已知函数f(x)=sin²x+根号3sinxcosx+2cos²x,x∈R求函数的最小正周期和单调地增区间x∈【0,π/2】时求函数的值域x∈【0,π/2】时求函数的单调递增区间
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![已知函数f(x)=sin²x+根号3sinxcosx+2cos²x,x∈R求函数的最小正周期和单调地增区间x∈【0,π/2】时求函数的值域x∈【0,π/2】时求函数的单调递增区间](/uploads/image/z/660569-41-9.jpg?t=%E5%B7%B2%E7%9F%A5%E5%87%BD%E6%95%B0f%EF%BC%88x%EF%BC%89%3Dsin%26%23178%3Bx%2B%E6%A0%B9%E5%8F%B73sinxcosx%2B2cos%26%23178%3Bx%2Cx%E2%88%88R%E6%B1%82%E5%87%BD%E6%95%B0%E7%9A%84%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%9C%B0%E5%A2%9E%E5%8C%BA%E9%97%B4x%E2%88%88%E3%80%900%2C%CF%80%2F2%E3%80%91%E6%97%B6%E6%B1%82%E5%87%BD%E6%95%B0%E7%9A%84%E5%80%BC%E5%9F%9Fx%E2%88%88%E3%80%900%2C%CF%80%2F2%E3%80%91%E6%97%B6%E6%B1%82%E5%87%BD%E6%95%B0%E7%9A%84%E5%8D%95%E8%B0%83%E9%80%92%E5%A2%9E%E5%8C%BA%E9%97%B4)
已知函数f(x)=sin²x+根号3sinxcosx+2cos²x,x∈R求函数的最小正周期和单调地增区间x∈【0,π/2】时求函数的值域x∈【0,π/2】时求函数的单调递增区间
已知函数f(x)=sin²x+根号3sinxcosx+2cos²x,x∈R
求函数的最小正周期和单调地增区间
x∈【0,π/2】时求函数的值域
x∈【0,π/2】时求函数的单调递增区间
已知函数f(x)=sin²x+根号3sinxcosx+2cos²x,x∈R求函数的最小正周期和单调地增区间x∈【0,π/2】时求函数的值域x∈【0,π/2】时求函数的单调递增区间
f(x)=(1-cos2x)/2+√3/2sin2x+1+cos2x
=√3/2sin2x+1/2cos2x+3/2
=sin(2x+π/6)+3/2
的最小正周期 T=2π/2=π
2kπ-π/2
f(x)=sin^2x+√3sinxcosx+2cos^2x
=1+1/2+1/2*cos2x+√3/2*sin2x
=3/2+sin(2x+π/6)
(1)f(x)的最小正周期π
单调递增区间 2kπ-2π/2<=2x+π/6<=2kπ+π/2
即x属于[kπ-π/3,kπ+π/6]
(2)
x∈【0,π/2】时...
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f(x)=sin^2x+√3sinxcosx+2cos^2x
=1+1/2+1/2*cos2x+√3/2*sin2x
=3/2+sin(2x+π/6)
(1)f(x)的最小正周期π
单调递增区间 2kπ-2π/2<=2x+π/6<=2kπ+π/2
即x属于[kπ-π/3,kπ+π/6]
(2)
x∈【0,π/2】时求函数的值域
2x+π/6属于[π/2,7π/6]
sin(2x+π/6)属于[-1/2,1]
f(x)属于[1,5/2]
(3)
f(x)单调递增区间为[kπ-π/3,kπ+π/6]
x的定义域为[0,π/2]
令k=0,则可得f(x)递增区间为[0,π/6]
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f(x)=sin²x+根号3sinxcosx+2cos²x
=1/2(2cos²x-1)+1/2+(sin²x+cos²x)+√3/2(2sinxcosx)
=1/2cos2x+√3/2sin2x+3/2
=sin(2x+π/6)+3...
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f(x)=sin²x+根号3sinxcosx+2cos²x
=1/2(2cos²x-1)+1/2+(sin²x+cos²x)+√3/2(2sinxcosx)
=1/2cos2x+√3/2sin2x+3/2
=sin(2x+π/6)+3/2
所以可得:最小正周期T=2π/2=π
单调增区间为:[kπ-π/3,kπ+π/6] (注:k为任意整数)
x∈【0,π/2】时求函数的值域
当x=π/6时有最大值为:5/2
当x=π/2时有最小值为:1
所以它在此区间上的值域为[1,5/2]
x∈【0,π/2】时求函数的单调递增区间
在[0,π/6] 上单调递增,
在[π/6,π/2] 上单调递减
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