Rolle's Theorem and Mean Value Theorem

Rolle's Theorem

• For a function $f$ continuous on $[a, b]$ and differentiable on $(a, b)$, with $f(a) = f(b)$, there exists a point $c$ in $(a, b)$ such that $f'(c) = 0$.

Mean Value Theorem

• Extends Rolle's Theorem by considering the average change of a function across an interval where the endpoints are not necessarily equal.
• Guarantees a point where the derivative matches the average change.