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Rolle's Theoremright In calculus, Rolle's theorem states that if a function f is continuous on a closed interval a,b and differentiable on the open interval (a,b), and f(a) = f(b) then there is some number c in the open interval (a,b) such that - f '(c) = 0.
Intuitively, this means that if a smooth curve is equal at two points then there must be a stationary point somewhere between them. All the assumptions are necessary. For example, if f(x) = |x|, the absolute value of x, then we have that f(-1) = f(1), but there is no x between -1 and 1 for which f '(x) = 0. The theorem was first stated by Michel Rolle, and published in 1691. Rolle's Theorem is used in proving the mean value theorem, which can be seen as a generalisation of it. Proof The idea of the proof is to argue that if f(a) = f(b) then f must attain either a maximum or a minimum somewhere between a and b, and f ' (x) = 0 at either of these points. Now, by assumption, f is continuous on b, and by the continuity property is bounded and attains both its maximum and its minimum at points of b. If these are both attained at endpoints of b then f is constant on b and so f ' (x) = 0 at every point of (a, b). Suppose then that the maximum is obtained at an interior point x of (a, b) (the argument for the minimum is very similar). We wish to show that f ' (x) = 0. We shall examine the left-hand and right-hand derivatives separately. For y just below x, (f(x) − f(y)) / (x − y) is non-negative, since x is a maximum. Thus the limit limy->x− is non-negative. (Note that we assume that f is differentiable to guarantee that the left-hand and right-hand derivatives exist; it does not follow from the other assumptions). For y just greater than x, (f(x) − f(y)) / (x − y) is non-positive. Thus limy->x+ is non-positive. Finally, since f is differentiable at x, these two limits must be equal and hence are both 0. This implies that f ' (x) = 0. Generalization The theorem is usually stated in the form above, but it is actually valid in a slightly more general setting: We only need to assume that f : b -> R is continuous on b, that f(a) = f(b), and that for every x in (a, b) the limit limh->0 (f(x + h) − f(x))/h exists or is equal to +/− infinity. See also External links
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