f(x)＝2cos²x＋2√3sinxcosx＋1＝(2cos²x－1)＋√3×(2sinxcosx)＋2
＝cos(2x)＋√3sin(2x)＋2＝2[1/2×cos(2x)＋√3/2×sin(2x)]＋2
＝2sin(2x＋π/6)＋2
∵x∈[π/6,π/2] ∴2x∈[π/3,π] ∴2x＋π/6∈[π/2,7π/6]
∴sin(2x＋π/6)∈[﹣1/2,1] ∴f(x)∈[1,4]