f(x)=2cos²x+2根號(hào)3sinxcosx
= (cos2x+1) + √3sin2x
= √3sin2x + cos2x + 1
= 2(sin2xcosπ/6+cos2xsinπ/6) +1
= 2sin(2x+π/6) + 1
最小正周期 = 2π/2 = π
2x+π/6∈(2kπ+π/2,2kπ+3π/2),其中k∈Z時(shí)單調(diào)減
所以單調(diào)減區(qū)間：x∈（kπ+π/6,kπ+2π/3),其中k∈Z
x∈[0,π/2]時(shí)：
2x+π/6∈【π/3,4π/3】
其中,2x+π/6∈【π/3,π/2）時(shí)單調(diào)增,2x+π/6∈【π/2,4π/3）時(shí)單調(diào)減
∴單調(diào)增區(qū)間：x∈∈【π/12,π/6）,單調(diào)減區(qū)間x∈【π/6,7π/12）
2x+π/6=π/2時(shí),最大值f(x)max = 2+1 = 3
2x+π/6=4π/3時(shí),最小值f(x)min = 2*(-√3/2)+1 = 1-√3