已知OA上向右箭头=(sin3分之x,根号3cos3分之x),OB上向右箭头=(cos3分之x,cos3分之x)(x属于R),f(x)=OA上...已知OA上向右箭头=(sin3分之x,根号3cos3分之x),OB上向右箭头=(cos3分之x,cos3分之x)(x属于R),f(x)=OA上向右箭头乘OB向右箭头,若x属于(0,3分之派],求函数f(x)的值域?急

x∈(0,派/3),则(2x+派)/3∈（派/3,5派/9),sin(2x+派)/3∈(√3/2,1] f(x)=OA向量•OB向量=sin(x/3)×cosx/3+√3cos(x/3)cosx/3
=[sin(2x/3)]/2 +√3(1+cos2x/3)/2 =sin(2x+派)/3 +√3/2∈(√3,√3/2+1]

f(x)=sin(x/3)cos(x/3)+(√3)cos²(x/3)
=sin(2x/3+π/3)+sin(π/3)
∵ 0∴ π/3∴ √3/2f(x)的值域为(√3，1+√3/2)

由题意：f(x)=sin(x/3)cos(x/3) + √3[cos(x/3)]^2 =sin(2x/3 + π/3) + (√3)/2
∵x属于(0,3分之派]
∴sin(2x/3 + π/3)可以取到最大值1
f(x)的值域(√3 ,1+(√3/2)]