 34.5656848442796$$
$$x_{24} = 22.0058475927713$$
$$x_{25} = 18.8675971617309$$
$$x_{26} = 69.1184539759405$$
$$x_{27} = 25.1450734377105$$
$$x_{28} = 6.36781151369107$$
$$x_{29} = 53.4117815402062$$
$$x_{30} = 94.2501135627054$$
$$x_{31} = 78.5427340593526$$
$$x_{32} = 56.5530498251275$$
$$x_{33} = 87.967133489911$$
Signos de extremos en los puntos:
(62.835696654150134, 4.14049274584816)
(75.40129166426813, 4.32280406143887)
(28.284911304772503, -3.34214152179011)
(47.1293968198114, -3.85283851894378)
(59.69435700308747, -4.08920318061012)
(31.425156335012787, 3.44746188086714)
(50.27056033759003, 3.91736911923955)
(9.471702186779549, -2.24583383410247)
(100.53312237774094, 4.61047651900993)
(43.98830494609213, 3.78385551437724)
(84.82565643761893, -4.44058240090197)
(15.731027775220827, -2.7549021263166)
(37.70641802817207, 3.62973343908044)
(97.39161475746043, -4.57872860340226)
(12.597692197680386, 2.53227099874907)
(1.2728506982714773, 0.0708232692475832)
(91.10861952519349, -4.5120390660658)
(81.68418951289463, 4.40284344444233)
(40.847303449590925, -3.70976003369716)
(72.25986421564514, -4.28024647479203)
(65.9770636783598, -4.18927974348899)
(3.3799161420872266, -1.18342849059061)
(34.56568484427963, -3.54274330777479)
(22.00584759277127, -3.09097426796676)
(18.86759716173087, 2.93696797853021)
(69.11845397594048, 4.23579704883419)
(25.14507343771052, 3.22441678455125)
(6.367811513691074, 1.8446308321891)
(53.41178154020617, -3.9779872921632)
(94.25011356270541, 4.54593964955556)
(78.5427340593526, -4.3636242855634)
(56.553049825127495, 4.03514038997721)
(87.96713348991098, 4.4769488290828)
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:
$$x_{1} = 28.2849113047725$$
$$x_{2} = 47.1293968198114$$
$$x_{3} = 59.6943570030875$$
$$x_{4} = 9.47170218677955$$
$$x_{5} = 84.8256564376189$$
$$x_{6} = 15.7310277752208$$
$$x_{7} = 97.3916147574604$$
$$x_{8} = 91.1086195251935$$
$$x_{9} = 40.8473034495909$$
$$x_{10} = 72.2598642156451$$
$$x_{11} = 65.9770636783598$$
$$x_{12} = 3.37991614208723$$
$$x_{13} = 34.5656848442796$$
$$x_{14} = 22.0058475927713$$
$$x_{15} = 53.4117815402062$$
$$x_{16} = 78.5427340593526$$
Puntos máximos de la función:
$$x_{16} = 62.8356966541501$$
$$x_{16} = 75.4012916642681$$
$$x_{16} = 31.4251563350128$$
$$x_{16} = 50.27056033759$$
$$x_{16} = 100.533122377741$$
$$x_{16} = 43.9883049460921$$
$$x_{16} = 37.7064180281721$$
$$x_{16} = 12.5976921976804$$
$$x_{16} = 1.27285069827148$$
$$x_{16} = 81.6841895128946$$
$$x_{16} = 18.8675971617309$$
$$x_{16} = 69.1184539759405$$
$$x_{16} = 25.1450734377105$$
$$x_{16} = 6.36781151369107$$
$$x_{16} = 94.2501135627054$$
$$x_{16} = 56.5530498251275$$
$$x_{16} = 87.967133489911$$
Decrece en los intervalos
$$\left[97.3916147574604, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, 3.37991614208723\right]$$