设函数f (x)=√3cos2ωx+sinωxcosωx+a(其中ω>0,a∈r),且f (x)的图象在y轴右侧的第一个最高点的横坐标为π/6.(Ⅰ)求ω的值;(Ⅱ)如果f (x)在区间[-π/3,5π/6]上的最小值为√3.求a的值.
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![设函数f (x)=√3cos2ωx+sinωxcosωx+a(其中ω>0,a∈r),且f (x)的图象在y轴右侧的第一个最高点的横坐标为π/6.(Ⅰ)求ω的值;(Ⅱ)如果f (x)在区间[-π/3,5π/6]上的最小值为√3.求a的值.](/uploads/image/z/6897185-17-5.jpg?t=%E8%AE%BE%E5%87%BD%E6%95%B0f+%28x%29%EF%BC%9D%E2%88%9A3cos2%CF%89x%EF%BC%8Bsin%CF%89xcos%CF%89x%EF%BC%8Ba%EF%BC%88%E5%85%B6%E4%B8%AD%CF%89%EF%BC%9E0%2Ca%E2%88%88r%EF%BC%89%2C%E4%B8%94f+%28x%29%E7%9A%84%E5%9B%BE%E8%B1%A1%E5%9C%A8y%E8%BD%B4%E5%8F%B3%E4%BE%A7%E7%9A%84%E7%AC%AC%E4%B8%80%E4%B8%AA%E6%9C%80%E9%AB%98%E7%82%B9%E7%9A%84%E6%A8%AA%E5%9D%90%E6%A0%87%E4%B8%BA%CF%80%2F6%EF%BC%8E%EF%BC%88%E2%85%A0%EF%BC%89%E6%B1%82%CF%89%E7%9A%84%E5%80%BC%EF%BC%9B%EF%BC%88%E2%85%A1%EF%BC%89%E5%A6%82%E6%9E%9Cf+%28x%29%E5%9C%A8%E5%8C%BA%E9%97%B4%5B%EF%BC%8D%CF%80%2F3%2C5%CF%80%2F6%5D%E4%B8%8A%E7%9A%84%E6%9C%80%E5%B0%8F%E5%80%BC%E4%B8%BA%E2%88%9A3%EF%BC%8E%E6%B1%82a%E7%9A%84%E5%80%BC%EF%BC%8E)
设函数f (x)=√3cos2ωx+sinωxcosωx+a(其中ω>0,a∈r),且f (x)的图象在y轴右侧的第一个最高点的横坐标为π/6.(Ⅰ)求ω的值;(Ⅱ)如果f (x)在区间[-π/3,5π/6]上的最小值为√3.求a的值.
设函数f (x)=√3cos2ωx+sinωxcosωx+a(其中ω>0,a∈r),且f (x)的图象在y轴右侧的第一个最高点的横坐标为π/6.(Ⅰ)求ω的值;(Ⅱ)如果f (x)在区间[-π/3,5π/6]上的最小值为√3.求a的值.
设函数f (x)=√3cos2ωx+sinωxcosωx+a(其中ω>0,a∈r),且f (x)的图象在y轴右侧的第一个最高点的横坐标为π/6.(Ⅰ)求ω的值;(Ⅱ)如果f (x)在区间[-π/3,5π/6]上的最小值为√3.求a的值.
(Ⅰ)
f (x)=√3cos2ωx+sinωxcosωx+a
=√3cos2ωx+(1/2)sin2ωx+a
=[(√3)^2+(1/2)^2]sin[2ωx+arctan(√3/(1/2))]+a
=(13/4)sin(2ωx+arctan2√3)+a
∴2ω×(π/6)+arctan2√3=π/2
ω=3/2+3arctan2√3/π
如果题目是f (x)=√3cos2ωx+2sinωxcosωx+a的话,应该得到f (x)=2sin(2ωx+arctan√3)+a=2sin(2ωx+π/3)+a
则2ω×(π/6)+π/3=π/2
ω=1/2
(Ⅱ)
为了方便,我姑且按ω=1/2算了.都是一个意思
ω=1/2则f (x)=2sin(2×(1/2)x+π/3)+a=2sin(x+π/3)+a
x∈[-π/3,5π/6]则x+π/3∈[0,7π/6].
令X=x+π/3,则在X∈[0,7π/6]上,sinX的最小值是sin(7π/6)=-1/2(画正弦函数图像可知).则f (x)在区间[-π/3,5π/6]上的最小值为f (x)=2sinX+a=2×(-1/2)+a=a-1=√3
∴a=1+√3
X