已知函数f(x)=4cos^2x-4√3sinxcosx-1(x∈R)
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已知函数f(x)=4cos^2x-4√3sinxcosx-1(x∈R)
(1)求出函数f(x)的最小正周期
(2)求出函数f(x)的单调递增区间
(3)求出函数f(x)的对称轴
(1)求出函数f(x)的最小正周期
(2)求出函数f(x)的单调递增区间
(3)求出函数f(x)的对称轴
f(x)=4cos^2x-4√3sinxcosx-1
=8cosx(1/2cosx-√3/2sinx)-1
=8cosx(sinπ/6cosx-cosπ/6sinx)-1
=8cosxsin(π/6-x)-1
=4(sinπ/6-sin(2x-π/6)-1
=2-4sin(2x-π/6)-1
=1-4sin(2x-π/6)
(1)最小正周期T=2π/ω=π
(2)单调递增区间
2Kπ+π/2≤2x-π/6≤2Kπ+3π/2
2Kπ+2π/3≤2x≤2Kπ+5π/3
Kπ+π/3≤x≤Kπ+5π/6
(3)2x-π/6=Kπ+π/2
2x=Kπ+2π/3
x=Kπ/2+π/3
对称轴为x=Kπ/2+π/3
=8cosx(1/2cosx-√3/2sinx)-1
=8cosx(sinπ/6cosx-cosπ/6sinx)-1
=8cosxsin(π/6-x)-1
=4(sinπ/6-sin(2x-π/6)-1
=2-4sin(2x-π/6)-1
=1-4sin(2x-π/6)
(1)最小正周期T=2π/ω=π
(2)单调递增区间
2Kπ+π/2≤2x-π/6≤2Kπ+3π/2
2Kπ+2π/3≤2x≤2Kπ+5π/3
Kπ+π/3≤x≤Kπ+5π/6
(3)2x-π/6=Kπ+π/2
2x=Kπ+2π/3
x=Kπ/2+π/3
对称轴为x=Kπ/2+π/3
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