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$$- \frac{\sin{\left(\frac{x}{2} \right)} \operatorname{sign}{\left(\frac{\cos{\left(\frac{x}{2} \right)}}{2} + 3 \right)}}{4} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = 81.6814089933346$$
$$x_{2} = -43.9822971502571$$
$$x_{3} = -100.530964914873$$
$$x_{4} = 56.5486677646163$$
$$x_{5} = -31.4159265358979$$
$$x_{6} = -56.5486677646163$$
$$x_{7} = 12.5663706143592$$
$$x_{8} = 43.9822971502571$$
$$x_{9} = 100.530964914873$$
$$x_{10} = -106.814150222053$$
$$x_{11} = 6.28318530717959$$
$$x_{12} = -87.9645943005142$$
$$x_{13} = 69.1150383789755$$
$$x_{14} = 87.9645943005142$$
$$x_{15} = 18.8495559215388$$
$$x_{16} = 25.1327412287183$$
$$x_{17} = -25.1327412287183$$
$$x_{18} = 37.6991118430775$$
$$x_{19} = 0$$
$$x_{20} = 50.2654824574367$$
$$x_{21} = -6.28318530717959$$
$$x_{22} = -62.8318530717959$$
$$x_{23} = 75.398223686155$$
$$x_{24} = -69.1150383789755$$
$$x_{25} = 94.2477796076938$$
$$x_{26} = -18.8495559215388$$
$$x_{27} = -50.2654824574367$$
$$x_{28} = -226.194671058465$$
$$x_{29} = -37.6991118430775$$
$$x_{30} = -81.6814089933346$$
$$x_{31} = 62.8318530717959$$
$$x_{32} = 31.4159265358979$$
$$x_{33} = -75.398223686155$$
$$x_{34} = -12.5663706143592$$
$$x_{35} = -94.2477796076938$$
Signos de extremos en los puntos:
(81.68140899333463, 2.5)
(-43.982297150257104, 2.5)
(-100.53096491487338, 3.5)
(56.548667764616276, 2.5)
(-31.41592653589793, 2.5)
(-56.548667764616276, 2.5)
(12.566370614359172, 3.5)
(43.982297150257104, 2.5)
(100.53096491487338, 3.5)
(-106.81415022205297, 2.5)
(6.283185307179586, 2.5)
(-87.96459430051421, 3.5)
(69.11503837897546, 2.5)
(87.96459430051421, 3.5)
(18.84955592153876, 2.5)
(25.132741228718345, 3.5)
(-25.132741228718345, 3.5)
(37.69911184307752, 3.5)
(0, 7/2)
(50.26548245743669, 3.5)
(-6.283185307179586, 2.5)
(-62.83185307179586, 3.5)
(75.39822368615503, 3.5)
(-69.11503837897546, 2.5)
(94.2477796076938, 2.5)
(-18.84955592153876, 2.5)
(-50.26548245743669, 3.5)
(-226.1946710584651, 3.5)
(-37.69911184307752, 3.5)
(-81.68140899333463, 2.5)
(62.83185307179586, 3.5)
(31.41592653589793, 2.5)
(-75.39822368615503, 3.5)
(-12.566370614359172, 3.5)
(-94.2477796076938, 2.5)
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:
$$x_{1} = 81.6814089933346$$
$$x_{2} = -43.9822971502571$$
$$x_{3} = 56.5486677646163$$
$$x_{4} = -31.4159265358979$$
$$x_{5} = -56.5486677646163$$
$$x_{6} = 43.9822971502571$$
$$x_{7} = -106.814150222053$$
$$x_{8} = 6.28318530717959$$
$$x_{9} = 69.1150383789755$$
$$x_{10} = 18.8495559215388$$
$$x_{11} = -6.28318530717959$$
$$x_{12} = -69.1150383789755$$
$$x_{13} = 94.2477796076938$$
$$x_{14} = -18.8495559215388$$
$$x_{15} = -81.6814089933346$$
$$x_{16} = 31.4159265358979$$
$$x_{17} = -94.2477796076938$$
Puntos máximos de la función:
$$x_{17} = -100.530964914873$$
$$x_{17} = 12.5663706143592$$
$$x_{17} = 100.530964914873$$
$$x_{17} = -87.9645943005142$$
$$x_{17} = 87.9645943005142$$
$$x_{17} = 25.1327412287183$$
$$x_{17} = -25.1327412287183$$
$$x_{17} = 37.6991118430775$$
$$x_{17} = 0$$
$$x_{17} = 50.2654824574367$$
$$x_{17} = -62.8318530717959$$
$$x_{17} = 75.398223686155$$
$$x_{17} = -50.2654824574367$$
$$x_{17} = -226.194671058465$$
$$x_{17} = -37.6991118430775$$
$$x_{17} = 62.8318530717959$$
$$x_{17} = -75.398223686155$$
$$x_{17} = -12.5663706143592$$
Decrece en los intervalos
$$\left[94.2477796076938, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -106.814150222053\right]$$