设偶函数f(x)=sin (ωx+φ)+cos(ωx+φ)(ω>0,|φ|

题目

设偶函数f(x)=sin (ωx+φ)+cos(ωx+φ)(ω>0,|φ|

答案

f(x)=sin (ωx+φ)+cos(ωx+φ)
=√2sin (ωx+φ+π/4)
∵周期是π,∴ω=2
∵f(-x)=f(x),∴φ+π/4=π/2,∴φ=π/4
∴f(x)=√2sin(2x+π/2)=√2cos2x
∴在[0,π/2]上单调递减,