已知函数f(x)=sin^2ωx+√3cosωxcos(π/2-ωx)(ω>0)且函数y=f(x)的图像相邻两条对称轴之间的距离为π/2,(1)求f(π/6)的值 (2)若函数f(kx+π/12)(k>0)在[-π/6,π/3]上单调递增,求k的取值范围
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![已知函数f(x)=sin^2ωx+√3cosωxcos(π/2-ωx)(ω>0)且函数y=f(x)的图像相邻两条对称轴之间的距离为π/2,(1)求f(π/6)的值 (2)若函数f(kx+π/12)(k>0)在[-π/6,π/3]上单调递增,求k的取值范围](/uploads/image/z/5226558-6-8.jpg?t=%E5%B7%B2%E7%9F%A5%E5%87%BD%E6%95%B0f%28x%29%3Dsin%5E2%CF%89x%2B%E2%88%9A3cos%CF%89xcos%28%CF%80%2F2-%CF%89x%29%28%CF%89%3E0%29%E4%B8%94%E5%87%BD%E6%95%B0y%3Df%28x%29%E7%9A%84%E5%9B%BE%E5%83%8F%E7%9B%B8%E9%82%BB%E4%B8%A4%E6%9D%A1%E5%AF%B9%E7%A7%B0%E8%BD%B4%E4%B9%8B%E9%97%B4%E7%9A%84%E8%B7%9D%E7%A6%BB%E4%B8%BA%CF%80%2F2%2C%EF%BC%881%EF%BC%89%E6%B1%82f%28%CF%80%2F6%EF%BC%89%E7%9A%84%E5%80%BC+%EF%BC%882%EF%BC%89%E8%8B%A5%E5%87%BD%E6%95%B0f%28kx%2B%CF%80%2F12%EF%BC%89%EF%BC%88k%3E0%29%E5%9C%A8%5B-%CF%80%2F6%2C%CF%80%2F3%5D%E4%B8%8A%E5%8D%95%E8%B0%83%E9%80%92%E5%A2%9E%2C%E6%B1%82k%E7%9A%84%E5%8F%96%E5%80%BC%E8%8C%83%E5%9B%B4)
已知函数f(x)=sin^2ωx+√3cosωxcos(π/2-ωx)(ω>0)且函数y=f(x)的图像相邻两条对称轴之间的距离为π/2,(1)求f(π/6)的值 (2)若函数f(kx+π/12)(k>0)在[-π/6,π/3]上单调递增,求k的取值范围
已知函数f(x)=sin^2ωx+√3cosωxcos(π/2-ωx)(ω>0)
且函数y=f(x)的图像相邻两条对称轴之间的距离为π/2,(1)求f(π/6)的值 (2)若函数f(kx+π/12)(k>0)在[-π/6,π/3]上单调递增,求k的取值范围
已知函数f(x)=sin^2ωx+√3cosωxcos(π/2-ωx)(ω>0)且函数y=f(x)的图像相邻两条对称轴之间的距离为π/2,(1)求f(π/6)的值 (2)若函数f(kx+π/12)(k>0)在[-π/6,π/3]上单调递增,求k的取值范围
f(x)=sin^2ωx+√3cosωxcos(π/2-ωx)(ω>0)
=(1-cos2ωx)/2+(√3/2)sin2ωx
=sin(2ωx-π/6)+1/2
∵函数y=f(x)的图像相邻两条对称轴之间的距离为π/2即半周期=π/2
∴T=2π/2ω=π,∴ω=1,
∴f(x)=sin(2x-π/6)+1/2
(1)f(π/6)=sin(π/3-π/6)+1/2=sinπ/6+1/2=1/2+1/2=1
(2)函数f(kx+π/12)=sin(2kx+π/6-π/6)+1/2=sin2kx+1/2,其在[-π/2,π/2]上单调递增,
∴-π/2
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