已知函数f(x)=2asin²x-2根号3倍asinxcosx+b的定义域[0,π／2],值域为[-5,4],求实数a,b

题目

已知函数f(x)=2asin²x-2根号3倍asinxcosx+b的定义域[0,π／2],值域为[-5,4],求实数a,b

答案

2sin^2x=1-cos2x
2sinxcosx=sin2x
f(x)=a-acos2x-a√3sin2x+b
=a+b-2a(cos2xcosπ/3+sin2xsinπ/3)
=a+b-2acos(2x-π/3)
-5