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$$- \sin{\left(\left|{x}\right| \right)} \operatorname{sign}{\left(x \right)} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = 12.5663706143592$$
$$x_{2} = 53.4070751110265$$
$$x_{3} = -97.3893722612836$$
$$x_{4} = 37.6991118430775$$
$$x_{5} = 97.3893722612836$$
$$x_{6} = 78.5398163397448$$
$$x_{7} = -59.6902604182061$$
$$x_{8} = -65.9734457253857$$
$$x_{9} = 0$$
$$x_{10} = -31.4159265358979$$
$$x_{11} = -113.097335529233$$
$$x_{12} = -50.2654824574367$$
$$x_{13} = -21.9911485751286$$
$$x_{14} = 6.28318530717959$$
$$x_{15} = -34.5575191894877$$
$$x_{16} = -94.2477796076938$$
$$x_{17} = -69.1150383789755$$
$$x_{18} = -15.707963267949$$
$$x_{19} = 21.9911485751286$$
$$x_{20} = 69.1150383789755$$
$$x_{21} = 62.8318530717959$$
$$x_{22} = 50.2654824574367$$
$$x_{23} = 81.6814089933346$$
$$x_{24} = 100.530964914873$$
$$x_{25} = -40.8407044966673$$
$$x_{26} = 9.42477796076938$$
$$x_{27} = -87.9645943005142$$
$$x_{28} = 34.5575191894877$$
$$x_{29} = 65.9734457253857$$
$$x_{30} = -62.8318530717959$$
$$x_{31} = -18.8495559215388$$
$$x_{32} = -28.2743338823081$$
$$x_{33} = -267.035375555132$$
$$x_{34} = -232.477856365645$$
$$x_{35} = -56.5486677646163$$
$$x_{36} = -53.4070751110265$$
$$x_{37} = -37.6991118430775$$
$$x_{38} = -25.1327412287183$$
$$x_{39} = -100.530964914873$$
$$x_{40} = -9.42477796076938$$
$$x_{41} = 40.8407044966673$$
$$x_{42} = -91.106186954104$$
$$x_{43} = -2642.07942166902$$
$$x_{44} = -75.398223686155$$
$$x_{45} = 18.8495559215388$$
$$x_{46} = 87.9645943005142$$
$$x_{47} = 59.6902604182061$$
$$x_{48} = -6.28318530717959$$
$$x_{49} = 25.1327412287183$$
$$x_{50} = 47.1238898038469$$
$$x_{51} = 91.106186954104$$
$$x_{52} = 28.2743338823081$$
$$x_{53} = 56.5486677646163$$
$$x_{54} = -43.9822971502571$$
$$x_{55} = -47.1238898038469$$
$$x_{56} = -3.14159265358979$$
$$x_{57} = 31.4159265358979$$
$$x_{58} = 94.2477796076938$$
$$x_{59} = -12.5663706143592$$
$$x_{60} = 75.398223686155$$
$$x_{61} = -72.2566310325652$$
$$x_{62} = -84.8230016469244$$
$$x_{63} = 84.8230016469244$$
$$x_{64} = 72.2566310325652$$
$$x_{65} = -81.6814089933346$$
$$x_{66} = 43.9822971502571$$
$$x_{67} = -78.5398163397448$$
$$x_{68} = 15.707963267949$$
$$x_{69} = 3.14159265358979$$
Signos de extremos en los puntos:
(12.566370614359172, 1)
(53.40707511102649, -1)
(-97.3893722612836, -1)
(37.69911184307752, 1)
(97.3893722612836, -1)
(78.53981633974483, -1)
(-59.69026041820607, -1)
(-65.97344572538566, -1)
(0, 1)
(-31.41592653589793, 1)
(-113.09733552923255, 1)
(-50.26548245743669, 1)
(-21.991148575128552, -1)
(6.283185307179586, 1)
(-34.55751918948773, -1)
(-94.2477796076938, 1)
(-69.11503837897546, 1)
(-15.707963267948966, -1)
(21.991148575128552, -1)
(69.11503837897546, 1)
(62.83185307179586, 1)
(50.26548245743669, 1)
(81.68140899333463, 1)
(100.53096491487338, 1)
(-40.840704496667314, -1)
(9.42477796076938, -1)
(-87.96459430051421, 1)
(34.55751918948773, -1)
(65.97344572538566, -1)
(-62.83185307179586, 1)
(-18.84955592153876, 1)
(-28.274333882308138, -1)
(-267.0353755551324, -1)
(-232.4778563656447, 1)
(-56.548667764616276, 1)
(-53.40707511102649, -1)
(-37.69911184307752, 1)
(-25.132741228718345, 1)
(-100.53096491487338, 1)
(-9.42477796076938, -1)
(40.840704496667314, -1)
(-91.106186954104, -1)
(-2642.079421669016, -1)
(-75.39822368615503, 1)
(18.84955592153876, 1)
(87.96459430051421, 1)
(59.69026041820607, -1)
(-6.283185307179586, 1)
(25.132741228718345, 1)
(47.1238898038469, -1)
(91.106186954104, -1)
(28.274333882308138, -1)
(56.548667764616276, 1)
(-43.982297150257104, 1)
(-47.1238898038469, -1)
(-3.141592653589793, -1)
(31.41592653589793, 1)
(94.2477796076938, 1)
(-12.566370614359172, 1)
(75.39822368615503, 1)
(-72.25663103256524, -1)
(-84.82300164692441, -1)
(84.82300164692441, -1)
(72.25663103256524, -1)
(-81.68140899333463, 1)
(43.982297150257104, 1)
(-78.53981633974483, -1)
(15.707963267948966, -1)
(3.141592653589793, -1)
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:
$$x_{1} = 53.4070751110265$$
$$x_{2} = -97.3893722612836$$
$$x_{3} = 97.3893722612836$$
$$x_{4} = 78.5398163397448$$
$$x_{5} = -59.6902604182061$$
$$x_{6} = -65.9734457253857$$
$$x_{7} = -21.9911485751286$$
$$x_{8} = -34.5575191894877$$
$$x_{9} = -15.707963267949$$
$$x_{10} = 21.9911485751286$$
$$x_{11} = -40.8407044966673$$
$$x_{12} = 9.42477796076938$$
$$x_{13} = 34.5575191894877$$
$$x_{14} = 65.9734457253857$$
$$x_{15} = -28.2743338823081$$
$$x_{16} = -267.035375555132$$
$$x_{17} = -53.4070751110265$$
$$x_{18} = -9.42477796076938$$
$$x_{19} = 40.8407044966673$$
$$x_{20} = -91.106186954104$$
$$x_{21} = -2642.07942166902$$
$$x_{22} = 59.6902604182061$$
$$x_{23} = 47.1238898038469$$
$$x_{24} = 91.106186954104$$
$$x_{25} = 28.2743338823081$$
$$x_{26} = -47.1238898038469$$
$$x_{27} = -3.14159265358979$$
$$x_{28} = -72.2566310325652$$
$$x_{29} = -84.8230016469244$$
$$x_{30} = 84.8230016469244$$
$$x_{31} = 72.2566310325652$$
$$x_{32} = -78.5398163397448$$
$$x_{33} = 15.707963267949$$
$$x_{34} = 3.14159265358979$$
Puntos máximos de la función:
$$x_{34} = 12.5663706143592$$
$$x_{34} = 37.6991118430775$$
$$x_{34} = 0$$
$$x_{34} = -31.4159265358979$$
$$x_{34} = -113.097335529233$$
$$x_{34} = -50.2654824574367$$
$$x_{34} = 6.28318530717959$$
$$x_{34} = -94.2477796076938$$
$$x_{34} = -69.1150383789755$$
$$x_{34} = 69.1150383789755$$
$$x_{34} = 62.8318530717959$$
$$x_{34} = 50.2654824574367$$
$$x_{34} = 81.6814089933346$$
$$x_{34} = 100.530964914873$$
$$x_{34} = -87.9645943005142$$
$$x_{34} = -62.8318530717959$$
$$x_{34} = -18.8495559215388$$
$$x_{34} = -232.477856365645$$
$$x_{34} = -56.5486677646163$$
$$x_{34} = -37.6991118430775$$
$$x_{34} = -25.1327412287183$$
$$x_{34} = -100.530964914873$$
$$x_{34} = -75.398223686155$$
$$x_{34} = 18.8495559215388$$
$$x_{34} = 87.9645943005142$$
$$x_{34} = -6.28318530717959$$
$$x_{34} = 25.1327412287183$$
$$x_{34} = 56.5486677646163$$
$$x_{34} = -43.9822971502571$$
$$x_{34} = 31.4159265358979$$
$$x_{34} = 94.2477796076938$$
$$x_{34} = -12.5663706143592$$
$$x_{34} = 75.398223686155$$
$$x_{34} = -81.6814089933346$$
$$x_{34} = 43.9822971502571$$
Decrece en los intervalos
$$\left[97.3893722612836, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -2642.07942166902\right]$$