AP Calculus BC Practice Test

Rolle's Theorem tells us that when a function is continuous on a closed interval and differentiable on the open interval, and the endpoint values are equal, there must be an interior point where the tangent is horizontal. In other words, there exists a c in the open interval such that the derivative at c is zero. Why that must be the case: a continuous function on a closed interval attains its maximum and minimum there. If the endpoints have the same value, at least one interior point must achieve a local maximum or minimum. At any interior point where the function has a local extremum, Fermat’s theorem says the derivative must be zero; the tangent line is horizontal, so f'(c) = 0. The other possibilities—derivative positive, derivative negative, or derivative undefined—contradict either the interior extremum condition or the differentiability requirement in the interval. Since the theorem guarantees differentiability on the open interval, the derivative cannot be undefined at c, and the local extremum forces the derivative to be zero rather than strictly positive or negative. Note: the statement relies on the endpoint condition f(a) = f(b); without that, the conclusion need not hold.