f(x)=√3sinωxcosωx+cos²ωx
=√3/2*2sinωxcosωx+cos²ωx
=√3/2*sin2ωx+(cos2ωx+1)/2.正弦二倍角,余弦二倍角
=√3/2*sin2ωx+1/2*cos2ωx+1/2
=sin(2ωx+π/6)+1/2
最大值=1+1/2=3/2
最小值=-1+1/2=-1/2
值域是[-1/2,3/2]
(2)最小正周期为π/2=2π/2ω
∴ω=2
f(x)=sin(4x+π/6)+1/2
x∈[0,π/2]
4x+π/6∈[π/6,13π/6]
sinX在[π/2,3π/2]上是减函数
∴4x+π/6∈[π/2,3π/2]
x∈[π/12,π/3]
f(x)减区间是[π/12,π/3]