Rolle's Theorem

Statement: If f is continuous on [a,b], differentiable on (a,b), and f(a) = f(b), then there exists c in (a,b) where f'(c) = 0.

In English: If a function starts and ends at the same height, it must have a horizontal tangent somewhere in between.

This is a special case of MVT where the average rate of change is zero (because f(a) = f(b)).

AP usage: "Show that f'(c) = 0 for some c in (2, 5)." Verify f(2) = f(5), check continuity and differentiability, cite Rolle's Theorem.

Reference:

Wikipedia: Rolle's Theorem

https://en.wikipedia.org/wiki/Rolle%27s_theorem