Rolle's Theorem
Rolle's theorem implies that if we have a function f that is continuous on an interval [a, b] and f(a) = f(b), then there always exists at least one critical point of the function in (a, b). The theorem is usually written symbolically as: Assume f is a function that is continuous on [a, b] and differentiable on (a, b). If f(a) = f(b) then there is a number c in (a, b) such that the derivative of f at c, f'(c) = 0. From Rolle's theorem it follows that c is at least one critical point of f in (a, b).
This means that at some point (c, f(c)) on the graph of f that the slope of the tangent line is 0. There may be more than one such point c but the theorem provides that there exist at least one. The line tangent to point (c, f(c)) is parallel to the line joining (a, f(a)) to (b, f(b)).
Rolle's theorem is named after the French mathematician Michel Rolle who developed the theorem during the 17th century. Rolle's theorem is very closely related to the mean value theorem. In the mean value theorem f(a) does not have to equal f(b) but there is still at lest one point in between these two on f where the slope of the tangent line is parallel to that of the line connecting (a, f(a)) and (b, f(b)). The mean value theorem is a generalized form of Rolle's theorem that is applicable to a wider variety of situations but has a similar meaning.
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