Rolle's theorem is one of the foundational theorems in differential calculus. It is a special case of, and in fact is equivalent to, the mean value theorem, which in turn is an essential ingredient in the proof of the fundamental theorem of calculus. Rolle's theorem provides valuable insights into the behavior of differentiable functions and is widely used in calculus, analysis, and various applications in mathematics and science. Rolle’s theorem is a variation or a case of Lagrange’s mean value theorem. The mean value theorem follows two conditions, while Rolle’s theorem follows three conditions. This topic will help you understand Rolle’s theorem, its geometrical interpretation, and how it is different from the mean value theorem. We will also study numerical examples related to Rolle’s theorem. What Is Rolle’s Theorem? Rolle’s Theorem is a theorem stating that if a continuous function attains two ... Rolle’s theorem, in analysis, special case of the mean-value theorem of differential calculus. Rolle’s theorem states that if a function f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b) such that f(a) = f(b), then f′(x) = 0 for some x with a ≤ x ≤ b.
