已知f(x)=sin²x+根号3sinxcosx+2cos²x,x∈R,求f(x)的最小正周期和它的单调增区间

问题描述:

已知f(x)=sin²x+根号3sinxcosx+2cos²x,x∈R,求f(x)的最小正周期和它的单调增区间

f(x) = sin²x + √3sinxcosx + 2cos²x
= (1 - cos2x)/2 + (√3/2)sin2x + (1 + cos2x)
= 3/2 + (√3/2)sin2x + (1/2)cos2x
= 3/2 + sin(2x + π/6)
蕞小正周期T = 2π/2 = π
min:1/2,max:5/2
2kπ - π/2 ≤ 2x + π/6 ≤ 2kπ + π/2
==> kπ - π/3 ≤ x ≤ kπ + π/6
单调增区间:x∈[kπ - π/3,kπ + π/6]