一、 f(x)=a.b.
a.b=2cosxcosx-√3sin2x*1 ( x∈R)..
=2cos^2x-√3sin2x..
=1+cos2x-√3sin2x.
=2[(1/2)cos2x-√3/2sin2x]+1
=2sin(π/6-2x)
=-2sin(2x-π/6).
∴f(x)=a.b=-2sin(2x-π/6),在[4kπ+5π/6.(4k+3)π-π/6]区间单调递减.
二、f(A)=-2sin(2A-π/6)=-1.
sin(2A-π/6)=1/2.
2A-π/6=π/6,或2A-π/6=5π/6,
2A=π/3,A=π/6；
或,2A-π/6=5π/6,
2A=5π/6+π/6=π.A=π/2.(舍去).
∴A=π/6.
向量AB.向量AC=|AB||AC|cosA=|AB||ACcosπ/6=3.
|AB||AC|=2√3.
S△ABC=(1/2)|AB||AC|sinA.
=(1/2)*2√3*(1/2).
=√3/2.(面积单位).----即为所求.