f(x) has three solutions meaning that one root x belongs to (0,pi/6) and the other (pi/6 , pi/2)
So basically in the expression
x^2 + 2(1-a)x + a-1=0 one root lies between (0,1/2) and the other lies between (1/2,1)
So basically the roots lie between (0,1) and that f(1/2)<0 .
Is there doubt in further solution ? @pratyaksh_tyagi