已知函数f（x）=cosx^2 +s√3sinxcosx-sinx^2

题目

已知函数f（x）=cosx^2 +s√3sinxcosx-sinx^2
（1）求f(π/4)的值
（2）求函数f（x）的最小正周期、最大值和单调区间.
要完整过程,

答案

(1)
∵ （cosx)^2-(sinx)^2=cos2x,2sinxcosx=sin2x
∴f（x）=cosx^2 +2√3sinxcosx-sinx^2
=√3sin2x+cos2x
=2(√3/2sin2x+1/2*cos2x)
=2sin(2x+π/6)
f(π/4)=2sin(π/2+π/6)=2cosπ/6=√3
(2)
f(x)的最小正周期T=2π/2=π
f(x)最大值为2
由2kπ-π/2≤2x+π/6≤2kπ+π/2,k∈Z
得kπ-π/3≤x≤kπ+π/6,k∈Z
∴f(x)单调递增区间为[kπ-π/3,kπ+π/6],k∈Z
由2kπ+π/2≤2x+π/6≤2kπ+3π/2,k∈Z
得kπ+π/6≤x≤kπ+2π/3,k∈Z
∴f(x)单调递增区间为[kπ+π/6,kπ+2π/3],k∈Z