Mean Value Theorem

Some tangent matches the secant slope: f'(c) = (f(b) - f(a))/(b - a).

The idea

Theorem (Mean Value Theorem).

Let $f$ be continuous on the closed interval $[a, b]$ and differentiable on the open interval $(a, b)$. Then there is some $c$ in $(a, b)$ with $f'(c) = \frac{f(b) - f(a)}{b - a}.$

The right-hand side is the average rate of change of $f$ across the interval, which is also the slope of the secant line joining $(a, f(a))$ to $(b, f(b))$. The left-hand side is an instantaneous rate of change, the slope of a tangent. The theorem says the average is attained exactly, at least once, somewhere inside the interval: drive sixty miles in an hour, and at some instant the speedometer read exactly sixty.

The two hypotheses are placed on different intervals. Continuity is required on the closed interval, endpoints included, so the curve cannot jump as it arrives at an end. Differentiability is required only on the open interval, because the conclusion concerns an interior point; the curve may reach an endpoint with a corner or a vertical tangent. Drop either hypothesis and the conclusion can fail.

We use the theorem less to produce a particular $c$ than to pass from information about $f'$ to information about the values of $f$: if $f' > 0$ at every interior point, the equation forces $f(b) > f(a)$, which is how a positive derivative proves a function increasing.

Ways to work on it

• Walkthrough. The statement, its geometric meaning, and a worked example.
• Proof. Prove the MVT by reducing it to Rolle's theorem with an auxiliary tilt.
• Practice. Find the guaranteed point by matching the tangent slope to the secant slope.
• Hardest. MVT applications: monotonicity, Lipschitz bounds, and when the hypotheses fail.

Not sure where to start? Take the ten-question placement test.