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:

f'(c)=[f(b)-f(a)]/(b-a)

eg: a=6, b=7

AROC=(2.5-4)/(1)=-1.5

then we have

so which of course belongs to [6,7]