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 derivada3xsin(x)−3cos(x)+3cos(3x)=0Resolvermos esta ecuaciónRaíces de esta ecuación
x1=65.9734457253857x2=−15.707963267949x3=−8.57750497498977⋅10−8x4=37.6991118430775x5=−69.1150383789755x6=−34.5575191894877x7=−91.106186954104x8=0x9=6.28318530717959x10=−47.1238898038469x11=−78.5398163397448x12=−28.2743338823081x13=97.3893722612836x14=3.14159265358979x15=40.8407044966673x16=62.8318530717959x17=−81.6814089933346x18=43.9822971502571x19=−84.8230016469244x20=100.530964914873x21=69.1150383789755x22=−94.2477796076938x23=91.106186954104x24=78.5398163397448x25=207.345115136926x26=47.1238898038469x27=81.6814089933346x28=−72.2566310325652x29=−6.28318530717959x30=28.2743338823081x31=−100.530964914873x32=−65.9734457253857x33=94.2477796076938x34=31.4159265358979x35=50.2654824574367x36=−21.9911485751286x37=12.5663706143592x38=−1.23728839368491x39=15.707963267949x40=−75.398223686155x41=72.2566310325652x42=18.8495559215388x43=−37.6991118430775x44=−50.2654824574367x45=−3.14159265358979x46=−87.9645943005142x47=−25.1327412287183x48=−40.8407044966673x49=53.4070751110265x50=9.42477796076938x51=−43.9822971502571x52=−56.5486677646163x53=−97.3893722612836x54=−59.6902604182061x55=−12.5663706143592x56=−18.8495559215388x57=84.8230016469244x58=25.1327412287183x59=21.9911485751286x60=59.6902604182061x61=−53.4070751110265x62=−31.4159265358979x63=−9.42477796076938x64=−62.8318530717959x65=56.5486677646163x66=75.398223686155x67=34.5575191894877x68=87.9645943005142Signos de extremos en los puntos:
(65.97344572538566, 197.920337176157)
(-15.707963267948966, -47.1238898038469)
(-8.577504974989769e-08, 1.90582413132218e-21)
(37.69911184307752, -113.097335529233)
(-69.11503837897546, 207.345115136926)
(-34.55751918948773, -103.672557568463)
(-91.106186954104, -273.318560862312)
(0, 0)
(6.283185307179586, -18.8495559215388)
(-47.1238898038469, -141.371669411541)
(-78.53981633974483, -235.619449019234)
(-28.274333882308138, -84.8230016469244)
(97.3893722612836, 292.168116783851)
(3.141592653589793, 9.42477796076938)
(40.840704496667314, 122.522113490002)
(62.83185307179586, -188.495559215388)
(-81.68140899333463, 245.044226980004)
(43.982297150257104, -131.946891450771)
(-84.82300164692441, -254.469004940773)
(100.53096491487338, -301.59289474462)
(69.11503837897546, -207.345115136926)
(-94.2477796076938, 282.743338823081)
(91.106186954104, 273.318560862312)
(78.53981633974483, 235.619449019234)
(207.34511513692635, -622.035345410779)
(47.1238898038469, 141.371669411541)
(81.68140899333463, -245.044226980004)
(-72.25663103256524, -216.769893097696)
(-6.283185307179586, 18.8495559215388)
(28.274333882308138, 84.8230016469244)
(-100.53096491487338, 301.59289474462)
(-65.97344572538566, -197.920337176157)
(94.2477796076938, -282.743338823081)
(31.41592653589793, -94.2477796076938)
(50.26548245743669, -150.79644737231)
(-21.991148575128552, -65.9734457253857)
(12.566370614359172, -37.6991118430775)
(-1.2372883936849146, 1.75497647171138)
(15.707963267948966, 47.1238898038469)
(-75.39822368615503, 226.194671058465)
(72.25663103256524, 216.769893097696)
(18.84955592153876, -56.5486677646163)
(-37.69911184307752, 113.097335529233)
(-50.26548245743669, 150.79644737231)
(-3.141592653589793, -9.42477796076938)
(-87.96459430051421, 263.893782901543)
(-25.132741228718345, 75.398223686155)
(-40.840704496667314, -122.522113490002)
(53.40707511102649, 160.221225333079)
(9.42477796076938, 28.2743338823081)
(-43.982297150257104, 131.946891450771)
(-56.548667764616276, 169.646003293849)
(-97.3893722612836, -292.168116783851)
(-59.69026041820607, -179.070781254618)
(-12.566370614359172, 37.6991118430775)
(-18.84955592153876, 56.5486677646163)
(84.82300164692441, 254.469004940773)
(25.132741228718345, -75.398223686155)
(21.991148575128552, 65.9734457253857)
(59.69026041820607, 179.070781254618)
(-53.40707511102649, -160.221225333079)
(-31.41592653589793, 94.2477796076938)
(-9.42477796076938, -28.2743338823081)
(-62.83185307179586, 188.495559215388)
(56.548667764616276, -169.646003293849)
(75.39822368615503, -226.194671058465)
(34.55751918948773, 103.672557568463)
(87.96459430051421, -263.893782901543)
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=−15.707963267949x2=37.6991118430775x3=−34.5575191894877x4=−91.106186954104x5=6.28318530717959x6=−47.1238898038469x7=−78.5398163397448x8=−28.2743338823081x9=62.8318530717959x10=43.9822971502571x11=−84.8230016469244x12=100.530964914873x13=69.1150383789755x14=207.345115136926x15=81.6814089933346x16=−72.2566310325652x17=−65.9734457253857x18=94.2477796076938x19=31.4159265358979x20=50.2654824574367x21=−21.9911485751286x22=12.5663706143592x23=18.8495559215388x24=−3.14159265358979x25=−40.8407044966673x26=−97.3893722612836x27=−59.6902604182061x28=25.1327412287183x29=−53.4070751110265x30=−9.42477796076938x31=56.5486677646163x32=75.398223686155x33=87.9645943005142Puntos máximos de la función:
x33=65.9734457253857x33=−69.1150383789755x33=97.3893722612836x33=3.14159265358979x33=40.8407044966673x33=−81.6814089933346x33=−94.2477796076938x33=91.106186954104x33=78.5398163397448x33=47.1238898038469x33=−6.28318530717959x33=28.2743338823081x33=−100.530964914873x33=−1.23728839368491x33=15.707963267949x33=−75.398223686155x33=72.2566310325652x33=−37.6991118430775x33=−50.2654824574367x33=−87.9645943005142x33=−25.1327412287183x33=53.4070751110265x33=9.42477796076938x33=−43.9822971502571x33=−56.5486677646163x33=−12.5663706143592x33=−18.8495559215388x33=84.8230016469244x33=21.9911485751286x33=59.6902604182061x33=−31.4159265358979x33=−62.8318530717959x33=34.5575191894877Decrece en los intervalos
[207.345115136926,∞)Crece en los intervalos
(−∞,−97.3893722612836]