已知函数f(x)=sin(wx+π/3) (w>0) 若f(π/6)=f(π/3),且f(x)在区间(π/6,π/3)内有最大值,无最小值,则w=

Question

已知函数f(x)=sin(wx+π/3) (w>0) 若f(π/6)=f(π/3),且f(x)在区间(π/6,π/3)内有最大值,无最小值,则w=

Answer 1

f(π/6)=f(π/3),说明函数图像关于直线x=(π/6+π/3)/2(即x=π/4)对称.
f(x)在区间(π/6,π/3)内有最大值,无最小值,所以x=π/4时取到最大值.
且知函数周期大于π/3-π/6=π/6.
x=π/4时取到最大值,则wπ/4+π/3=2kπ+π/2,w=8k+2/3.k∈Z.
又周期为2π/w>π/6,0故k=0时,w=2/3或k=1时,w=26/3适合题意.