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Understanding Rolle's Theorem: A Key Concept in Calculus

Rolle's Theorem is a fundamental principle in calculus, named after the French mathematician Michel Rolle, who published it in 1691. This theorem lays a crucial foundation for further exploration of mathematical analysis and is essential in proving the Mean Value Theorem, which follows from its concepts.

The Conditions of Rolle's Theorem

For a function to satisfy the conditions of Rolle's Theorem, three primary criteria must be met:

Continuity: The function ( f(x) ) must be continuous over the closed interval ([a, b]). This means that the endpoints ( a ) and ( b ) are included in this interval, and there are no interruptions in the graph within this range.