Cauchy’s theorem compares the rate of change of two functions.

Statement

For 𝑓,𝑔: continuous on [𝑎,𝑏] and differentiable on (𝑎,𝑏), there exists some 𝑐(𝑎,𝑏) such that

𝑓(𝑐)𝑔(𝑐)=𝑓(𝑏)𝑓(𝑎)𝑔(𝑏)𝑔(𝑎)

With 𝑔(𝑥)=𝑥, we get Lagrange’s mean value theorem

References