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.

Therefore,

|f(x)|>0

Thus, by the inequalities of integration we have,

INTEGRAL (a,b) |f(x)|dx>0