mean value theorem

Suppose you drive 120 miles in 2 hours. Your average speed was 60 mph, even if you sped up and slowed down along the way. The mean value theorem says something stronger and almost obvious: at some instant during the trip, your speedometer must have read exactly 60. Your instantaneous speed had to match your average speed at least once, because you cannot stay entirely above or entirely below your own average the whole time.

Formally: if f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one point c in (a, b) where f'(c) = (f(b) - f(a)) / (b - a). The right side is the average rate of change — the slope of the straight line joining the two endpoints — and f'(c) is the instantaneous rate of change. The theorem promises a point where the tangent line is parallel to that connecting line.

Both conditions earn their keep. Continuity on the whole closed interval and differentiability inside cannot be dropped: the function f(x) = |x| on [-1, 1] has average slope 0, yet its derivative is never 0 because of the corner at x = 0, where f is not differentiable. The theorem is the quiet workhorse behind many results — for instance, it is how we prove that a function with derivative 0 everywhere on an interval must be constant.