Mean Value Theorem

The Mean Value Theorem (MVT) is a foundational result in calculus. If $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$, there exists at least one point $c \in (a, b)$ such that

$f'(c) = \frac{f(b) - f(a)}{b - a}.$

Geometrically: the tangent line at $c$ is parallel to the secant line through $(a, f(a))$ and $(b, f(b))$.

Intuition (driving analogy): if you cover 60 miles in 1 hour, your average speed is 60 mph; the MVT guarantees that at some moment your instantaneous speed was exactly 60 mph.

The MVT is the engine behind:

• Increasing/decreasing test ($f' > 0 \implies$ increasing).
• The Fundamental Theorem of Calculus proof.
• Error bounds in numerical methods (Taylor's theorem with remainder).
• Uniqueness theorems for differential equations.

A special case ($f(a) = f(b)$) is Rolle's theorem: there's a $c$ where $f'(c) = 0$.