已知函数f(x)=2cos²x+2根号3sinxcosx.求 (1)函数f(x)的周期.(2)函数f(x)的单调递减区间.(3)函数f(x)在区间[0,π/2]上的单调区间及最值.
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![已知函数f(x)=2cos²x+2根号3sinxcosx.求 (1)函数f(x)的周期.(2)函数f(x)的单调递减区间.(3)函数f(x)在区间[0,π/2]上的单调区间及最值.](/uploads/image/z/1108830-30-0.jpg?t=%E5%B7%B2%E7%9F%A5%E5%87%BD%E6%95%B0f%28x%29%3D2cos%26%23178%3Bx%2B2%E6%A0%B9%E5%8F%B73sinxcosx.%E6%B1%82+%EF%BC%881%EF%BC%89%E5%87%BD%E6%95%B0f%EF%BC%88x%EF%BC%89%E7%9A%84%E5%91%A8%E6%9C%9F.%EF%BC%882%EF%BC%89%E5%87%BD%E6%95%B0f%EF%BC%88x%EF%BC%89%E7%9A%84%E5%8D%95%E8%B0%83%E9%80%92%E5%87%8F%E5%8C%BA%E9%97%B4.%EF%BC%883%EF%BC%89%E5%87%BD%E6%95%B0f%EF%BC%88x%EF%BC%89%E5%9C%A8%E5%8C%BA%E9%97%B4%5B0%2C%CF%80%2F2%5D%E4%B8%8A%E7%9A%84%E5%8D%95%E8%B0%83%E5%8C%BA%E9%97%B4%E5%8F%8A%E6%9C%80%E5%80%BC.)
已知函数f(x)=2cos²x+2根号3sinxcosx.求 (1)函数f(x)的周期.(2)函数f(x)的单调递减区间.(3)函数f(x)在区间[0,π/2]上的单调区间及最值.
已知函数f(x)=2cos²x+2根号3sinxcosx.求 (1)函数f(x)的周期.(2)函数f(x)的单调递减区间.
(3)函数f(x)在区间[0,π/2]上的单调区间及最值.
已知函数f(x)=2cos²x+2根号3sinxcosx.求 (1)函数f(x)的周期.(2)函数f(x)的单调递减区间.(3)函数f(x)在区间[0,π/2]上的单调区间及最值.
f(x)=2cos²x+2根号3sinxcosx
= (cos2x+1) + √3sin2x
= √3sin2x + cos2x + 1
= 2(sin2xcosπ/6+cos2xsinπ/6) +1
= 2sin(2x+π/6) + 1
最小正周期 = 2π/2 = π
2x+π/6∈(2kπ+π/2,2kπ+3π/2),其中k∈Z时单调减
所以单调减区间:x∈(kπ+π/6,kπ+2π/3),其中k∈Z
x∈[0,π/2]时:
2x+π/6∈【π/3,4π/3】
其中,2x+π/6∈【π/3,π/2)时单调增,2x+π/6∈【π/2,4π/3)时单调减
∴单调增区间:x∈∈【π/12,π/6),单调减区间x∈【π/6,7π/12)
2x+π/6=π/2时,最大值f(x)max = 2+1 = 3
2x+π/6=4π/3时,最小值f(x)min = 2*(-√3/2)+1 = 1-√3
f(x)=1-cos2x+√3sin2x+1
=2+2sin(2x-π/6)
最小正周期是π,递增区间是2kπ-π/2≤2x-π/6≤2kπ+π/2,即增区间是[kπ-π/6,kπ+π/3],其中k是整数。
f(x)在区间[0,π/2]的最大值是x=π/3时取得的,是3,最小值是x=0时取得是,是1。