Question

求一数学三角函数题将函数f(x)=(2√3cosx-2sinx)/(5+2cos²x-2√3sinxcosx)+2的图像

Answer 1

(1) f(x)=4cos(x+π/6)/(6 + cos2x - √3sin2x )+2

=4cos(x+π/6)/(6 + 2cos(2x+π/3))+2

=2cos(x+π/6)/(3 + cos(2x+π/3))+2

(2)又f(x)的图像按向量a=（π/6,-2）平移,得到函数g(x)的图像

所以g(x)=2cosx/(3+cos2x)

易知g(x)的定义域为R,则在R上

g'(x)=(-2sinx(3+cos2x)+4cosxsin2x)/(3+cos2x)^2

=-2sinx(3+cos2x-4(cosx)^2)/(3+cos2x)^2

-4(sinx)^3/(3+cos2x)^2

易知sinx在[-π/2,0）上小于0；在（0,π/2]上是大于0,

(3+cos2x)^2恒大于0

所以在[-π/2,0）上,g'(x)>0,在（0,π/2]上g'(x)