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$$\sin{\left(x \right)} \operatorname{sign}{\left(\cos{\left(x \right)} \right)} - \sin{\left(x \right)} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = 18$$
$$x_{2} = 50$$
$$x_{3} = -72.2566310325652$$
$$x_{4} = 91.106186954104$$
$$x_{5} = -59.6902604182061$$
$$x_{6} = 70$$
$$x_{7} = 30$$
$$x_{8} = 24$$
$$x_{9} = 94$$
$$x_{10} = -100$$
$$x_{11} = -56$$
$$x_{12} = 40.8407044966673$$
$$x_{13} = -6$$
$$x_{14} = 38$$
$$x_{15} = 72.2566310325652$$
$$x_{16} = -47.1238898038469$$
$$x_{17} = 6$$
$$x_{18} = 82$$
$$x_{19} = -88$$
$$x_{20} = 59.6902604182061$$
$$x_{21} = -44$$
$$x_{22} = 84.8230016469244$$
$$x_{23} = 0$$
$$x_{24} = -74$$
$$x_{25} = -14$$
$$x_{26} = 38.7483330486114$$
$$x_{27} = -53.4070751110265$$
$$x_{28} = -62$$
$$x_{29} = -21.9911485751286$$
$$x_{30} = 99.2691857797929$$
$$x_{31} = 56$$
$$x_{32} = -24$$
$$x_{33} = 64$$
$$x_{34} = -34.5575191894877$$
$$x_{35} = 12$$
$$x_{36} = -50$$
$$x_{37} = -3.14159265358979$$
$$x_{38} = 53.4070751110265$$
$$x_{39} = -70$$
$$x_{40} = -75.7312933201993$$
$$x_{41} = -68$$
$$x_{42} = 74$$
$$x_{43} = -92.6907188312909$$
$$x_{44} = -65.9734457253857$$
$$x_{45} = -76$$
$$x_{46} = -82$$
$$x_{47} = 82.5136949447994$$
$$x_{48} = 76$$
$$x_{49} = -95.75$$
$$x_{50} = -26$$
$$x_{51} = -38$$
$$x_{52} = -58$$
$$x_{53} = 3.14159265358979$$
$$x_{54} = -7.75$$
$$x_{55} = -78.5398163397448$$
$$x_{56} = 26$$
$$x_{57} = 47.1238898038469$$
$$x_{58} = -18$$
$$x_{59} = 28.2743338823081$$
$$x_{60} = 100$$
$$x_{61} = 88$$
$$x_{62} = -20$$
$$x_{63} = -91.106186954104$$
$$x_{64} = -94$$
$$x_{65} = -12$$
$$x_{66} = -51.75$$
$$x_{67} = -48.8578078716517$$
$$x_{68} = 80.25$$
$$x_{69} = -30$$
$$x_{70} = -9.42477796076938$$
$$x_{71} = 65.9734457253857$$
$$x_{72} = 20$$
$$x_{73} = 14$$
$$x_{74} = 36.25$$
$$x_{75} = 32$$
$$x_{76} = -15.707963267949$$
$$x_{77} = 97.3893722612836$$
$$x_{78} = -32$$
$$x_{79} = 44$$
$$x_{80} = 62$$
$$x_{81} = 15.707963267949$$
$$x_{82} = 9.42477796076938$$
$$x_{83} = -84.8230016469244$$
$$x_{84} = -64$$
$$x_{85} = 68$$
$$x_{86} = 21.9911485751286$$
$$x_{87} = 34.5575191894877$$
$$x_{88} = -97.3893722612836$$
$$x_{89} = -32.0322958555426$$
$$x_{90} = -5.04363182667898$$
$$x_{91} = -40.8407044966673$$
$$x_{92} = -28.2743338823081$$
$$x_{93} = 58$$
$$x_{94} = 78.5398163397448$$
$$x_{95} = 11.7052375414856$$
$$x_{96} = 55.4742156532357$$
Signos de extremos en los puntos:
(18, 0)
(50, 0)
(-72.25663103256524, -2)
(91.106186954104, -2)
(-59.69026041820607, -2)
(70, 0)
(30, 0)
(24, 0)
(94, 0)
(-100, 0)
(-56, 0)
(40.840704496667314, -2)
(-6, 0)
(38, 0)
(72.25663103256524, -2)
(-47.1238898038469, -2)
(6, 0)
(82, 0)
(-88, 0)
(59.69026041820607, -2)
(-44, 0)
(84.82300164692441, -2)
(0, 0)
(-74, 0)
(-14, 0)
(38.748333048611386, 0)
(-53.40707511102649, -2)
(-62, 0)
(-21.991148575128552, -2)
(99.2691857797929, 0)
(56, 0)
(-24, 0)
(64, 0)
(-34.55751918948773, -2)
(12, 0)
(-50, 0)
(-3.141592653589793, -2)
(53.40707511102649, -2)
(-70, 0)
(-75.73129332019926, 0)
(-68, 0)
(74, 0)
(-92.69071883129092, 0)
(-65.97344572538566, -2)
(-76, 0)
(-82, 0)
(82.51369494479937, 0)
(76, 0)
(-95.75, 0)
(-26, 0)
(-38, 0)
(-58, 0)
(3.141592653589793, -2)
(-7.75, 0)
(-78.53981633974483, -2)
(26, 0)
(47.1238898038469, -2)
(-18, 0)
(28.274333882308138, -2)
(100, 0)
(88, 0)
(-20, 0)
(-91.106186954104, -2)
(-94, 0)
(-12, 0)
(-51.75, 0)
(-48.8578078716517, 0)
(80.25, 0)
(-30, 0)
(-9.42477796076938, -2)
(65.97344572538566, -2)
(20, 0)
(14, 0)
(36.25, 0)
(32, 0)
(-15.707963267948966, -2)
(97.3893722612836, -2)
(-32, 0)
(44, 0)
(62, 0)
(15.707963267948966, -2)
(9.42477796076938, -2)
(-84.82300164692441, -2)
(-64, 0)
(68, 0)
(21.991148575128552, -2)
(34.55751918948773, -2)
(-97.3893722612836, -2)
(-32.03229585554262, 0)
(-5.043631826678977, 0)
(-40.840704496667314, -2)
(-28.274333882308138, -2)
(58, 0)
(78.53981633974483, -2)
(11.705237541485587, 0)
(55.47421565323565, 0)
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:
$$x_{1} = -72.2566310325652$$
$$x_{2} = 91.106186954104$$
$$x_{3} = -59.6902604182061$$
$$x_{4} = 40.8407044966673$$
$$x_{5} = 72.2566310325652$$
$$x_{6} = -47.1238898038469$$
$$x_{7} = 59.6902604182061$$
$$x_{8} = 84.8230016469244$$
$$x_{9} = -53.4070751110265$$
$$x_{10} = -21.9911485751286$$
$$x_{11} = -34.5575191894877$$
$$x_{12} = -3.14159265358979$$
$$x_{13} = 53.4070751110265$$
$$x_{14} = -65.9734457253857$$
$$x_{15} = 3.14159265358979$$
$$x_{16} = -78.5398163397448$$
$$x_{17} = 47.1238898038469$$
$$x_{18} = 28.2743338823081$$
$$x_{19} = -91.106186954104$$
$$x_{20} = -9.42477796076938$$
$$x_{21} = 65.9734457253857$$
$$x_{22} = -15.707963267949$$
$$x_{23} = 97.3893722612836$$
$$x_{24} = 15.707963267949$$
$$x_{25} = 9.42477796076938$$
$$x_{26} = -84.8230016469244$$
$$x_{27} = 21.9911485751286$$
$$x_{28} = 34.5575191894877$$
$$x_{29} = -97.3893722612836$$
$$x_{30} = -40.8407044966673$$
$$x_{31} = -28.2743338823081$$
$$x_{32} = 78.5398163397448$$
La función no tiene puntos máximos
Decrece en los intervalos
$$\left[97.3893722612836, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -97.3893722612836\right]$$