Rolle’s theorem

Definition

Rolle's Theorem states that if a function $f$ is continuous on the closed interval $[a, b]$, differentiable on the open interval $(a, b)$, and $f(a) = f(b)$, then there exists at least one number $c$ in $(a, b)$ such that $f'(c) = 0$.

5 Must Know Facts For Your Next Test

1. Rolle's Theorem requires three conditions: continuity on $[a, b]$, differentiability on $(a, b)$, and equal function values at the endpoints ($f(a) = f(b)$).
2. If any of the conditions for Rolle's Theorem are not met, the theorem does not apply.
3. The theorem guarantees at least one value $c$ in $(a,b)$ where the derivative of the function is zero ($f'(c) = 0$).
4. Rolle’s Theorem is a special case of the Mean Value Theorem where the average rate of change between two points is zero.
5. Graphically, Rolle's Theorem implies that there must be a horizontal tangent line at some point within the interval.

Review Questions

• What are the three conditions required for Rolle’s Theorem to hold?
• Why is it necessary for a function to be continuous on $[a,b]$ and differentiable on $(a,b)$ for Rolle’s Theorem?
• How can you interpret Rolle’s Theorem graphically?

Related terms

Mean Value Theorem:

If a function is continuous on $[a,b]$ and differentiable on $(a,b)$, then there exists some number $c$ in $(a,b)$ such that $$f'(c) = \frac{f(b) - f(a)}{b - a}$$.

Critical Point: A point in the domain of a function where its derivative is zero or undefined.

Continuous Function: A function without breaks or jumps; formally, it means that $\lim_{{x \to c}} f(x) = f(c)$ for all points in its domain.