good. that was what i was looking for.

 

If _a^b f(x) = 0, then by mean value theorem, there exists c(a,b) so that f'(c) = 0

 

Another way to understand is that if _a^b f(x) = 0, then f(x) neither be +ve for all x in the interval nor -ve for all x. So there's got to be x in the interval for which it is zero.