已知函数f(x)=sin(2ωx-π/6)-4sin^2ωx+2 其图像与x轴相邻两个交点的距离为π/2.

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  • f(x)=sin(2ωx-π/6)-4sin²ωx+2

    =√3/2sin2ωx-1/2cos2ωx-4sin²ωx+2

    =√3/2sin2ωx-1/2cos2ωx+2(cos2ωx-1)+2

    =√3/2sin2ωx+(3/2)cos2ωx

    =√3sin(2ωx+φ),tanφ=√3→φ=π/3

    ∵相邻两个交点的距离为π/2,最小正周期=π

    ∴2π/2ω=π→ω=1

    解析式:f(x)=√3sin(2x+π/3)

    (2)g(x)=√3sin(2x+π/3+m) 三角函数在平移的时候是左加右减

    将(-π/3,0)代入:

    0=√3sin(-2π/3+π/3+m)

    m=2kπ+π/3 最小值,m=π/3

    ∴g(x)=√3sin(2x+2π/3)

    g'(x)=2√3cos(2x+2π/3)

    驻点:2x+2π/3=2kπ±π/2

    即x=kπ-π/12、x=kπ-7π/12

    在[-π/6,7π/12]上的单调递增区间为x∈[-π/6,-π/12) (g'(x)>0)