Question

已知f（x）=根号2cos （x/2）-（a-1）sin（x/2）为偶函数

Answer 1

∵f(x)是偶函数,∴f(-x)=f(x)

即√2cos(-x/2)-(a-1)sin(-x/2)=√2cos(x/2)-(a-1)sin(x/2)

即√2cos(x/2)+(a-1)sin(x/2)=√2cos(x/2)-(a-1)sin(x/2)

∴2(a-1)sin(x/2)=0,∴a=1

f(x)=√2cos(x/2),所以最小正周期为T=2π/(1/2)=4π