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$$\frac{x \left(\sin{\left(x \right)} - \cos{\left(x \right)}\right) \cos^{2}{\left(x \right)}}{\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)^{2}} + \frac{- 2 x \sin{\left(x \right)} \cos{\left(x \right)} + \cos^{2}{\left(x \right)}}{\sin{\left(x \right)} + \cos{\left(x \right)}} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = 17.2787595947439$$
$$x_{2} = -23.5619449019235$$
$$x_{3} = -14.1371669411541$$
$$x_{4} = 51.8362787842316$$
$$x_{5} = -17.2787595947439$$
$$x_{6} = -10.9955742875643$$
$$x_{7} = -36.1283155162826$$
$$x_{8} = -95.8185759344887$$
$$x_{9} = -26.7035375555132$$
$$x_{10} = 26.7035375555132$$
$$x_{11} = 89.5353906273091$$
$$x_{12} = 23.5619449019235$$
$$x_{13} = 14.1371669411541$$
$$x_{14} = 42.4115008234622$$
$$x_{15} = 95.8185759344887$$
$$x_{16} = -61.261056745001$$
$$x_{17} = 58.1194640914112$$
$$x_{18} = 36.1283155162826$$
$$x_{19} = 102.101761241668$$
$$x_{20} = 29.845130209103$$
$$x_{21} = -73.8274273593601$$
$$x_{22} = 48.6946861306418$$
$$x_{23} = -4.71238898038469$$
$$x_{24} = 70.6858347057703$$
$$x_{25} = -7.85398163397448$$
$$x_{26} = -51.8362787842316$$
$$x_{27} = -76.9690200129499$$
$$x_{28} = -89.5353906273091$$
$$x_{29} = -39.2699081698724$$
$$x_{30} = 80.1106126665397$$
$$x_{31} = -42.4115008234622$$
$$x_{32} = 45.553093477052$$
$$x_{33} = 20.4203522483337$$
$$x_{34} = 64.4026493985908$$
$$x_{35} = -32.9867228626928$$
$$x_{36} = 67.5442420521806$$
$$x_{37} = -20.4203522483337$$
$$x_{38} = -80.1106126665397$$
$$x_{39} = 7.85398163397448$$
$$x_{40} = -45.553093477052$$
$$x_{41} = 76.9690200129499$$
$$x_{42} = -1.5707963267949$$
$$x_{43} = 39.2699081698724$$
$$x_{44} = -67.5442420521806$$
$$x_{45} = -105.243353895258$$
$$x_{46} = -83.2522053201295$$
$$x_{47} = -29.845130209103$$
$$x_{48} = -86.3937979737193$$
$$x_{49} = 98.9601685880785$$
$$x_{50} = 73.8274273593601$$
$$x_{51} = -58.1194640914112$$
$$x_{52} = 92.6769832808989$$
$$x_{53} = 54.9778714378214$$
$$x_{54} = 86.3937979737193$$
$$x_{55} = 0.623960630317383$$
$$x_{56} = 1.5707963267949$$
$$x_{57} = -54.9778714378214$$
$$x_{58} = -64.4026493985908$$
Signos de extremos en los puntos:
(17.278759594743864, -2.10139136502907e-29)
(-23.56194490192345, -1.73402701495235e-29)
(-14.137166941154069, 4.29347676987173e-30)
(51.83627878423159, 3.09398107171563e-30)
(-17.278759594743864, -2.10139136502906e-29)
(-10.995574287564276, -2.02011321271057e-30)
(-36.12831551628262, -3.66424875021483e-28)
(-95.81857593448869, 3.67652016504401e-28)
(-26.703537555513243, 1.44419018202913e-29)
(26.703537555513243, 1.44419018202913e-29)
(89.53539062730911, 2.60267852044684e-27)
(23.56194490192345, -1.73402701495236e-29)
(14.137166941154069, 4.29347676987172e-30)
(42.411500823462205, -4.98859428281589e-28)
(95.81857593448869, 3.676520165044e-28)
(-61.26105674500097, -5.29879683037423e-28)
(58.119464091411174, 1.39112146798308e-29)
(36.12831551628262, -3.6642487502148e-28)
(102.10176124166829, 2.45314668072883e-27)
(29.845130209103036, -1.12127665170555e-29)
(-73.82742735936014, -4.43565443427592e-28)
(48.6946861306418, -5.73178094238609e-28)
(-4.71238898038469, -1.59017658143397e-31)
(70.68583470577035, 6.77618297499812e-29)
(-7.853981633974483, 7.36192861774987e-31)
(-51.83627878423159, 3.09398107171563e-30)
(-76.96902001294994, 2.66242463704609e-27)
(-89.53539062730911, 2.60267852044681e-27)
(-39.269908169872416, 2.36773935254175e-30)
(80.11061266653972, -1.9228326430437e-27)
(-42.411500823462205, -4.98859428281592e-28)
(45.553093477052, 1.74530768724744e-35)
(20.420352248333657, 1.96251734458305e-29)
(64.40264939859077, 2.61429235475569e-27)
(-32.98672286269283, 7.93556229873748e-30)
(67.54424205218055, -1.3132184568469e-27)
(-20.420352248333657, 1.96251734458305e-29)
(-80.11061266653972, -1.92283264304372e-27)
(7.853981633974483, 7.36192861774987e-31)
(-45.553093477052, 1.74530768724744e-35)
(76.96902001294994, 2.66242463704612e-27)
(-1.5707963267948966, 5.8895428941999e-33)
(39.269908169872416, 2.36773935254175e-30)
(-67.54424205218055, -1.31321845684691e-27)
(-105.24335389525807, -3.63739373812053e-27)
(-83.25220532012952, 1.79644538434511e-28)
(-29.845130209103036, -1.12127665170554e-29)
(-86.39379797371932, -3.32328051180094e-28)
(98.96016858807849, -2.14260182065912e-28)
(73.82742735936014, -4.43565443427594e-28)
(-58.119464091411174, 1.39112146798308e-29)
(92.6769832808989, -2.6915268448779e-27)
(54.977871437821385, -5.58036659424785e-28)
(86.39379797371932, -3.32328051180096e-28)
(0.6239606303173832, 0.294427989456615)
(1.5707963267948966, 5.8895428941999e-33)
(-54.977871437821385, -5.58036659424781e-28)
(-64.40264939859077, 2.61429235475566e-27)
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:
$$x_{1} = -14.1371669411541$$
$$x_{2} = 51.8362787842316$$
$$x_{3} = -95.8185759344887$$
$$x_{4} = -26.7035375555132$$
$$x_{5} = 26.7035375555132$$
$$x_{6} = 89.5353906273091$$
$$x_{7} = 14.1371669411541$$
$$x_{8} = 95.8185759344887$$
$$x_{9} = 58.1194640914112$$
$$x_{10} = 102.101761241668$$
$$x_{11} = 70.6858347057703$$
$$x_{12} = -7.85398163397448$$
$$x_{13} = -51.8362787842316$$
$$x_{14} = -76.9690200129499$$
$$x_{15} = -89.5353906273091$$
$$x_{16} = -39.2699081698724$$
$$x_{17} = 45.553093477052$$
$$x_{18} = 20.4203522483337$$
$$x_{19} = 64.4026493985908$$
$$x_{20} = -32.9867228626928$$
$$x_{21} = -20.4203522483337$$
$$x_{22} = 7.85398163397448$$
$$x_{23} = -45.553093477052$$
$$x_{24} = 76.9690200129499$$
$$x_{25} = -1.5707963267949$$
$$x_{26} = 39.2699081698724$$
$$x_{27} = -83.2522053201295$$
$$x_{28} = -58.1194640914112$$
$$x_{29} = 1.5707963267949$$
$$x_{30} = -64.4026493985908$$
Puntos máximos de la función:
$$x_{30} = 17.2787595947439$$
$$x_{30} = -23.5619449019235$$
$$x_{30} = -17.2787595947439$$
$$x_{30} = -10.9955742875643$$
$$x_{30} = -36.1283155162826$$
$$x_{30} = 23.5619449019235$$
$$x_{30} = 42.4115008234622$$
$$x_{30} = -61.261056745001$$
$$x_{30} = 36.1283155162826$$
$$x_{30} = 29.845130209103$$
$$x_{30} = -73.8274273593601$$
$$x_{30} = 48.6946861306418$$
$$x_{30} = -4.71238898038469$$
$$x_{30} = 80.1106126665397$$
$$x_{30} = -42.4115008234622$$
$$x_{30} = 67.5442420521806$$
$$x_{30} = -80.1106126665397$$
$$x_{30} = -67.5442420521806$$
$$x_{30} = -105.243353895258$$
$$x_{30} = -29.845130209103$$
$$x_{30} = -86.3937979737193$$
$$x_{30} = 98.9601685880785$$
$$x_{30} = 73.8274273593601$$
$$x_{30} = 92.6769832808989$$
$$x_{30} = 54.9778714378214$$
$$x_{30} = 86.3937979737193$$
$$x_{30} = 0.623960630317383$$
$$x_{30} = -54.9778714378214$$
Decrece en los intervalos
$$\left[102.101761241668, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -95.8185759344887\right]$$