18.已知函数f(x)=sin2x+2cos^2x-1 ,求函数f(x)在区间【π/4,3π/4]上的最大值和最小值.
问题描述:
18.已知函数f(x)=sin2x+2cos^2x-1 ,求函数f(x)在区间【π/4,3π/4]上的最大值和最小值.
答
f(x)
=sin2x+2(cosx)^2-1=sin2x+cos2x=√2[sin2xcos(π/4)+cos2xsin(π/4)]
=√2sin(2x+π/4).
∵π/4≦x≦3π/4,∴π/2≦2x≦3π/2,∴π/2+π/4≦2x+π/4≦3π/2+π/4.
∴f(x)的最大值=√2sin(π/2+π/4)=√2cos(π/4)=1.
f(x)的最小值=√2sin(3π/2)=-√2.