一只函数f(x)=根号3sin2x+2cos²x=m在区间[0,2分之π]上的最大值为6

1,f(x)=√3sin2x+2cos²x+m
=√3sin2x+1+cos2x+m
=2sin(2x+π/6)+m+1
∵0≤x≤π/2 ∴π/6≤2x+π/6≤7π/6
那么f(x)max=2+m+1=6,m=3
令2x+π/6=(2k+1)π,那么x=kπ+5π/12
那么对称中心为(kπ+5π/12,4) (k∈Z)
2,设点(x,y)在f1(x)上,那么点(-x,y)在f(x)上
那么y=f1(x)=f(-x)=2sin(-2x+π/6)+4=-2sin(2x-π/6)+4
那么f2(x)=-2sin[2(x-π/4)-π/6]+4=-2sin(2x-2π/3)+4
令2kπ-π/2≤2x-2π/3≤2kπ+π/2
kπ+π/12≤x≤kπ+7π/12
那么函数f2(x)的单调递减区间为[kπ+π/12,kπ+7π/12] (k∈Z)

f(x)=√3sin2x+2cos²x+m
=√3sin2x+cos2x+1+m
=2sin(2x+π/6)+m
x∈[0,π/2]
2x+π/6∈[π/6,7π/6]
f(x)的最大值为 2+m=6
得 m=4
所以
f(x)=2sin(2x+π/6)+4
对称中心：
2x+π/6=kπ...