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$$x \left(\tan^{2}{\left(\left|{x}\right| \right)} + 1\right) \sin{\left(x \right)} \operatorname{sign}{\left(x \right)} + \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \tan{\left(\left|{x}\right| \right)} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = 65.9734457253857$$
$$x_{2} = -15.707963267949$$
$$x_{3} = 37.6991118430775$$
$$x_{4} = -69.1150383789755$$
$$x_{5} = -34.5575191894877$$
$$x_{6} = -91.106186954104$$
$$x_{7} = 0$$
$$x_{8} = 6.28318530717959$$
$$x_{9} = -47.1238898038469$$
$$x_{10} = -78.5398163397448$$
$$x_{11} = -28.2743338823081$$
$$x_{12} = 97.3893722612836$$
$$x_{13} = 3.14159265358979$$
$$x_{14} = 40.8407044966673$$
$$x_{15} = 62.8318530717959$$
$$x_{16} = -81.6814089933346$$
$$x_{17} = 43.9822971502571$$
$$x_{18} = -84.8230016469244$$
$$x_{19} = 100.530964914873$$
$$x_{20} = 69.1150383789755$$
$$x_{21} = -94.2477796076938$$
$$x_{22} = 91.106186954104$$
$$x_{23} = 78.5398163397448$$
$$x_{24} = 47.1238898038469$$
$$x_{25} = 81.6814089933346$$
$$x_{26} = -72.2566310325652$$
$$x_{27} = -6.28318530717959$$
$$x_{28} = 28.2743338823081$$
$$x_{29} = -100.530964914873$$
$$x_{30} = -65.9734457253857$$
$$x_{31} = 94.2477796076938$$
$$x_{32} = 31.4159265358979$$
$$x_{33} = 50.2654824574367$$
$$x_{34} = -21.9911485751286$$
$$x_{35} = 12.5663706143592$$
$$x_{36} = 15.707963267949$$
$$x_{37} = -75.398223686155$$
$$x_{38} = 72.2566310325652$$
$$x_{39} = 18.8495559215388$$
$$x_{40} = -37.6991118430775$$
$$x_{41} = -50.2654824574367$$
$$x_{42} = -3.14159265358979$$
$$x_{43} = -87.9645943005142$$
$$x_{44} = -25.1327412287183$$
$$x_{45} = -40.8407044966673$$
$$x_{46} = 53.4070751110265$$
$$x_{47} = 9.42477796076938$$
$$x_{48} = -43.9822971502571$$
$$x_{49} = -56.5486677646163$$
$$x_{50} = -97.3893722612836$$
$$x_{51} = -59.6902604182061$$
$$x_{52} = -12.5663706143592$$
$$x_{53} = -18.8495559215388$$
$$x_{54} = 84.8230016469244$$
$$x_{55} = 25.1327412287183$$
$$x_{56} = 21.9911485751286$$
$$x_{57} = 59.6902604182061$$
$$x_{58} = -53.4070751110265$$
$$x_{59} = -31.4159265358979$$
$$x_{60} = -9.42477796076938$$
$$x_{61} = -62.8318530717959$$
$$x_{62} = 56.5486677646163$$
$$x_{63} = 75.398223686155$$
$$x_{64} = 34.5575191894877$$
$$x_{65} = 87.9645943005142$$
Signos de extremos en los puntos:
(65.97344572538566, -6.34844983898999e-29)
(-15.707963267948966, -5.8895428941999e-30)
(37.69911184307752, 8.14170409694193e-29)
(-69.11503837897546, 1.3448904736186e-27)
(-34.55751918948773, -1.68111309202325e-28)
(-91.106186954104, -1.39624614979795e-34)
(0, 0)
(6.283185307179586, 3.76930745228793e-31)
(-47.1238898038469, -1.38722161196188e-28)
(-78.53981633974483, -1.8941914820334e-29)
(-28.274333882308138, -3.43478141589738e-29)
(97.3893722612836, -4.58542475390885e-27)
(3.141592653589793, -4.71163431535992e-32)
(40.840704496667314, -1.57001387566644e-28)
(62.83185307179586, 3.76930745228793e-28)
(-81.68140899333463, 1.25601110053315e-27)
(43.982297150257104, 1.29287245613476e-28)
(-84.82300164692441, -3.99087542625273e-27)
(100.53096491487338, 1.54390833245714e-27)
(69.11503837897546, 1.3448904736186e-27)
(-94.2477796076938, 1.10977728956951e-27)
(91.106186954104, -1.39624614979795e-34)
(78.53981633974483, -1.8941914820334e-29)
(47.1238898038469, -1.38722161196188e-28)
(81.68140899333463, 1.25601110053315e-27)
(-72.25663103256524, -2.93139900017185e-27)
(-6.283185307179586, 3.76930745228793e-31)
(28.274333882308138, -3.43478141589738e-29)
(-100.53096491487338, 1.54390833245714e-27)
(-65.97344572538566, -6.34844983898999e-29)
(94.2477796076938, 1.10977728956951e-27)
(31.41592653589793, 4.71163431535992e-29)
(50.26548245743669, 1.92988541557142e-28)
(-21.991148575128552, -1.61609057016845e-29)
(12.566370614359172, 3.01544596183035e-30)
(15.707963267948966, -5.8895428941999e-30)
(-75.39822368615503, 6.51336327755355e-28)
(72.25663103256524, -2.93139900017185e-27)
(18.84955592153876, 1.01771301211774e-29)
(-37.69911184307752, 8.14170409694193e-29)
(-50.26548245743669, 1.92988541557142e-28)
(-3.141592653589793, -4.71163431535992e-32)
(-87.96459430051421, 1.03429796490781e-27)
(-25.132741228718345, 2.41235676946428e-29)
(-40.840704496667314, -1.57001387566644e-28)
(53.40707511102649, -1.15535214562331e-28)
(9.42477796076938, -1.27214126514718e-30)
(-43.982297150257104, 1.29287245613476e-28)
(-56.548667764616276, 2.7478251327179e-28)
(-97.3893722612836, -4.58542475390885e-27)
(-59.69026041820607, -8.97021321364436e-29)
(-12.566370614359172, 3.01544596183035e-30)
(-18.84955592153876, 1.01771301211774e-29)
(84.82300164692441, -3.99087542625273e-27)
(25.132741228718345, 2.41235676946428e-29)
(21.991148575128552, -1.61609057016845e-29)
(59.69026041820607, -8.97021321364436e-29)
(-53.40707511102649, -1.15535214562331e-28)
(-31.41592653589793, 4.71163431535992e-29)
(-9.42477796076938, -1.27214126514718e-30)
(-62.83185307179586, 3.76930745228793e-28)
(56.548667764616276, 2.7478251327179e-28)
(75.39822368615503, 6.51336327755355e-28)
(34.55751918948773, -1.68111309202325e-28)
(87.96459430051421, 1.03429796490781e-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:
$$x_{1} = 37.6991118430775$$
$$x_{2} = -69.1150383789755$$
$$x_{3} = 0$$
$$x_{4} = 6.28318530717959$$
$$x_{5} = 62.8318530717959$$
$$x_{6} = -81.6814089933346$$
$$x_{7} = 43.9822971502571$$
$$x_{8} = 100.530964914873$$
$$x_{9} = 69.1150383789755$$
$$x_{10} = -94.2477796076938$$
$$x_{11} = 81.6814089933346$$
$$x_{12} = -6.28318530717959$$
$$x_{13} = -100.530964914873$$
$$x_{14} = 94.2477796076938$$
$$x_{15} = 31.4159265358979$$
$$x_{16} = 50.2654824574367$$
$$x_{17} = 12.5663706143592$$
$$x_{18} = -75.398223686155$$
$$x_{19} = 18.8495559215388$$
$$x_{20} = -37.6991118430775$$
$$x_{21} = -50.2654824574367$$
$$x_{22} = -87.9645943005142$$
$$x_{23} = -25.1327412287183$$
$$x_{24} = -43.9822971502571$$
$$x_{25} = -56.5486677646163$$
$$x_{26} = -12.5663706143592$$
$$x_{27} = -18.8495559215388$$
$$x_{28} = 25.1327412287183$$
$$x_{29} = -31.4159265358979$$
$$x_{30} = -62.8318530717959$$
$$x_{31} = 56.5486677646163$$
$$x_{32} = 75.398223686155$$
$$x_{33} = 87.9645943005142$$
Puntos máximos de la función:
$$x_{33} = 65.9734457253857$$
$$x_{33} = -15.707963267949$$
$$x_{33} = -34.5575191894877$$
$$x_{33} = -91.106186954104$$
$$x_{33} = -47.1238898038469$$
$$x_{33} = -78.5398163397448$$
$$x_{33} = -28.2743338823081$$
$$x_{33} = 97.3893722612836$$
$$x_{33} = 3.14159265358979$$
$$x_{33} = 40.8407044966673$$
$$x_{33} = -84.8230016469244$$
$$x_{33} = 91.106186954104$$
$$x_{33} = 78.5398163397448$$
$$x_{33} = 47.1238898038469$$
$$x_{33} = -72.2566310325652$$
$$x_{33} = 28.2743338823081$$
$$x_{33} = -65.9734457253857$$
$$x_{33} = -21.9911485751286$$
$$x_{33} = 15.707963267949$$
$$x_{33} = 72.2566310325652$$
$$x_{33} = -3.14159265358979$$
$$x_{33} = -40.8407044966673$$
$$x_{33} = 53.4070751110265$$
$$x_{33} = 9.42477796076938$$
$$x_{33} = -97.3893722612836$$
$$x_{33} = -59.6902604182061$$
$$x_{33} = 84.8230016469244$$
$$x_{33} = 21.9911485751286$$
$$x_{33} = 59.6902604182061$$
$$x_{33} = -53.4070751110265$$
$$x_{33} = -9.42477796076938$$
$$x_{33} = 34.5575191894877$$
Decrece en los intervalos
$$\left[100.530964914873, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -100.530964914873\right]$$