Statement

Lagrange’s MVT states that if 𝑓:ℝ→ℝ is continuous on [𝑎,𝑏] and differentiable on (𝑎,𝑏), then, for some 𝑐∈(𝑎,𝑏),

𝑓′(𝑐)=𝑓(𝑏)−𝑓(𝑎)𝑏−𝑎

and (appropriately)

min(𝑎,𝑏)𝑓′≤𝑓′(𝑐)≤max(𝑎,𝑏)𝑓′

A more general version of Lagrange’s MVT is Cauchy’s mean value theorem.

References