This is the exact, classical statement of Rolle's Theorem:
If a real-valued function $f$ 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 $f'(c) = 0$.

Detailed analysis of options:

• Lagrange's Mean Value Theorem states that $f'(c) = \frac{f(b) - f(a)}{b - a}$ (without requiring $f(a) = f(b)$).
• Rolle's Theorem is the specific case where $f(a) = f(b)$, implying $f'(c) = 0$.
• Fundamental Theorem of Calculus relates differentiation and integration.
• Fundamental Theorem of Algebra states that every non-zero single-variable polynomial with complex coefficients has at least one complex root.