mean value theorem

On a road trip averaging 60 miles per hour over an hour, common sense says that at some single instant your speedometer must have read exactly 60. Your average speed is achieved as an instantaneous speed somewhere along the way. The mean value theorem is the rigorous form of this certainty.

Statement: if f is continuous on [a, b] and differentiable on (a, b), then there exists c in (a, b) with f'(c) = (f(b) - f(a)) / (b - a). Geometrically, somewhere the tangent line is parallel to the secant line through the endpoints. Proof: subtract off that secant to form g(x) = f(x) - [f(a) + ((f(b) - f(a))/(b - a))(x - a)]; then g(a) = g(b), so Rolle's theorem gives g'(c) = 0, which rearranges to the claim.

The mean value theorem is the workhorse of differential calculus, far more than a curiosity: it is the bridge from a derivative's local values to a function's global behavior. From it follow the facts that a function with zero derivative on an interval is constant, that two antiderivatives differ by a constant, the monotonicity tests, and crucial error estimates. As with Rolle's theorem, it only asserts that some c exists, not which one.