作业帮已知向量a=(5√3cosx,cosx),b=(sinx,2cosx),设函数f(x)=a*b+|b|^2+3/2函数f(x)=a*b+|b|^2+3/2(1)当x∈[π/6,π/2],求函数f(x)的值域(2)当x∈[π/6,π/2]时,若f(x)=8,求函数f(x-π/12)的值
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![已知向量a=(5√3cosx,cosx),b=(sinx,2cosx),设函数f(x)=a*b+|b|^2+3/2函数f(x)=a*b+|b|^2+3/2(1)当x∈[π/6,π/2],求函数f(x)的值域(2)当x∈[π/6,π/2]时,若f(x)=8,求函数f(x-π/12)的值](/uploads/image/z/988572-12-2.jpg?t=%E5%B7%B2%E7%9F%A5%E5%90%91%E9%87%8Fa%3D%285%E2%88%9A3cosx%2Ccosx%29%2Cb%3D%28sinx%2C2cosx%29%2C%E8%AE%BE%E5%87%BD%E6%95%B0f%28x%29%3Da%2Ab%2B%7Cb%7C%5E2%2B3%2F2%E5%87%BD%E6%95%B0f%28x%29%3Da%2Ab%2B%7Cb%7C%5E2%2B3%2F2%281%29%E5%BD%93x%E2%88%88%5B%CF%80%2F6%2C%CF%80%2F2%5D%2C%E6%B1%82%E5%87%BD%E6%95%B0f%28x%29%E7%9A%84%E5%80%BC%E5%9F%9F%282%29%E5%BD%93x%E2%88%88%5B%CF%80%2F6%2C%CF%80%2F2%5D%E6%97%B6%2C%E8%8B%A5f%28x%29%3D8%2C%E6%B1%82%E5%87%BD%E6%95%B0f%28x-%CF%80%2F12%29%E7%9A%84%E5%80%BC)
已知向量a=(5√3cosx,cosx),b=(sinx,2cosx),设函数f(x)=a*b+|b|^2+3/2函数f(x)=a*b+|b|^2+3/2(1)当x∈[π/6,π/2],求函数f(x)的值域(2)当x∈[π/6,π/2]时,若f(x)=8,求函数f(x-π/12)的值
已知向量a=(5√3cosx,cosx),b=(sinx,2cosx),设函数f(x)=a*b+|b|^2+3/2
函数f(x)=a*b+|b|^2+3/2
(1)当x∈[π/6,π/2],求函数f(x)的值域
(2)当x∈[π/6,π/2]时,若f(x)=8,求函数f(x-π/12)的值
已知向量a=(5√3cosx,cosx),b=(sinx,2cosx),设函数f(x)=a*b+|b|^2+3/2函数f(x)=a*b+|b|^2+3/2(1)当x∈[π/6,π/2],求函数f(x)的值域(2)当x∈[π/6,π/2]时,若f(x)=8,求函数f(x-π/12)的值
f(x)=5√3sinxcosx+2cos^2x+sin^2x+4cos^2x+3/2
=5√3/2sin2x+6cos^2+sin^2x+3/2
=5√3/2sin2x+6cos^2x-3+3+sin^2x-1/2+1/2+3/2
=5√3/2sin2x+3cos2x-1/2cos2x+5
=5√3/2sin2x+5/2cos2x+5
=5sin(2x+π/6)+5
(1)
2x+π/6=π/2+2kπ（k∈Z）
x=π/6+kπ
2x+π/6=3π/2+2kπ（k∈Z）
x=2π/3+kπ
f（x）在[π/6+kπ,2π/3+kπ]单调递减
f（x）的值域为[5/2,10]
(2)
x∈[π/6,π/2]
2x+π/6∈[π/2,7π/6]
5sin(2x+π/6)+5=8
sin(2x+π/6)=3/5
cos(2x+π/6)=-4/5
sin[2(x-π/12)+π/6]
=sin[2x-π/6+π/6]
=sin(2x+π/6)cosπ/6-cos(2x+π/6)sinπ/6
=3/5*√3/2-(-4/5)*1/2
=3√3/10+4/10
=(3√3+4)/10