已知函数y=Asin(Wx+φ)(A>0,w>0,-π/2

题目

已知函数y=Asin(Wx+φ)(A>0,w>0,-π/2

答案

1.同一个周期内,当x=π/4时,y取最大值2,当X=7π/12,y取最大值为-2
A=2
7π/12-π/4=π/3=T/2 (T为周期) T=2π/3
T=2π/w=2π/3 w=3
f(x)=2sin(3x+φ) 过（π/4,2）
sin(3π/4+φ)=1 3π/4+φ=π/2 φ=-π/4
(1) f(x)=2sin(3x-π/4)
(2) 2sin(3x-π/4)=√3
sin(3x-π/4)=√3/2
3x-π/4=2kπ+π/3或3x-π/4=2kπ+2π/3
x=2kπ/3+7π/36或x=2kπ/3+11π/36
x∈[0,2π]
所以x=7π/36,11π/36,31π/36,55π/36,35π/36,59π/36
(3) 有6个根
x1+x2=π/2
6个根之和=3π/2