已知函数f(x)=2sin(π-x)cosx+2sin²(3π/2-x)-1(1)求函数f(x)的最小正周期和单调递增区间(2)求函数f(x)在区间[π/4,3π/4]上的最大值和最小值
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![已知函数f(x)=2sin(π-x)cosx+2sin²(3π/2-x)-1(1)求函数f(x)的最小正周期和单调递增区间(2)求函数f(x)在区间[π/4,3π/4]上的最大值和最小值](/uploads/image/z/2556425-65-5.jpg?t=%E5%B7%B2%E7%9F%A5%E5%87%BD%E6%95%B0f%EF%BC%88x%EF%BC%89%3D2sin%EF%BC%88%CF%80-x%EF%BC%89cosx%2B2sin%26%23178%3B%EF%BC%883%CF%80%2F2-x%EF%BC%89-1%EF%BC%881%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%E9%80%92%E5%A2%9E%E5%8C%BA%E9%97%B4%EF%BC%882%EF%BC%89%E6%B1%82%E5%87%BD%E6%95%B0f%EF%BC%88x%EF%BC%89%E5%9C%A8%E5%8C%BA%E9%97%B4%5B%CF%80%2F4%2C3%CF%80%2F4%5D%E4%B8%8A%E7%9A%84%E6%9C%80%E5%A4%A7%E5%80%BC%E5%92%8C%E6%9C%80%E5%B0%8F%E5%80%BC)
已知函数f(x)=2sin(π-x)cosx+2sin²(3π/2-x)-1(1)求函数f(x)的最小正周期和单调递增区间(2)求函数f(x)在区间[π/4,3π/4]上的最大值和最小值
已知函数f(x)=2sin(π-x)cosx+2sin²(3π/2-x)-1
(1)求函数f(x)的最小正周期和单调递增区间
(2)求函数f(x)在区间[π/4,3π/4]上的最大值和最小值
已知函数f(x)=2sin(π-x)cosx+2sin²(3π/2-x)-1(1)求函数f(x)的最小正周期和单调递增区间(2)求函数f(x)在区间[π/4,3π/4]上的最大值和最小值
由诱导公式,f(x)=2sinxcosx+2cos²x-1
由倍角公式:f(x)=sin2x+cos2x
由辅助角公式:f(x)=√2sin(2x+π/4)
(1)T=2π/2=π
递增区间:-π/2+2kπ
f(x)=2sin(π-x)cosx+2sin²(3π/2-x)-1
=2sinxcosx+2cos²x-1
=sin2x+cos2x
=√2sin(2x+π/4)
(1)
f(x)的最小正周期是T=2π/2=π
令2kπ-π/2<2x+π/4<2kπ+π/2,k∈Z
2kπ-3π/4<2x<2kπ+π/4,k...
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f(x)=2sin(π-x)cosx+2sin²(3π/2-x)-1
=2sinxcosx+2cos²x-1
=sin2x+cos2x
=√2sin(2x+π/4)
(1)
f(x)的最小正周期是T=2π/2=π
令2kπ-π/2<2x+π/4<2kπ+π/2,k∈Z
2kπ-3π/4<2x<2kπ+π/4,k∈Z
kπ-3π/8
(2)
x∈[π/4,3π/4]
2x∈[π/2,3π/2]
2x+π/4∈[3π/4,7π/4]
sin(2x+π/4)∈[-1,√2/2]
√2sin(2x+π/4)∈[-√2,1]所以最大值是1,最小值是-√2
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