f(x)=0.5cos(2x+π/6)+sinxcosx
=0.5[cos2xcos(π/6)-sin2xsin(π/6)]+0.5sin2x
=0.5[(√3/2)cos2x+(1/2)sin2x]
=0.5[sin(π/3)cos2x+cos(π/3)sin2x]
=0.5sin(2x+π/3)
令sin(2x+π/3)=±1得2x+π/3=kπ+π/2,可求得对称轴方程为x=(k/2)π+π/12 (k∈Z)
f(α) =0.5sin(2α+π/3)=√2/4,可得sin(2α+π/3)=√2/2
因为α∈(-π/2,0),所以2α+π/3∈(-2π/3,π/3),结合上面的结果得2α+π/3=π/4,解得
α= - π/6