Rolle's Theorem

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The following result concerning the existence of stationary points of a function f:


Let f be a function which is continuous on [a, b] and differentiable in (a, b), such that f(a)=f(b). (Some authors require that f(a)=f(b)=0.) Then there is a number c with a < c < b such that f′(c)=0. The result stated in the theorem can be expressed as a statement about the graph of f: with appropriate conditions on f, between any two points on the graph y=f(x) that are level with each other, there must be a stationary point; that is, a point at which the tangent is horizontal. The theorem is, in fact, a special case of the Mean Value Theorem; however, it is normal to establish Rolle's Theorem first and deduce the Mean Value Theorem from it. A rigorous proof relies on the non-elementary result that a continuous function on a closed interval attains its bounds.

Subjects: Mathematics.

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