Rolle’s Theorem | A Fundamental Principle in Calculus with Applications in Optimization and Critical Point Analysis

Rolle’s Theorem

Rolle’s Theorem is a mathematical theorem that deals with differentiable functions on a closed interval

Rolle’s Theorem is a mathematical theorem that deals with differentiable functions on a closed interval. It states that if a function f(x) is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), and if f(a) = f(b), then there exists at least one value c in the open interval (a, b) such that f'(c) = 0.

In simpler terms, Rolle’s Theorem essentially asserts that if a function is continuous on a closed interval and has the same values at its endpoints, then there must exist a point within the interval where the derivative of the function is zero.

Visually, this theorem implies that for a curve that starts and ends at the same height, there must be at least one point where the curve is flat or horizontal (where the derivative is zero) somewhere in between its endpoints.

Rolle’s Theorem is a special case of the more general Mean Value Theorem. It is named after the French mathematician Michel Rolle who first formulated it in the late 17th century.

This theorem has important applications in various fields of calculus, especially in optimization problems, where it can be used to prove the existence of critical points (where the derivative is zero) within a given interval. It also serves as an intermediate step in the proof of many core theorems in calculus.

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