Question

已知函数f(x)=2cos平方x加根号3sin2x加a,若x属于[0,2分之派]，且|f(x)|小于2,求实数a的取值范围？急

Answer 1

f(x)=2cos^2x+√3sin2x+a
=cos2x+1+√3sin2x+a
=2sin(2x+∏/6)+1+a
因x∈[0,∏/2],故2x+∏/6∈[∏/6,7∏/6]，sin(2x+∏/6)∈[-1/2,1]
故 f(x)∈[a,3+a]，由|a|＜2 , 且|3+a|＜2 ，解得 -2＜a＜-1

Answer 2

f(x)=√3sin2x+cos2x+1+a
=2(sin2xcosπ/6+cos2xsinπ/6)+1+a
=2sin(2x+π/6)+1+a
0<=x<=π/2
π/6<=2x+π/6<=7π/6
所以-1/2<=sin(2x+π/6)<=1
a<=f(x)<=3+a

-2<f(x)<2
所以-2<a,3+a<2
-2<a<-1