Para hallar los extremos hay que resolver la ecuación
dxdf(x)=0(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
dxdf(x)=primera derivada∣sin(x)∣2sin2(x)−2cos2(x)+2sin(x)cos2(x)sign(sin(x))=0Resolvermos esta ecuaciónRaíces de esta ecuación
x1=40.8407044966673x2=15.707963267949x3=−285.884931476671x4=97.3893722612836x5=−56.5486677646163x6=53.4070751110265x7=87.9645943005142x8=−97.3893722612836x9=62.8318530717959x10=43.9822971502571x11=−21.9911485751286x12=65.9734457253857x13=−50.2654824574367x14=−94.2477796076938x15=−75.398223686155x16=12.5663706143592x17=−34.5575191894877x18=21.9911485751286x19=−9.42477796076935x20=−47.1238898038469x21=−43.9822971502571x22=28.2743338823081x23=−6.28318530717959x24=−25.1327412287183x25=−3.14159265358979x26=650.309679293087x27=59.6902604182061x28=−65.9734457253857x29=72.2566310325652x30=−59.6902604182061x31=31.4159265358979x32=−427.256600888212x33=34.5575191894877x34=94.2477796076938x35=−12.5663706143592x36=−53.4070751110265x37=81.6814089933346x38=−91.106186954104x39=−100.530964914873x40=−31.4159265358979x41=78.5398163397448x42=84.8230016469244x43=100.530964914873x44=−69.1150383789755x45=9.42477796076938x46=−28.2743338823081x47=−78.5398163397448x48=−87.9645943005142x49=−81.6814089933346x50=56.5486677646163x51=−15.707963267949x52=37.6991118430775x53=−37.6991118430775x54=18.8495559215388x55=50.2654824574367x56=−72.2566310325652x57=75.398223686155x58=6.28318530717959Signos de extremos en los puntos:
(40.840704496667314, 0)
(15.707963267948962, 1)
(-285.88493147667117, 0)
(97.3893722612836, 0)
(-56.548667764616276, 0)
(53.40707511102649, 0)
(87.96459430051421, 1)
(-97.3893722612836, 1)
(62.83185307179586, 1)
(43.982297150257104, 1)
(-21.991148575128552, 0)
(65.97344572538566, 0)
(-50.26548245743669, 0)
(-94.2477796076938, 1)
(-75.39822368615503, 5.55111512312578e-17)
(12.566370614359196, 0)
(-34.55751918948773, 1)
(21.991148575128552, 1)
(-9.424777960769353, 0)
(-47.1238898038469, 1)
(-43.982297150257104, 0)
(28.274333882308138, 1)
(-6.283185307179588, 1)
(-25.132741228718345, 0)
(-3.141592653589793, 0)
(650.3096792930872, 0)
(59.69026041820606, 1)
(-65.97344572538566, 1)
(72.25663103256524, 1)
(-59.69026041820607, 1)
(31.41592653589793, 1)
(-427.2566008882119, 1)
(34.557519189487735, 0)
(94.2477796076938, 0)
(-12.566370614359172, 0)
(-53.40707511102648, 0)
(81.68140899333463, 0)
(-91.106186954104, 1)
(-100.53096491487338, 0)
(-31.415926535897928, 0)
(78.53981633974483, 0)
(84.82300164692441, 1)
(100.53096491487338, 1)
(-69.11503837897546, 1)
(9.42477796076938, 1)
(-28.27433388230814, 1)
(-78.53981633974483, 1)
(-87.96459430051421, 0)
(-81.68140899333463, 1)
(56.54866776461628, 0)
(-15.707963267948966, 5.55111512312578e-17)
(37.69911184307751, 1)
(-37.69911184307752, 5.55111512312578e-17)
(18.84955592153876, 1)
(50.26548245743669, 1)
(-72.25663103256524, 0)
(75.39822368615503, 1)
(6.283185307179586, 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:
x1=40.8407044966673x2=−285.884931476671x3=97.3893722612836x4=−56.5486677646163x5=53.4070751110265x6=−21.9911485751286x7=65.9734457253857x8=−50.2654824574367x9=−75.398223686155x10=12.5663706143592x11=−9.42477796076935x12=−43.9822971502571x13=−25.1327412287183x14=−3.14159265358979x15=650.309679293087x16=34.5575191894877x17=94.2477796076938x18=−12.5663706143592x19=−53.4070751110265x20=81.6814089933346x21=−100.530964914873x22=−31.4159265358979x23=78.5398163397448x24=−87.9645943005142x25=56.5486677646163x26=−15.707963267949x27=−37.6991118430775x28=−72.2566310325652Puntos máximos de la función:
x28=15.707963267949x28=87.9645943005142x28=−97.3893722612836x28=62.8318530717959x28=43.9822971502571x28=−94.2477796076938x28=−34.5575191894877x28=21.9911485751286x28=−47.1238898038469x28=28.2743338823081x28=−6.28318530717959x28=59.6902604182061x28=−65.9734457253857x28=72.2566310325652x28=−59.6902604182061x28=31.4159265358979x28=−427.256600888212x28=−91.106186954104x28=84.8230016469244x28=100.530964914873x28=−69.1150383789755x28=9.42477796076938x28=−28.2743338823081x28=−78.5398163397448x28=−81.6814089933346x28=37.6991118430775x28=18.8495559215388x28=50.2654824574367x28=75.398223686155x28=6.28318530717959Decrece en los intervalos
[650.309679293087,∞)Crece en los intervalos
(−∞,−285.884931476671]