已知函数f(x)=2sin²x+√3 sin2x-1若方程在f(x-π/6)+4sinx+1=a在x∈[π/6,π/2]上有解,求实数a的取值范围
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![已知函数f(x)=2sin²x+√3 sin2x-1若方程在f(x-π/6)+4sinx+1=a在x∈[π/6,π/2]上有解,求实数a的取值范围](/uploads/image/z/3939408-0-8.jpg?t=%E5%B7%B2%E7%9F%A5%E5%87%BD%E6%95%B0f%28x%29%3D2sin%26%23178%3Bx%2B%E2%88%9A3+sin2x-1%E8%8B%A5%E6%96%B9%E7%A8%8B%E5%9C%A8f%28x-%CF%80%2F6%29%2B4sinx%2B1%3Da%E5%9C%A8x%E2%88%88%5B%CF%80%2F6%2C%CF%80%2F2%5D%E4%B8%8A%E6%9C%89%E8%A7%A3%2C%E6%B1%82%E5%AE%9E%E6%95%B0a%E7%9A%84%E5%8F%96%E5%80%BC%E8%8C%83%E5%9B%B4)
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已知函数f(x)=2sin²x+√3 sin2x-1若方程在f(x-π/6)+4sinx+1=a在x∈[π/6,π/2]上有解,求实数a的取值范围
已知函数f(x)=2sin²x+√3 sin2x-1
若方程在f(x-π/6)+4sinx+1=a在x∈[π/6,π/2]上有解,求实数a的取值范围
已知函数f(x)=2sin²x+√3 sin2x-1若方程在f(x-π/6)+4sinx+1=a在x∈[π/6,π/2]上有解,求实数a的取值范围
f(x)=2sin^x+根号3sin2x-1
=-(1-2sin^2x)+根号3sin2x
=-cos2x+根号3sin2x
=2sin(2x-π/6)
所以
f(x-π/6)=2sin(2x-π/3-π/6)
=2sin(2x-π/2)
=-2cos2x
所以f(x-π/6)+4sinx+1
=-2cos2x+4sinx+1
=-2(1-sin^2x)+4sinx+1
=2sin^2x+4sinx-1
=2(sin^2x+2sinx+1-1)-1
=2(sinx+1)^2 -3=a
又因为x∈[π/6,π/2]
(sinx+1)^2=1/2(a+3)
sinx∈[1/2,1]
sinx+1∈[3/2,2]
(sinx+1)^2∈[9/4,4]
所以a∈[9/4,4]
明天帮你算,今天先睡了…