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Mathematics

The Mean Value Theorem and Why It Matters

Quick fact

The Mean Value Theorem was first stated by Joseph-Louis Lagrange in 1797, but a special case (Rolle's Theorem) was known much earlier. It's so central that it's used to prove many other calculus results, making it a hidden engine of the subject.

Why this is interesting

Imagine driving a car: your average speed over a trip is 60 mph. Did you ever hit exactly 60 mph? The Mean Value Theorem says yes — for any smooth journey, you must have hit that average at some instant. But why?