已知函数f(x)=2乘以根号3乘以sinxcosx+1-2Sin^2.x,求函数最小正周期和单调区间.
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![已知函数f(x)=2乘以根号3乘以sinxcosx+1-2Sin^2.x,求函数最小正周期和单调区间.](/uploads/image/z/3985867-19-7.jpg?t=%E5%B7%B2%E7%9F%A5%E5%87%BD%E6%95%B0f%EF%BC%88x%EF%BC%89%3D2%E4%B9%98%E4%BB%A5%E6%A0%B9%E5%8F%B73%E4%B9%98%E4%BB%A5sinxcosx%2B1-2Sin%5E2.x%2C%E6%B1%82%E5%87%BD%E6%95%B0%E6%9C%80%E5%B0%8F%E6%AD%A3%E5%91%A8%E6%9C%9F%E5%92%8C%E5%8D%95%E8%B0%83%E5%8C%BA%E9%97%B4.)
已知函数f(x)=2乘以根号3乘以sinxcosx+1-2Sin^2.x,求函数最小正周期和单调区间.
已知函数f(x)=2乘以根号3乘以sinxcosx+1-2Sin^2.x,求函数最小正周期和单调区间.
已知函数f(x)=2乘以根号3乘以sinxcosx+1-2Sin^2.x,求函数最小正周期和单调区间.
f(x)=√3sin2x+cos2x=2sin(2x+π/6).
最小正周期为π,
单调增区间是[kπ-π/3,kπ+π/6],k∈Z.
单调减区间是[kπ+π/6,kπ+2π/3],k∈Z.
f(x)=2*根号3*(sinxcosx+1-2Sin^2.x)
=2*根号3*[(sin2x)/2+cos2x]
=根号15*sin(2x+theta),其中cos(theta)=1/根号5,sin(theta)=2/根号5
所以最小正周期是pi
单调增加区间是[-pi/4-theta/2+n*pi,pi/4-theta/2+n*pi]
2sinxcosx=sin2x,cos2x=1-2Sin^2.x,f(x)=根号3乘以sin2x+cos2x=2(cosπ/6*sin2x+sinπ/6*cos2x)=2sin(2x+π/6).最小正周期为π,-π/2+2kπ<=2x+π/6<=π/2+2kπ时,递减π/2+2kπ<2x+π/6<3π/2+2kπ时,单调递减
f(x) =2sin(2x+π/6) T=2π/2=π
(-π/2,π/2)时单调增,x的范围是(-π/3,π/6)
(π/2,3π/2)时单调减,x的范围是(π/6,2π/3)
所以单调增区间(-π/3+Kπ,π/6+Kπ),单调减区间(π/6+Kπ,2π/3+Kπ).