f(x)=2sinxcosx+cos^2(x)
=sin2x+(1+cos2x)
=sin2x+cos2x+1
-π/2+2kπ≤2x+π/4≤π/2+2kπ
-3π/8+kπ≤x≤π/8+kπ
所以原函数的单调增区间为：【-3π/8+kπ,π/8+kπ】

2）
原式可化为：
√2sin(2x+π/4)+1≥2
sin(2x+π/4)≥√2/2
π/4+2kπ≤2x+π/4≤3π/4+2kπ
kπ≤x≤π/4+kπ