MVT

The Mean Value Theorem (MVT) is a result in calculus that relates the average rate of change of a function to its instantaneous rate of change at some point within an interval. It provides a precise condition under which a function’s slope at a single point matches the slope of the secant line connecting two endpoints.

Statement and conditions: If a function f is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one c in (a, b) such that f'(c) = (f(b) − f(a)) / (b − a). The right-hand side represents the average rate of change of f over [a, b], while f'(c) is the instantaneous rate of change at c. The conclusion guarantees the existence of a point where the tangent line is parallel to the secant line through (a, f(a)) and (b, f(b)).

Implications and related results: The theorem implies that if f' is zero for all x in (a, b), then f is constant on [a, b]. It also yields Rolle’s theorem as a special case when f(a) = f(b). Various generalizations exist, including Cauchy’s Mean Value Theorem, which applies to two functions and extends the idea of comparing average and instantaneous rates.

Examples and applications: A standard example is f(x) = x² on [0, 2], where the average slope is (4 − 0) / (2 − 0) = 2 and f'(x) = 2x. At x = 1, f'(1) = 2, illustrating the theorem. MVT is widely used in analysis to bound functions, prove inequalities, and justify numerical methods that rely on rate-of-change arguments.

Notes: The theorem requires continuity on the closed interval and differentiability on the interior; it does not specify where the point c lies, only that such a point exists.