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$$\left(x + 1\right) e^{x} + e^{x} - \frac{\sin{\left(x \right)}}{2} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = -56.5486677646163$$
$$x_{2} = -9.42357836392699$$
$$x_{3} = -97.3893722612836$$
$$x_{4} = -37.6991118430775$$
$$x_{5} = -31.4159265358993$$
$$x_{6} = -25.132741229281$$
$$x_{7} = -18.8495561410012$$
$$x_{8} = -50.2654824574367$$
$$x_{9} = -94.2477796076938$$
$$x_{10} = -21.9911485638765$$
$$x_{11} = -75.398223686155$$
$$x_{12} = -28.2743338822805$$
$$x_{13} = -53.4070751110265$$
$$x_{14} = -113.097335529233$$
$$x_{15} = -47.1238898038469$$
$$x_{16} = -43.9822971502571$$
$$x_{17} = -62.8318530717959$$
$$x_{18} = -65.9734457253857$$
$$x_{19} = -59.6902604182061$$
$$x_{20} = -91.106186954104$$
$$x_{21} = -3.04193881531373$$
$$x_{22} = -100.530964914873$$
$$x_{23} = -40.8407044966673$$
$$x_{24} = -69.1150383789755$$
$$x_{25} = -84.8230016469244$$
$$x_{26} = -15.7079591363057$$
$$x_{27} = -78.5398163397448$$
$$x_{28} = -87.9645943005142$$
$$x_{29} = -81.6814089933346$$
$$x_{30} = -12.5664443065461$$
$$x_{31} = -72.2566310325652$$
$$x_{32} = -232.477856365645$$
$$x_{33} = -34.5575191894877$$
$$x_{34} = -6.29899042622584$$
Signos de extremos en los puntos:
(-56.548667764616276, 0.5)
(-9.423578363926987, -0.500680234901783)
(-97.3893722612836, -0.5)
(-37.699111843077524, 0.499999999999998)
(-31.415926535899267, 0.499999999999309)
(-25.132741229281006, 0.499999999706508)
(-18.84955614100122, 0.499999883756347)
(-50.26548245743669, 0.5)
(-94.2477796076938, 0.5)
(-21.99114856387646, -0.500000005907473)
(-75.39822368615503, 0.5)
(-28.274333882280523, -0.500000000014334)
(-53.40707511102649, -0.5)
(-113.09733552923255, 0.5)
(-47.1238898038469, -0.5)
(-43.982297150257104, 0.5)
(-62.83185307179586, 0.5)
(-65.97344572538566, -0.5)
(-59.69026041820607, -0.5)
(-91.106186954104, -0.5)
(-3.0419388153137312, -0.595006057568337)
(-100.53096491487338, 0.5)
(-40.840704496667314, -0.5)
(-69.11503837897546, 0.5)
(-84.82300164692441, -0.5)
(-15.70795913630565, -0.500002216519741)
(-78.53981633974483, -0.5)
(-87.96459430051421, 0.5)
(-81.68140899333463, 0.5)
(-12.566444306546071, 0.499959665463574)
(-72.25663103256524, -0.5)
(-232.4778563656447, 0.5)
(-34.557519189487664, -0.500000000000033)
(-6.298990426225845, 0.490197160739698)
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:
$$x_{1} = -9.42357836392699$$
$$x_{2} = -97.3893722612836$$
$$x_{3} = -21.9911485638765$$
$$x_{4} = -28.2743338822805$$
$$x_{5} = -53.4070751110265$$
$$x_{6} = -47.1238898038469$$
$$x_{7} = -65.9734457253857$$
$$x_{8} = -59.6902604182061$$
$$x_{9} = -91.106186954104$$
$$x_{10} = -3.04193881531373$$
$$x_{11} = -40.8407044966673$$
$$x_{12} = -84.8230016469244$$
$$x_{13} = -15.7079591363057$$
$$x_{14} = -78.5398163397448$$
$$x_{15} = -72.2566310325652$$
$$x_{16} = -34.5575191894877$$
Puntos máximos de la función:
$$x_{16} = -56.5486677646163$$
$$x_{16} = -37.6991118430775$$
$$x_{16} = -31.4159265358993$$
$$x_{16} = -25.132741229281$$
$$x_{16} = -18.8495561410012$$
$$x_{16} = -50.2654824574367$$
$$x_{16} = -94.2477796076938$$
$$x_{16} = -75.398223686155$$
$$x_{16} = -113.097335529233$$
$$x_{16} = -43.9822971502571$$
$$x_{16} = -62.8318530717959$$
$$x_{16} = -100.530964914873$$
$$x_{16} = -69.1150383789755$$
$$x_{16} = -87.9645943005142$$
$$x_{16} = -81.6814089933346$$
$$x_{16} = -12.5664443065461$$
$$x_{16} = -232.477856365645$$
$$x_{16} = -6.29899042622584$$
Decrece en los intervalos
$$\left[-3.04193881531373, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -97.3893722612836\right]$$