已知函数f(x)=根号3乘cos2x+2sinxcosx 当x属于[0,2/派]求函数fx的值域

问题描述:

已知函数f(x)=根号3乘cos2x+2sinxcosx 当x属于[0,2/派]求函数fx的值域

f(x)=√3cos2x+2sinxcosx)=√3cos2x+sin2x=2sin(2x+π/3),当x∈[0,π/2]时,2x+π/3∈[π/3,4π/3],所以f(x)的值域为[-√3,2]

∵f(x)=√3*cos2x+2sinxcosx
=√3*cos2x+sin2x
=2(√3/2*cos2x+1/2*sin2x)
=2cos(2x-π/6)
又∵x∈[0,π/2]
∴2x-π/6∈[-π/6,5π/6]
∴f(x)∈[-√3,2]