已知向量a＝(2,sinx),b＝(sin^2x,2cosx),函数f(x)a乘b.求f(x)单调区间

已知向量a＝(2,sinx),b＝(sin^2x,2cosx),函数f(x)a乘b.求f(x)单调区间
已知向量a＝(2,sinx),b＝(sin^2x,2cosx),函数f(x)a乘b.求f(x)单调区间

已知向量a＝(2,sinx),b＝(sin^2x,2cosx),函数f(x)a乘b.求f(x)单调区间
f(x)=a.b=2sin²x+2sinxcosx
=1-cos2x+sin2x
=√2[sin(2x)*cos(π/4)-cos(2x)*sin(π/4)]+1
=√2sin(2x-π/4)+1
(1)增区间
2kπ-π/2≤2x-π/4≤2kπ+π/2
2kπ-π/4≤2x≤2kπ+3π/4
kπ-π/8≤x≤kπ+3π/8
增区间【kπ-π/8,kπ+3π/8】,k∈Z
(2)减区间
2kπ+π/2≤2x-π/4≤2kπ+3π/2
2kπ+3π/4≤2x≤2kπ+7π/4
kπ+3π/8≤x≤kπ+7π/8
减区间【kπ+3π/8,kπ+7π/8】,k∈Z