已知函数f(x)=2√3sinxcosx+1-2sin²x,x∈R.(1)求函数f(x)的最小正周期和单调区间;(2)将函数y=f(x)的图像向左平移π/6个单位,得到的函数y=g(x)的图像,求函数y=g(x)在区间{0,π/8}上
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![已知函数f(x)=2√3sinxcosx+1-2sin²x,x∈R.(1)求函数f(x)的最小正周期和单调区间;(2)将函数y=f(x)的图像向左平移π/6个单位,得到的函数y=g(x)的图像,求函数y=g(x)在区间{0,π/8}上](/uploads/image/z/11418572-20-2.jpg?t=%E5%B7%B2%E7%9F%A5%E5%87%BD%E6%95%B0f%EF%BC%88x%EF%BC%89%3D2%E2%88%9A3sinxcosx%2B1-2sin%26%23178%3Bx%2Cx%E2%88%88R.%281%EF%BC%89%E6%B1%82%E5%87%BD%E6%95%B0f%EF%BC%88x%EF%BC%89%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%8C%BA%E9%97%B4%EF%BC%9B%EF%BC%882%EF%BC%89%E5%B0%86%E5%87%BD%E6%95%B0y%3Df%EF%BC%88x%29%E7%9A%84%E5%9B%BE%E5%83%8F%E5%90%91%E5%B7%A6%E5%B9%B3%E7%A7%BB%CF%80%EF%BC%8F6%E4%B8%AA%E5%8D%95%E4%BD%8D%2C%E5%BE%97%E5%88%B0%E7%9A%84%E5%87%BD%E6%95%B0y%3Dg%EF%BC%88x%EF%BC%89%E7%9A%84%E5%9B%BE%E5%83%8F%2C%E6%B1%82%E5%87%BD%E6%95%B0y%3Dg%28x%29%E5%9C%A8%E5%8C%BA%E9%97%B4%EF%BD%9B0%2C%CF%80%EF%BC%8F8%EF%BD%9D%E4%B8%8A)
已知函数f(x)=2√3sinxcosx+1-2sin²x,x∈R.(1)求函数f(x)的最小正周期和单调区间;(2)将函数y=f(x)的图像向左平移π/6个单位,得到的函数y=g(x)的图像,求函数y=g(x)在区间{0,π/8}上
已知函数f(x)=2√3sinxcosx+1-2sin²x,x∈R.
(1)求函数f(x)的最小正周期和单调区间;
(2)将函数y=f(x)的图像向左平移π/6个单位,得到的函数y=g(x)的图像,求函数y=g(x)在区间{0,π/8}上的最小值.
已知函数f(x)=2√3sinxcosx+1-2sin²x,x∈R.(1)求函数f(x)的最小正周期和单调区间;(2)将函数y=f(x)的图像向左平移π/6个单位,得到的函数y=g(x)的图像,求函数y=g(x)在区间{0,π/8}上
(1)
函数
f(x)=2√3sinxcosx+1-2sin²x
=√3sin2x+cos2x
=2(√3/2sin2x+1/2cos2x)
=2sin(2x+π/6)
f(x)最小正周期T=2π/2=π
由2kπ-π/2≤2x+π/6≤2kπ+π/2,k∈Z
得kπ-π/3≤x≤kπ+π/6,k∈Z
f(x)递增区间为[kπ-π/3,kπ+π/6],k∈Z
递减区间为[kπ+π/6,kπ+2π/3],k∈Z
(2)
函数y=f(x)的图像向左平移π/6个单位,
得到g(x)=f(x+π/6)=2sin[2(x+π/6)+π/6]=2sin(2x+π/2)=2cos2x
x∈[0,π/8],2x∈[0,π/4]
2x=π/4时,cos2x取得最小值√2/2
∴g(x)的最小值为√2
f(x)=2√3sinxcosx+1-2sin²x
=√3sin2x+1-(1-cos2x)
=√3sin2x+cos2x
=2sin(2x+π/6)
T=2π/2=π
(2x+π/6)在[2kπ-π/2,2kπ+π/2]是增函数
x在[kπ-π/3,kπ+π/6]是增函数
(2x+π/6)在[2kπ+π/2,2kπ+3π/2]是减...
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f(x)=2√3sinxcosx+1-2sin²x
=√3sin2x+1-(1-cos2x)
=√3sin2x+cos2x
=2sin(2x+π/6)
T=2π/2=π
(2x+π/6)在[2kπ-π/2,2kπ+π/2]是增函数
x在[kπ-π/3,kπ+π/6]是增函数
(2x+π/6)在[2kπ+π/2,2kπ+3π/2]是减函数
x在[kπ+2π/3,kπ+7π/12]是减函数
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