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$$3^{x} \log{\left(3 \right)} - 2 \cos{\left(x \right)} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = 0.449911460262718$$
$$x_{2} = -39.2699081698724$$
$$x_{3} = -70.6858347057703$$
$$x_{4} = -20.4203522482344$$
$$x_{5} = -61.261056745001$$
$$x_{6} = -36.1283155162826$$
$$x_{7} = -48.6946861306418$$
$$x_{8} = -80.1106126665397$$
$$x_{9} = -10.9955774035164$$
$$x_{10} = -98.9601685880785$$
$$x_{11} = -64.4026493985908$$
$$x_{12} = -4.71547899046$$
$$x_{13} = -1.46012220760261$$
$$x_{14} = -89.5353906273091$$
$$x_{15} = -86.3937979737193$$
$$x_{16} = -54.9778714378214$$
$$x_{17} = -29.845130209103$$
$$x_{18} = -92.6769832808989$$
$$x_{19} = -67.5442420521806$$
$$x_{20} = -58.1194640914112$$
$$x_{21} = -51.8362787842316$$
$$x_{22} = -83.2522053201295$$
$$x_{23} = -76.9690200129499$$
$$x_{24} = -73.8274273593601$$
$$x_{25} = -17.2787595978754$$
$$x_{26} = -7.85388333255536$$
$$x_{27} = -14.1371668423735$$
$$x_{28} = -23.5619449019266$$
$$x_{29} = -95.8185759344887$$
$$x_{30} = -45.553093477052$$
$$x_{31} = -32.9867228626928$$
$$x_{32} = -26.7035375555131$$
$$x_{33} = -42.4115008234622$$
Signos de extremos en los puntos:
(0.44991146026271794, 0.769543037371475)
(-39.269908169872416, 2)
(-70.68583470577035, 2)
(-20.420352248234384, 2.00000000018072)
(-61.26105674500097, -2)
(-36.12831551628262, -2)
(-48.6946861306418, -2)
(-80.11061266653972, -2)
(-10.995577403516354, -1.99999432746667)
(-98.96016858807849, -2)
(-64.40264939859077, 2)
(-4.715478990460005, -1.99436516403475)
(-1.4601222076026097, 2.1888325249929)
(-89.53539062730911, 2)
(-86.39379797371932, -2)
(-54.977871437821385, -2)
(-29.84513020910304, -1.99999999999999)
(-92.6769832808989, -2)
(-67.54424205218055, -2)
(-58.119464091411174, 2)
(-51.83627878423159, 2)
(-83.25220532012952, 2)
(-76.96902001294994, 2)
(-73.82742735936014, -2)
(-17.278759597875354, -1.99999999429919)
(-7.853883332555358, 2.00017894595198)
(-14.137166842373466, 2.00000017982795)
(-23.561944901926598, -1.99999999999427)
(-95.81857593448869, 2)
(-45.553093477052, 2)
(-32.98672286269283, 2)
(-26.703537555513144, 2.00000000000018)
(-42.411500823462205, -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:
$$x_{1} = 0.449911460262718$$
$$x_{2} = -61.261056745001$$
$$x_{3} = -36.1283155162826$$
$$x_{4} = -48.6946861306418$$
$$x_{5} = -80.1106126665397$$
$$x_{6} = -10.9955774035164$$
$$x_{7} = -98.9601685880785$$
$$x_{8} = -4.71547899046$$
$$x_{9} = -86.3937979737193$$
$$x_{10} = -54.9778714378214$$
$$x_{11} = -29.845130209103$$
$$x_{12} = -92.6769832808989$$
$$x_{13} = -67.5442420521806$$
$$x_{14} = -73.8274273593601$$
$$x_{15} = -17.2787595978754$$
$$x_{16} = -23.5619449019266$$
$$x_{17} = -42.4115008234622$$
Puntos máximos de la función:
$$x_{17} = -39.2699081698724$$
$$x_{17} = -70.6858347057703$$
$$x_{17} = -20.4203522482344$$
$$x_{17} = -64.4026493985908$$
$$x_{17} = -1.46012220760261$$
$$x_{17} = -89.5353906273091$$
$$x_{17} = -58.1194640914112$$
$$x_{17} = -51.8362787842316$$
$$x_{17} = -83.2522053201295$$
$$x_{17} = -76.9690200129499$$
$$x_{17} = -7.85388333255536$$
$$x_{17} = -14.1371668423735$$
$$x_{17} = -95.8185759344887$$
$$x_{17} = -45.553093477052$$
$$x_{17} = -32.9867228626928$$
$$x_{17} = -26.7035375555131$$
Decrece en los intervalos
$$\left[0.449911460262718, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -98.9601685880785\right]$$