已知向量a=(根号3sinx,cosx),b=(cosx,cosx),函数f(x)=2a·b-1(1)求f(x)的对称轴,及单调递增区间;(2)当x∈[π/6,π/2]时,若f(x)=1,求x.(3)当x∈[0,π/2]时,求f(x)的值域.
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![已知向量a=(根号3sinx,cosx),b=(cosx,cosx),函数f(x)=2a·b-1(1)求f(x)的对称轴,及单调递增区间;(2)当x∈[π/6,π/2]时,若f(x)=1,求x.(3)当x∈[0,π/2]时,求f(x)的值域.](/uploads/image/z/1700442-18-2.jpg?t=%E5%B7%B2%E7%9F%A5%E5%90%91%E9%87%8Fa%3D%28%E6%A0%B9%E5%8F%B73sinx%2Ccosx%29%2Cb%3D%28cosx%2Ccosx%29%2C%E5%87%BD%E6%95%B0f%28x%29%3D2a%C2%B7b-1%EF%BC%881%EF%BC%89%E6%B1%82f%EF%BC%88x%EF%BC%89%E7%9A%84%E5%AF%B9%E7%A7%B0%E8%BD%B4%2C%E5%8F%8A%E5%8D%95%E8%B0%83%E9%80%92%E5%A2%9E%E5%8C%BA%E9%97%B4%EF%BC%9B%EF%BC%882%EF%BC%89%E5%BD%93x%E2%88%88%5B%CF%80%2F6%2C%CF%80%2F2%5D%E6%97%B6%2C%E8%8B%A5f%EF%BC%88x%EF%BC%89%3D1%2C%E6%B1%82x.%EF%BC%883%EF%BC%89%E5%BD%93x%E2%88%88%5B0%2C%CF%80%2F2%5D%E6%97%B6%2C%E6%B1%82f%EF%BC%88x%EF%BC%89%E7%9A%84%E5%80%BC%E5%9F%9F.)
已知向量a=(根号3sinx,cosx),b=(cosx,cosx),函数f(x)=2a·b-1(1)求f(x)的对称轴,及单调递增区间;(2)当x∈[π/6,π/2]时,若f(x)=1,求x.(3)当x∈[0,π/2]时,求f(x)的值域.
已知向量a=(根号3sinx,cosx),b=(cosx,cosx),函数f(x)=2a·b-1
(1)求f(x)的对称轴,及单调递增区间;
(2)当x∈[π/6,π/2]时,若f(x)=1,求x.
(3)当x∈[0,π/2]时,求f(x)的值域.
已知向量a=(根号3sinx,cosx),b=(cosx,cosx),函数f(x)=2a·b-1(1)求f(x)的对称轴,及单调递增区间;(2)当x∈[π/6,π/2]时,若f(x)=1,求x.(3)当x∈[0,π/2]时,求f(x)的值域.
(1) f(x)=2a·b-1=2√3sinxcosx+2cos²x-1
=√3sin2x+cos2x
=sin(x+π/6)
当x+π/6=π/2+kπ 即 对称轴:x=π/3+kπ
当x+π/6∈[-π/2+2kπ,π/2+2kπ]时,f(x)单调递增,解得:x∈[-2π/3+2kπ,π/3+2kπ]
当x+π/6∈[π/2+2kπ,3/2π+2kπ]时,f(x)单调递减,解得:x∈[π/3+2kπ,4/3π+2kπ]
(2)因为 f(x)=1
所以 x+π/6=π/2+2kπ
x=π/3+2kπ
因为x∈[π/6,π/2]
所以x=π/3
(3)因为x∈[0,π/2]
所以 x+π/6∈[π/6,2/3π]
由函数图像可知:f(x)max=sin(π/2)=1,f(x)min=sin(2/3π)=-√3/2
所以,值域为:[-√3/2,1]