# Rolle's theorem

Rolle's theorem Part of a series of articles about Calculus Fundamental theorem Leibniz integral rule Limits of functionsContinuity Mean value theoremRolle's theorem show Differential show Integral show Series show Vector show Multivariable show Advanced show Specialized show Miscellaneous vte If a real-valued function f is continuous on a closed interval [a, b], differentiable on the open interval (a, b), and f (a) = f (b), then there exists a c in the open interval (a, b) such that f ′(c) = 0.

In calculus, Rolle's theorem or Rolle's lemma essentially states that any real-valued 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. The theorem is named after Michel Rolle.

Contents 1 Standard version of the theorem 2 History 3 Examples 3.1 First example 3.2 Second example 4 Generalization 4.1 Remarks 5 Proof of the generalized version 6 Generalization to higher derivatives 6.1 Proof 7 Generalizations to other fields 8 See also 9 References 10 Further reading 11 External links Standard version of the theorem If a real-valued 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 {displaystyle f'(c)=0.} 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 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.[1] The name "Rolle's theorem" was first used by Moritz Wilhelm Drobisch of Germany in 1834 and by Giusto Bellavitis of Italy in 1846.[2] The theorem, that derivatives are zero at a maximum, was stated by Bhaskara II in his Siddhanta Shiromani, completed in 1150.[3][4] Bhaskara II was part of the Kerala school of mathematics, and also theorized the derivatives of trigonometric functions.[4] Examples First example A semicircle of radius r.

For a radius r > 0, consider the function {displaystyle f(x)={sqrt {r^{2}-x^{2}}},quad xin [-r,r].} 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 The graph of the absolute value function.

If differentiability fails at an interior point of the interval, the conclusion of Rolle's theorem may not hold. Consider the absolute value function {displaystyle f(x)=|x|,qquad xin [-1,1].} Then f (−1) = f (1), but there is no c between −1 and 1 for which the f ′(c) 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 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 real-valued, 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 right-hand limit {displaystyle f'(x^{+}):=lim _{hto 0^{+}}{frac {f(x+h)-f(x)}{h}}} and the left-hand limit {displaystyle f'(x^{-}):=lim _{hto 0^{-}}{frac {f(x+h)-f(x)}{h}}} 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 {displaystyle f'(c^{+})quad {text{and}}quad f'(c^{-})} is ≥ 0 and the other one is ≤ 0 (in the extended real line). If the right- and left-hand 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 left-hand 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 one-sided derivatives are monotonically increasing:[5] {displaystyle f'(x^{-})leq f'(x^{+})leq f'(y^{-}),qquad x 0, {displaystyle {frac {f(c+h)-f(c)}{h}}leq 0,} hence {displaystyle f'(c^{+}):=lim _{hto 0^{+}}{frac {f(c+h)-f(c)}{h}}leq 0,} 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 {displaystyle {frac {f(c+h)-f(c)}{h}}geq 0,} hence {displaystyle f'(c^{-}):=lim _{hto 0^{-}}{frac {f(c+h)-f(c)}{h}}geq 0,} where the limit might be plus infinity. Finally, when the above right- and left-hand limits agree (in particular when f is differentiable), then the derivative of f at c must be zero. (Alternatively, we can apply Fermat's stationary point 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 a1 < b1 ≤ a2 < b2 ≤ ⋯ ≤ an < bn in [a, b] such that f (ak) = f (bk) 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 red curve is the graph of function with 3 roots in the interval [−3, 2]. Thus its second derivative (graphed in green) also has a root in the same interval. 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 left-hand limits defined above with f (n − 1) in place of f. Particularly, this version of the theorem asserts that if a function differentiable enough times has n roots (so they have the same value, that is 0), then there is an internal point where f (n − 1) vanishes. Proof The proof uses mathematical induction. The case n = 1 is simply the standard version of Rolle's theorem. For n > 1, take as the induction hypothesis that the generalization is true for n − 1. We want to prove it for n. Assume the function f satisfies the hypotheses of the theorem. By the standard version of Rolle's theorem, for every integer k from 1 to n, there exists a ck in the open interval (ak, bk) such that f ′(ck) = 0. Hence, the first derivative satisfies the assumptions on the n − 1 closed intervals [c1, c2], …, [cn − 1, cn]. 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 factors (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.[citation needed] More general fields do not always have differentiable functions, but they do always have 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, x3 − x = x(x − 1)(x + 1) factors over the rationals, but its derivative, {displaystyle 3x^{2}-1=3left(x-{tfrac {1}{sqrt {3}}}right)left(x+{tfrac {1}{sqrt {3}}}right),} does not. The question of which fields satisfy Rolle's property was raised in (Kaplansky 1972).[6] For finite fields, the answer is that only F2 and F4 have Rolle's property.[7][8] For a complex version, see Voorhoeve index.

See also Mean value theorem Intermediate value theorem Linear interpolation Gauss–Lucas theorem References ^ Besenyei, A. (September 17, 2012). "A brief history of the mean value theorem" (PDF). ^ See Cajori, Florian (1999). A History of Mathematics. p. 224. ISBN 9780821821022. ^ Seshadri, Dr.Sridhar; N.Shivakumar (2014-03-01). "A NOTE ON LEADING MATHEMATICIAN BHASKARA II OF 12TH CENTURY". International Journal of Information Technology and Decision Making. ^ Jump up to: a b Agarwal, Ravi P.; Sen, Syamal K. (2014-11-11). Creators of Mathematical and Computational Sciences. Springer. p. 13. ISBN 978-3-319-10870-4. ^ Artin, Emil (1964) [1931], The Gamma Function, translated by Butler, Michael, Holt, Rinehart and Winston, pp. 3–4 ^ Kaplansky, Irving (1972), Fields and Rings ^ Craven, Thomas; Csordas, George (1977), "Multiplier sequences for fields", Illinois J. Math., 21 (4): 801–817, doi:10.1215/ijm/1256048929 ^ 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 Further reading Leithold, Louis (1972). The Calculus, with Analytic Geometry (2nd ed.). New York: Harper & Row. pp. 201–207. ISBN 0-06-043959-9. Taylor, Angus E. (1955). Advanced Calculus. Boston: Ginn and Company. pp. 30–37. External links "Rolle theorem", Encyclopedia of Mathematics, EMS Press, 2001 [1994] Rolle's and Mean Value Theorems at cut-the-knot. Mizar system proof: http://mizar.org/version/current/html/rolle.html#T2 Wikimedia Commons has media related to Rolle's theorem. Categories: Theorems in real analysisTheorems in calculus

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