已知函数f(x)=3sin^2x /2+2倍根号3sinx/2cosx/2+cos^2x/2
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解决时间 2021-03-17 02:43
- 提问者网友：
- 2021-03-16 19:58
已知函数f(x)=3sin^2x /2+2倍根号3sinx/2cosx/2+cos^2x/2
最佳答案
- 五星知识达人网友：山河有幸埋战骨
- 2021-03-16 20:21
f(x)=3sin²(x/2)+2√3sin(x/2)cos(x/2)+cos²(x/2)
=2sin²(x/2)+√3[2sin(x/2)cos(x/2)]+[sin²(x/2)+cos²(x/2)]
=1-cosx +√3sinx +1
=√3sinx-cosx+2
=2[(√3/2)sinx-(1/2)cosx]+2
=2[sinxcos(π/6)-cosxsin(π/6)]+2
=2sin(x-π/6)
最小正周期T=2π/1=2π
2kπ-π/2≤x-π/6≤2kπ+π/2 (k∈Z)时，f(x)单调递增，此时2kπ-π/3≤x≤2kπ+2π/3 (k∈Z)
函数的单调递增区间为[2kπ-π/3，2kπ+2π/3] (k∈Z)
=2sin²(x/2)+√3[2sin(x/2)cos(x/2)]+[sin²(x/2)+cos²(x/2)]
=1-cosx +√3sinx +1
=√3sinx-cosx+2
=2[(√3/2)sinx-(1/2)cosx]+2
=2[sinxcos(π/6)-cosxsin(π/6)]+2
=2sin(x-π/6)
最小正周期T=2π/1=2π
2kπ-π/2≤x-π/6≤2kπ+π/2 (k∈Z)时，f(x)单调递增，此时2kπ-π/3≤x≤2kπ+2π/3 (k∈Z)
函数的单调递增区间为[2kπ-π/3，2kπ+2π/3] (k∈Z)
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