已知函数f(x)=cox(2x-π/3)+2sin(x-π/4)sin(x+π/4)(1)求函数f(x)的最小正周期和图像的对称轴方程（2）求函数f(x)在区间[-π/12，π/2]上的值域(3)求函数f(x)在区间[0,π]上的增区间

Question

已知函数f(x)=cox(2x-π/3)+2sin(x-π/4)sin(x+π/4)
(1)求函数f(x)的最小正周期和图像的对称轴方程（2）求函数f(x)在区间[-π/12，π/2]上的值域(3)求函数f(x)在区间 [0,π]上的增区间

Answer 1

f(x)=cos(2x-π/3)+2sin(x-π/4)cos(π/2-x-π/4)
=cos(2x-π/3)+2sin(x-π/4)cos(x-π/4)
=cos(2x-π/3)+sin(2x-π/2)
=cos(2x-π/3)-cos2x
=-2sin(2x-π/3)sin(-π/6)
=sin(2x-π/6)
f(x)=sin(2x-π/6)
1.最小正周期=2π/2=π
对称轴2x-π/6=kπ+π/2 x=kπ/2 +π/3 k∈Z
2.x=-π/12时, f(x)min=-√3 /2
x=π/3时, f(x)max=1
值域[-√3 /2,1]
3.增区间[0,π/3]