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

Statement

For 𝑓,𝑔:ℝ→ℝ 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