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