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
$$
f'(c) = \frac{f(b) – f(a)}{b-a}.
$$

E 4.2 Exercises