已知函数f(x)=2cosxsin(x+π/6)-sin²x+cos²x,
来源:学生作业帮 编辑:作业帮 分类:数学作业 时间:2024/07/19 06:24:04
已知函数f(x)=2cosxsin(x+π/6)-sin²x+cos²x,
(1)求函数f(x)的单调递增区间;
(2)当x∈[-π/12,π/6],求函数f(x)的最大值、最小值及相应的x值
(1)求函数f(x)的单调递增区间;
(2)当x∈[-π/12,π/6],求函数f(x)的最大值、最小值及相应的x值
2cosxsin(x+π/6)-sin²x+cos²x
=2cosx﹙sinxcosπ/6+cosxsinπ/6﹚+cos2x
=√3cosxsinx+cos²x+cos2x
=√3sin2x/2+﹙cos2x+1﹚/2+cos2x
=√3sin2x/2+3cos2x/2+1/2
=√3(sin2x/2+√3cos2x/2﹚+1/2
=√3(sin﹙2x+π/3﹚+1/2
∵2x+π/3∈[2kπ-π/2,2kπ+π/2],k∈Z时,f(x)递增
∴函数f(x)的单调递增区间是[kπ-5π/12,kπ+π/12],k∈Z
当x∈[-π/12,π/6],2x+π/3∈[π/6,2π/3]
∴当2x+π/3=π/2即x=π/12时,f(x)的最大值是√3+1/2;
当2x+π/3=π/6即x=﹣π/12时,f(x)的最小值是√3/2+1/2.
=2cosx﹙sinxcosπ/6+cosxsinπ/6﹚+cos2x
=√3cosxsinx+cos²x+cos2x
=√3sin2x/2+﹙cos2x+1﹚/2+cos2x
=√3sin2x/2+3cos2x/2+1/2
=√3(sin2x/2+√3cos2x/2﹚+1/2
=√3(sin﹙2x+π/3﹚+1/2
∵2x+π/3∈[2kπ-π/2,2kπ+π/2],k∈Z时,f(x)递增
∴函数f(x)的单调递增区间是[kπ-5π/12,kπ+π/12],k∈Z
当x∈[-π/12,π/6],2x+π/3∈[π/6,2π/3]
∴当2x+π/3=π/2即x=π/12时,f(x)的最大值是√3+1/2;
当2x+π/3=π/6即x=﹣π/12时,f(x)的最小值是√3/2+1/2.
已知函数f(x)=2cosxsin(x+ π/3 )- √3 sin²x+sinxcosx
已知函数f(x)=cos(2x-π\3)+sin²x-cos²x
已知函数f(x)=2cosxsin(x+π/3)-√3sin²x+sinxcosx,求函数f(x)的最大值、最
已知函数f(x)=sin(2x+π/6)+sin(2x+π/6)+2cos²x
已知函数f(x)=2cosxsin(x+π/3)-根号3sin^2x+sinxcosx+2(x∈R),
已知函数f(x)=2cosxsin(x+60°)-根号3sin^2x+sinxcosx
函数f(x)=2cosxsin(x+π/3)-根号3sin²x+sinxcosx
已知函数f(x)=sin²(π/4+x)+cos²x+1/2求最值
已知函数f(x)=2cosxsin(x+兀/6)+cos^4x-sin^4x.求f(x)的最小正周期,若x属于[-兀/1
已知函数f(x)=cos(2x-π/3)+sin^2 x-cos^2 x
已知函数f(x)=2cosxsin(x+π/3)-根号3sin^2x+sinxcosx,求方程f(x)=x/50π的个数
已知函数f(x)=2cosxsin(x+π/3)-根号3sin^2x+sinxcosx 当x∈[0,π/4]时,f(x)