f(x)=3(cosx)^3+2sinxcosx+(sinx)^2
=sin2x+2(cosx)^2+1
=sin2x+cos2x+2
=√2sin(2x+π/4)+2

(1)最小正周期为T=2π/2=π,周期为kπ,k是不为0的整数.

(2)2kπ-π/2<2x+π/4<2kπ+π/2,则kπ-3π/82kπ+π/2<2x+π/4<2kπ+3π/2,则kπ+π/8
(3)当2x+π/4=2kπ+π/2,即x=kπ+π/8时,f(x)取得最大值为√2+2.