a,b,c成等比数列,∴a/b = b/c,b^2=ac
根据余弦定理,b^2=a^2+c^2-2accosB
cosB = (a^2+c^2-b^2)/(2ac) = (a^2+c^2-ac)/(2ac) = (a^2+c^2)/(2ac) -1/2 ≥ 1-1/2 = 1/2
cosB≥1/2
0 < B ≤ 60°
b^2 = a^2+c^2-2accosB ≤ a^2+c^2-ac = a^2+c^2-b^2
b^2 ≤ (a^2+c^2)/2
三角形ABC周长是6,∴a+b+c=6
a+c=6-b
a^2+c^2+2ac=36-12b+b^2
a^2+c^2+2b^2=36-12b+b^2
a^2+c^2=36-12b-b^2
b^2 ≤ (a^2+c^2)/2 = (36-12b-b^2)/2
2b^2 ≤ 36-12b-b^2
3b^2+12b-36≤0
b^2+4b-12≤0
(b+6)(b-2)≤0
b+6>0
b-2≤0
b≤2
角B最大值60°,b边最大值2