Rolle's theorem states that if $f(x)$ is continuous on $[a, b]$, differentiable on $(a, b)$, and $f(a) = f(b)$, then there exists at least one $c \in (a, b)$ such that:
Options:
(A) $f'(c) = 0$
(B) $f''(c) = 0$
(C) $f(c) = 0$
(D) $f'(c) = \frac{f(b) - f(a)}{b - a} = 1$
Answer Key:
A
Step-by-Step Solution & Explanation
Rolle's theorem guarantees a point of horizontal tangent $f'(c) = 0$ in the open interval $(a, b)$.