Rolle's theorem

Rolle's theorem states that a function continuous on a closed interval and differentiable in its interior, with equal endpoint values, has an interior point where its derivative is zero.

Extreme value theorem: A continuous function on a closed, bounded interval attains both a maximum and a minimum. It ensures the function reaches extrema that can force a stationary point.
Maximum and minimum: The greatest and least values attained by a function on a specified domain. If the function is not constant, one of its extrema differs from the shared endpoint value.
Critical point: A point in a function's domain where its derivative is zero or undefined. Rolle's theorem guarantees a critical point where the derivative exists and vanishes.
Michel Rolle: A French mathematician whose work included the theorem now bearing his name. His name became attached to the result, which he presented in an algebraic setting.
Fermat's theorem on stationary points: A differentiable function has derivative zero at any local extremum in the interior of its domain. It turns an interior extremum into the zero derivative required by the conclusion.
Interior point: A point lying inside a set rather than on its boundary. A nonconstant function with equal endpoint values must attain a differing extremum in the interval's interior.
Darboux's theorem: A theorem stating that derivatives have the intermediate value property, even when they are discontinuous. Rolle's theorem supplies a key step in standard proofs of this property.
Traité d'algèbre: Michel Rolle's 1690 algebra textbook, which presented methods for solving algebraic equations. Rolle's theorem appeared in this work as part of his treatment of equations.
Continuity: Continuity means a function's values vary without jumps at a point or throughout a domain. Continuity on the closed interval guarantees extrema are attained.
Stationary point: A point where a function's derivative is zero. The theorem guarantees at least one such point between the endpoints.

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