Infinitesimal calculus MOC

Rolle’s theorem

Let 𝑓 :[𝑎,𝑏] →ℝ be continuous with 𝑓(𝑎) =𝑓(𝑏). If for every 𝑥 ∈(𝑎,𝑏) the limits 𝑓′(𝑥+) =lim𝜉→𝑥+𝑓′(𝜉) and 𝑓(𝑥−) =lim𝜉→𝑥−𝑓′(𝜉) have 𝑓′(𝑥+),𝑓(𝑥−) ∈[ −∞,∞], then there exists some 𝑐 ∈(𝑎,𝑏) such that {𝑓′(𝑐−),𝑓′(𝑐+)} contains both a positive and negative (but possibly infinite) number. anal

Proof

By the extreme value theorem 𝑓 reaches either its maximum or minimum at 𝑐 ∈(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+)=lim𝜖→0+𝑓(𝑐+𝜖)−𝑓(𝑐)ℎ∈[−∞,0]

and similarly

𝑓′(𝑐−)=lim𝜖→0−𝑓(𝑐+𝜖)−𝑓(𝑐)ℎ∈[0,∞]

as required.

In case 𝑓 is 𝐶1 differentiable, Rolle’s theorem is equivalent to saying there exists some 𝑐 ∈(𝑎,𝑏) such that 𝑓′(𝑐) =0, which itself is a special case of the mean value theorem.

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