已知函数f(x)=sin(wx+π/3)(w>0),f(π/6)=f(π/2),且f(x)在区间(π/6,π/2)有最大值无最小值,则w

Question

已知函数f(x)=sin(wx+π/3)(w>0),f(π/6)=f(π/2),且f(x)在区间(π/6,π/2)有最大值无最小值,则w

Answer 1

已知函数f(x)=sin(wx+π/3)(w>0),f(π/6)=f(π/2)且f(x)在区间(π/6, π/2)内有最大值,无最小值,求w
解析：∵函数f(x)=sin(wx+π/3)(w>0),f(π/6)=f(π/2)且f(x)在区间(π/6, π/2)内有最大值,无最小值
则函数f(x)初相为π/3,离Y轴最近的极值点为最大值点
最大值点：wx+π/3=2kπ+π/2==>x=2kπ/w+π/(6w)
最小值点：wx+π/3=2kπ+3π/2==>x=2kπ/w+7π/(6w)
要使f(π/6)=f(π/2)且f(x)在区间(π/6,π/2)内有最大值,无最小值
须使x=π/6,x=π/2关于x=π/(6w),即x=π/3对称
令π/(6w)=π/3==>w=1/2
∴f(x)=sin(1/2x+π/3)