Rolle's Theorem

analysis

Definition

Rolle's Theorem

Let $f:[a,b]\to \mathbb{R}$ be a continuous function on the closed interval $[a,b]$ and differentiable on the open interval $(a,b)$. If $f(a)=f(b)$, then there exists at least one point $\xi \in (a,b)$ such that the derivative of $f$ at that point is zero:

$$f^{\prime }(\xi )=0$$

Geometrically, this means there is a point between $a$ and $b$ where the tangent to the curve is horizontal.

Relation to Other Theorems

Rolle’s theorem is a special case of the Mean Value Theorem. While the Mean Value Theorem guarantees a point where the instantaneous rate of change equals the average rate of change, Rolle’s theorem specifically addresses the case where the average rate of change is zero.