f is continous on [a.b]
The problem also says f(x) not = 0 for all x in [a,b]
Therefore, f(x)<0 on [a,b] or f(x)>0 on [a,b]
Because if not, then by Intermediate value theorem there is a point such that f(x)=0 which we assumed was false.
Thus, by the inequalities of integration we have,
INTEGRAL (a,b) |f(x)|dx>0