Question

已知函数f(x)=5sinxcosx-5根号3cos^2x+5根号3/2

已知函数f(x)=5sinxcosx-5根号3cos^2x+(5根号3)/2
将函数y=f（x）向左平移ψ（0<ψ<π/2）个单位长度，所的图像关于y轴对称，求ψ的值

Answer 1

解：
f(x)=5sinxcosx-5√3cos^2x+(5√3)/2
=(5sin2x)/2-5√3(1+cos2x)/2+(5√3)/2
=5[sin2x*(1/2)-cos2x*(√3/2)]
=5[sin2xcos(π/3)-cos2xsin(π/3)]
=5sin(2x-π/3)
对称轴 2x-π/3=kπ+π/2
2x=kπ+5π/6
x=kπ+5π/12
所以，在y轴右侧最靠近y轴的对称轴为x=5π/12
所以 将函数y=f（x）向左平移ψ（0<ψ<π/2）个单位长度，所的图像关于y轴对称
则 ψ=5π/12

Answer 2

向左平移m个单位 f(x)=y=-2sin(x+m-π/3) 关于y轴对称 则f(-x)=f(x) 所以-2sin(-x+m-π/3)=-2sin(x+m-π/3) sin(-x+m-π/3)=sin(x+m-π/3) 所以-x+m-π/3=2kπ+x+m-π/3或-x+m-π/3=2kπ+π-(x+m-π/3) -x+m-π/3=2kπ+x+m-π/3 x=-kπ 与m无关，且不是恒等，舍去 -x+m-π/3=2kπ+π-(x+m-π/3) -x+m-π/3=2kπ+π-x-m+π/3 2m=2kπ+5π/3 m=kπ+5π/6>0 所以m最小=5π/6