This can be seen from Lagrange's Mean Value Theorem.
in fact there should exist infinitely many such intervals.
Choose any interval [a,b] which satisfies continuity of the function in [a,b] and differentiability in (a,b) then you will always get a c (a,b) such that:
eg: a=6, b=7
then we have
so which of course belongs to [6,7]