cosA+2cos[(B+C)/2]
=cosA+2sin(A/2)
=1-2sin^2 (A/2)+2sin(A/2)
=-2[sin(A/2)-1/2]^2+3/2≤3/2
当且仅当sin(A/2)=1/2,A=π/3时取得
故A=π/3时,cosA+2cos[(B+C)/2]取得最大值3/2