Show that if the integral of a function over [a,b] is zero the the function is zero.
Would you use something like the mean value theorem to show this? It looks like the answer should be simple.
This is not true - more conditions are needed:
1) f needs to be continuous
2) f needs be either nonnegative or nonpositive, otherwise a simple counterexample is $\displaystyle \displaystyle \int_0^{2 \pi} sinxdx$
And then what I would do is assume toward contradiction that the function is not identically zero, use continuity to show that its not zero on an interval and thus the integral itself is not zero.