已知函数f（x）=3cos2x+2cosxsinx+sin2x,求f（x）的最大值和单调增区间

f（x）= 3cos²x + 2cosxsinx + sin²x = 2cos²x + 2cosxsinx + cos²x + sin²x = 2cos²x + sin2x + 1 = 2cos²x - 1 + sin2x + 1 + 1 = cos2x + sin2x + 2 = √2 sin（2x + π/4）+ 2 最大值：√2 + 2 增区间：2kπ - π/2 ≤ 2x + π/4 ≤ 2kπ + π/2 kπ - 3π/8 ≤ x ≤ kπ + π/8 减区间：2kπ + π/2 ≤ 2x + π/4 ≤ 2kπ + 3π/2 kπ + π/8 ≤ x ≤ kπ + 5π/8