# 4.2 | Rolle’s Theorem and the Mean Value Theorem

Rolle’s Theorem
Let $$f$$ be continuous on the closed interval $$[a,b]$$ and differentiable on the open interval $$(a,b)$$. If $$f(a) = f(b)$$, then there is at least one number $$c$$ in $$(a,b)$$ such that $$f'(c) = 0$$.
The Mean Value Theorem
If $$f$$ is continuous on the closed interval $$[a,b]$$ and differentiable on the open interval $$(a,b)$$, then there exists a number $$c$$ in $$(a,b)$$ such that
$$f'(c) = \frac{f(b) – f(a)}{b-a}.$$