f(x)=sin(2x+π/6)+sin(2x-π/6)+cos2x+a
= sin2xcosπ/6+cos2xsinπ/6 + sin2xcosπ/6-cos2xsinπ/6 + cos2x + a
= 2sin2xcosπ/6 + cos2x + a
= √3sin2x+cos2x+a
= 2(sin2xcosπ/6+cos2xsinπ/6) + a
= 2sin(2x+π/6) + a
最小正周期 = 2π/2 = π
2x+π/6∈（2kπ-π/2,2kπ+π/2）時(shí)單調(diào)增,∴單調(diào)遞增區(qū)間（kπ-π/3,kπ+π/6）,其中k∈Z
最大值2+a,此時(shí)2x+π/6=2kπ+π/2,即x=kπ+π/6,其中k∈Z
x∈[0,π/2]時(shí),2x+π/6∈[-π/6,5π/6],2x+π/6=-π/6時(shí)最小值2sin(-π/6)+a=-1+a=2,a=3