已知函数f(x)=sin(2x+π/3)+sin(2x-π/3)+2cos^2x-1,xER求1）化简f（x）解析式.〔2〕求函数最小正周期.（3）求函数最大的值及其相对应的x值.

Question

已知函数f(x)=sin(2x+π/3)+sin(2x-π/3)+2cos^2x-1,xER求1）化简f（x）解析式.〔2〕求函数最小正周期.（3）求函数最大的值及其相对应的x值.

Answer 1

f(x)=sin(2x+π/3)+sin(2x-π/3)+2cos^2x-1
=sin2xcosπ/3+cos2xsinπ/3+sin2xcosπ/3-cos2xsinπ/3+cos2x
=2sin2xcosπ/3+cos2x
=sin2x+cos2x
=√2*(√2/2*sin2x+√2/2*cos2x)
=√2*(sin2xcosπ/4+cos2xsinπ/4)
=√2*sin(2x+π/4)
T=2π/2=π
x∈[-π/4,π/4]
2x∈[-π/2,π/2]
2x+π/4∈[-π/4,3π/4]
-1