f(x)=2sinx/2cosx/2+αcos²x/2=sinx+0.5α(cosx+1)
把x=π/2代入f(x)=0得到
1+0.5α=0
α=-2
f(x)=sinx-cosx-1=√2sin(x-π/4)-1
当x∈[0,π]时x-π/4∈[-π/4,3π/4]
∴√2sin(x-π/4)∈[-1,√2]
∴f(x)的值域为{f(x)|-2≤f(x)≤√2-1}