A theorem stating that if a function is continuous on [a,b] and differentiable on (a,b), then there exists c in (a,b) such that f'(c) = (f(b)-f(a))/(b-a), linking the average rate of change to an instantaneous derivative.

Principle

Relate global increments to a local derivative via an existence result (often deduced from Rolle's theorem): the derivative at some interior point equals the secant slope over the interval, connecting local and global behavior.

Demonstration

For f(x)=x^2 on [1,3], the average rate (f(3)-f(1))/(3-1) = (9-1)/2 = 4, and there exists c with f'(c)=2c=4, namely c=2, showing the theorem concretely.

Misapplication

Applying the theorem when differentiability does not hold on the open interval or when endpoints are not handled correctly (e.g., assuming derivative exists at endpoints), or misusing it for nondifferentiable cusps.

Consequence

Provides error estimates, justifies Taylor expansions with remainder forms, underlies uniqueness results (if f'=0 then f constant), and supplies bounds on function increments using suprema of derivatives.

Reversal

Without differentiability the conclusion may fail, but generalized versions (Cauchy mean value theorem or Darboux-type results) can replace differentiability hypotheses or compare two functions' derivatives; conversely, the existence of such c does not determine global monotonicity alone.

Boundary

Requires continuity on the closed interval and differentiability on the open interval; does not apply to functions with interior nondifferentiable points, to maps between Banach spaces without an appropriate derivative notion, or to endpoints lacking one-sided derivatives.

Semantic Tension

Tension with Cauchy's mean value theorem and Taylor's theorem: Cauchy's form compares two functions and yields a ratio of derivatives, while Taylor gives higher-order local approximations; choosing the appropriate form depends on available smoothness and the desired estimate.

Synthesis

The mean value theorem identifies a point where the instantaneous rate equals the average rate over an interval, linking differential information to finite differences and serving as a cornerstone for estimates, uniqueness, and further generalizations.