7788DD
幼苗
共回答了25个问题采纳率:84% 举报
(1)
f(x)=2cosxsin(x+π/3)-√3sin²x+sinxcosx
=2cosx[sinx/2+√3cosx/2]-√3sin²x+sinxcosx
=sinxcosx+√3cos²x-√3sin²x+sinxcosx
=2sinxcosx+√3(cos²x-sin²x)
=sin(2x)+√3cos(2x)
=2[(1/2)sin(2x)+(√3/2)cos(2x)]
=2sin(2x+π/3)
最小正周期:Tmin=2π/2=π
(2)
当2kπ-π/2≤2x+π/3≤2kπ+π/2 (k∈Z)时,函数单调递增.此时kπ-5π/12≤x≤kπ+π/12
当2kπ+π/2≤2x+π/3≤2kπ+3π/2 (k∈Z)时,函数单调递减.此时kπ+π/12≤x≤kπ+7π/12
函数的单调递增区间为[kπ-5π/12,kπ+π/12];单调递减区间为[kπ+π/12,kπ+7π/12] (k∈Z)
(3)
x∈[0,π/4],π/3≤2x+π/3≤5π/6
2x+π/3=5π/6时,有f(x)min=1
2x+π/3=π/2时,有f(x)max=2
函数的值域为[1,2]
1年前
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