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.
来源:学生作业帮助网 编辑:六六作业网 时间:2024/12/23 03:27:08
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).