y=sin²x+sinxcosx+2 （x∈R)
=1/2(1-cos2x)+1/2sin2x +2
=1/2(sin2x-cos2x) +5/2
=√2/2(√2/2sin2x - √2/2cos2x) +5/2
=√2/2(sin2xcosπ/4 -cos2xsinπ/4) +5/2
=√2/2sin(2x-π/4) +5/2
而 sin(2x-π/4) 在 x∈R上的取值范围是[-1,1]
所以 y=√2/2sin(2x-π/4) +5/2
的取值范围是 [(5-√2)/2,(5+√2)/2].