Rolle's theorem

Let $𝑓 : [𝑎, 𝑏] \to ℝ$ be continuous with $𝑓 (𝑎) = 𝑓 (𝑏)$ . If for every $𝑥 \in (𝑎, 𝑏)$ the limits $𝑓' (𝑥 +) = l i m 𝜉 \to 𝑥 + 𝑓' (𝜉)$ and $𝑓 (𝑥 -) = l i m 𝜉 \to 𝑥 - 𝑓' (𝜉)$ have $𝑓' (𝑥 +), 𝑓 (𝑥 -) \in [- \infty, \infty]$ , then there exists some $𝑐 \in (𝑎, 𝑏)$ such that ${𝑓' (𝑐 -), 𝑓' (𝑐 +)}$ contains both a positive and negative (but possibly infinite) number. #m/thm/anal

Proof

By the extreme value theorem $𝑓$ reaches either its maximum or minimum at $𝑐 \in (𝑎, 𝑏)$ , for if both lay on the boundary then $𝑓$ would be constant. Without loss of generality assume $𝑓$ has a maximum at $𝑐$ (otherwise consider $- 𝑓$ ). Clearly

𝑓' (𝑐 +) = l i m 𝜖 \to 0 + 𝑓 (𝑐 + 𝜖) - 𝑓 (𝑐) ℎ \in [- \infty, 0]

and similarly

𝑓' (𝑐 -) = l i m 𝜖 \to 0 - 𝑓 (𝑐 + 𝜖) - 𝑓 (𝑐) ℎ \in [0, \infty]

as required.

In case $𝑓$ is $𝐶 1$ differentiable, Rolle's theorem is equivalent to saying there exists some $𝑐 \in (𝑎, 𝑏)$ such that $𝑓' (𝑐) = 0$ , which itself is a special case of the mean value theorem.

#state/tidy | #lang/en | #SemBr