题目

已知函数f(x)=Asin(wx+φ)(其中A>0,w>0,-π/2

提问时间：2020-12-06

答案

(1)
不妨取ωx₁+φ = -π/2,ωx₂+φ = π/2 (-π/2,π/2为相邻对称轴)
ωx₂+φ - (ωx₁+φ) = ω(x₂ - x₁) = ωπ/2 = π/2 - (-π/2) = π
ω = 2
最高点y = 4,A = 4
x = 5π/12,y取最大值,ω(5π/12) + φ = π/2,5π/6 + φ = π/2,φ = -π/3
f(x) = 4sin(2x - π/3)
f(x)周期为π
x = 5π/12 + nπ时,y取最大值
x = -π/12 + nπ时,y取最小值
x∈(-π/12 + nπ,5π/12 + nπ),单调增
x∈(5π/12 + nπ,11π/12 + nπ),单调减
(2)
5π/12在[π/4,π/2]内,函数f(x)的最大值=4
f(π/4) = 4sin(π/2 - π/3) = 4sin(π/6) = 2
f(π/2) = 4sin(π - π/3) = 2√3 > f(π/4)
最小值2