Rolle's theorem
Part of a series of articles about  
Calculus  





Specialized


In calculus, Rolle's theorem essentially states that any realvalued differentiable function that attains equal values at two distinct points must have at least one stationary point somewhere between them—that is, a point where the first derivative (the slope of the tangent line to the graph of the function) is zero.
Contents
Standard version of the theorem
If a realvalued function f is continuous on a proper closed interval [a, b], differentiable on the open interval (a, b), and f(a) = f(b), then there exists at least one c in the open interval (a, b) such that
 .
This version of Rolle's theorem is used to prove the mean value theorem, of which Rolle's theorem is indeed a special case. It is also the basis for the proof of Taylor's theorem.
History
Indian mathematician Bhāskara II (1114–1185) is credited with knowledge of Rolle's theorem.^{[1]} Although the theorem is named after Michel Rolle, Rolle's 1691 proof covered only the case of polynomial functions. His proof did not use the methods of differential calculus, which at that point in his life he considered to be fallacious. The theorem was first proved by Cauchy in 1823 as a corollary of a proof of the mean value theorem.^{[2]} The name "Rolle's theorem" was first used by Moritz Wilhelm Drobisch of Germany in 1834 and by Giusto Bellavitis of Italy in 1846.^{[3]}
Examples
First example
For a radius r > 0, consider the function
Its graph is the upper semicircle centered at the origin. This function is continuous on the closed interval [−r,r] and differentiable in the open interval (−r,r), but not differentiable at the endpoints −r and r. Since f(−r) = f(r), Rolle's theorem applies, and indeed, there is a point where the derivative of f is zero. Note that the theorem applies even when the function cannot be differentiated at the endpoints because it only requires the function to be differentiable in the open interval.
Second example
If differentiability fails at an interior point of the interval, the conclusion of Rolle's theorem may not hold. Consider the absolute value function
Then f(−1) = f(1), but there is no c between −1 and 1 for which the derivative is zero. This is because that function, although continuous, is not differentiable at x = 0. Note that the derivative of f changes its sign at x = 0, but without attaining the value 0. The theorem cannot be applied to this function, clearly, because it does not satisfy the condition that the function must be differentiable for every x in the open interval. However, when the differentiability requirement is dropped from Rolle's theorem, f will still have a critical number in the open interval (a,b), but it may not yield a horizontal tangent (as in the case of the absolute value represented in the graph).
Generalization
The second example illustrates the following generalization of Rolle's theorem:
Consider a realvalued, continuous function f on a closed interval [a,b] with f(a) = f(b). If for every x in the open interval (a,b) the righthand limit
and the lefthand limit
exist in the extended real line [−∞,∞], then there is some number c in the open interval (a,b) such that one of the two limits
is ≥ 0 and the other one is ≤ 0 (in the extended real line). If the right and lefthand limits agree for every x, then they agree in particular for c, hence the derivative of f exists at c and is equal to zero.
Remarks
 If f is convex or concave, then the right and lefthand derivatives exist at every inner point, hence the above limits exist and are real numbers.
 This generalized version of the theorem is sufficient to prove convexity when the onesided derivatives are monotonically increasing:^{[4]}
Proof of the generalized version
Since the proof for the standard version of Rolle's theorem and the generalization are very similar, we prove the generalization.
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, say at c, and the function must change from increasing to decreasing (or the other way around) at c. In particular, if the derivative exists, it must be zero at c.
By assumption, f is continuous on [a,b], and by the extreme value theorem attains both its maximum and its minimum in [a,b]. If these are both attained at the endpoints of [a,b], then f is constant on [a,b] and so the derivative of f is zero at every point in (a,b).
Suppose then that the maximum is obtained at an interior point c of (a,b) (the argument for the minimum is very similar, just consider −f ). We shall examine the above right and lefthand limits separately.
For a real h such that c + h is in [a,b], the value f(c + h) is smaller or equal to f(c) because f attains its maximum at c. Therefore, for every h > 0,
hence
where the limit exists by assumption, it may be minus infinity.
Similarly, for every h < 0, the inequality turns around because the denominator is now negative and we get
hence
where the limit might be plus infinity.
Finally, when the above right and lefthand limits agree (in particular when f is differentiable), then the derivative of f at c must be zero.
(or apply Fermat's theorem directly.)
Generalization to higher derivatives
We can also generalize Rolle's theorem by requiring that f has more points with equal values and greater regularity. Specifically, suppose that
 the function f is n − 1 times continuously differentiable on the closed interval [a,b] and the nth derivative exists on the open interval (a,b), and
 there are n intervals given by a_{1} < b_{1} ≤ a_{2} < b_{2} ≤ . . .≤ a_{n} < b_{n} in [a,b] such that f(a_{k}) = f(b_{k}) for every k from 1 to n.
Then there is a number c in (a,b) such that the nth derivative of f at c is zero.
The requirements concerning the nth derivative of f can be weakened as in the generalization above, giving the corresponding (possibly weaker) assertions for the right and lefthand limits defined above with f^{ (n−1)} in place of f.
Particularly, this version of the theorem asserts that a function differentiable enough times has roots (so they have the same value, that is 0), then there is an internal point where vanishes.
Proof
The proof uses mathematical induction. For n = 1 is simply the standard version of Rolle's theorem. As induction hypothesis, assume the generalization is true for n − 1. We want to prove it for n > 1. By the standard version of Rolle's theorem, for every integer k from 1 to n, there exists a c_{k} in the open interval (a_{k},b_{k}) such that f' (c_{k}) = 0. Hence the first derivative satisfies the assumptions with the n − 1 closed intervals [c_{1},c_{2}], . . ., [c_{n−1},c_{n}]. By the induction hypothesis, there is a c such that the (n − 1)st derivative of f' at c is zero.
Generalizations to other fields
Rolle's theorem is a property of differentiable functions over the real numbers, which are an ordered field. As such, it does not generalize to other fields, but the following corollary does: if a real polynomial splits (has all of its roots) over the real numbers, then its derivative does as well. One may call this property of a field Rolle's property. More general fields do not always have a notion of differentiable function, but they do have a notion of polynomials, which can be symbolically differentiated. Similarly, more general fields may not have an order, but one has a notion of a root of a polynomial lying in a field.
Thus Rolle's theorem shows that the real numbers have Rolle's property. Any algebraically closed field such as the complex numbers has Rolle's property. However, the rational numbers do not – for example, splits over the rationals, but its derivative, , does not. The question of which fields satisfy Rolle's property was raised in (Kaplansky 1972). For finite fields, the answer is that only and have Rolle's property; this was first proven via technical means in (Craven & Csordas 1977), and a simple proof is given in (Ballantine & Roberts 2002).
For a complex version, see Voorhoeve index.
See also
Notes
 ^ Gupta, R. C. Encyclopaedia of the History of Science, Technology, and Medicine in NonWesten Cultures. p. 156.
 ^ Besenyei, A. (September 17, 2012). "A brief history of the mean value theorem" (PDF).
 ^ See Cajori, Florian. A History of Mathematics. p. 224.
 ^ Artin, Emil (1964) [1931], The Gamma Function, trans. Michael Butler, Holt, Rinehart and Winston, pp. 3–4
References
 Kaplansky, Irving (1972), Fields and Rings
 Craven, Thomas; Csordas, George (1977), "Multiplier sequences for fields", Illinois J. Math., 21 (4): 801–817
 Ballantine, C.; Roberts, J. (January 2002), "A Simple Proof of Rolle's Theorem for Finite Fields", The American Mathematical Monthly, Mathematical Association of America, 109 (1): 72–74, doi:10.2307/2695770, JSTOR 2695770
External links
 Hazewinkel, Michiel, ed. (2001) [1994], "Rolle theorem", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 9781556080104
 Rolle's and Mean Value Theorems at cuttheknot.
 Mizar system proof: http://mizar.org/version/current/html/rolle.html#T2
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