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Rolle's theorem

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On stationary points between two equal values of a function

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If areal-valued functionf iscontinuous on aclosed interval[a,b],differentiable on theopen interval(a,b), andf(a) =f(b), then there exists ac in the open interval(a,b) such thatf′(c) = 0.

Inreal analysis, a branch ofmathematics,Rolle's theorem orRolle's lemma essentially states that any real-valueddifferentiable function that attains equal values at two distinct points must have at least one point, somewhere between them, at which the slope of the tangent line is zero. Such a point is known as astationary point. It is a point at which the first derivative of the function is zero. The theorem is named afterMichel Rolle.

Standard version of the theorem

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If areal-valuedfunctionf iscontinuous on a properclosed interval[a, b],differentiable on theopen interval(a,b), andf(a) =f(b), then there exists at least onec in the open interval(a,b) such that $f'(c)=0.$

This version of Rolle's theorem is used to prove themean value theorem, of which Rolle's theorem is indeed a special case. It is also the basis for the proof ofTaylor's theorem.

History

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Although the theorem is named afterMichel Rolle, Rolle's 1691 proof covered only the case of polynomial functions. His proof did not use the methods ofdifferential calculus, which at that point in his life he considered to be fallacious. The theorem was first proved byCauchy in 1823 as a corollary of a proof of themean value theorem.^[1] The name "Rolle's theorem" was first used byMoritz Wilhelm Drobisch of Germany in 1834 and byGiusto Bellavitis of Italy in 1846.^[2]

Examples

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Differentiability is not needed at the endpoints: Half circle

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For a radiusr > 0, consider the function $f(x)={\sqrt {r^{2}-x^{2}}},\quad x\in [-r,r].$

Itsgraph is the uppersemicircle 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 andr. Sincef(−r) =f(r), Rolle's theorem applies, and indeed, there is a point where the derivative off is zero. 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.

Differentiability is needed within the open interval: Absolute value

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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 theabsolute value function $f(x)=|x|,\quad x\in [-1,1].$

Thenf(−1) =f(1), but there is noc between −1 and 1 for which thef′(c) is zero. This is because that function, although continuous, is not differentiable atx = 0. The derivative off changes its sign atx = 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 everyx in the open interval. However, when the differentiability requirement is dropped from Rolle's theorem,f will still have acritical 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).

Functions with zero derivative

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Rolle's theorem implies that adifferentiable function whose derivative is⁠ $0 {\displaystyle 0}$ ⁠ in an interval is constant in this interval.

Indeed, ifa andb are two points in an interval where a functionf is differentiable, then the function $g(x)=f(x)-f(a)-{\frac {f(b)-f(a)}{b-a}}(x-a)$ satisfies the hypotheses of Rolle's theorem on the interval⁠ $[a,b]$ ⁠.

If the derivative of⁠ $f {\displaystyle f}$ ⁠ is zero everywhere, the derivative of⁠ $g {\displaystyle g}$ ⁠ is $g'(x)=-{\frac {f(b)-f(a)}{b-a}},$ and Rolle's theorem implies that there is⁠ $c\in (a,b)$ ⁠ such that $0=g'(c)=-{\frac {f(b)-f(a)}{b-a}}.$

Hence,⁠ $f(a)=f(b)$ ⁠ for every⁠ $a {\displaystyle a}$ ⁠ and⁠ $b {\displaystyle b}$ ⁠, and the function⁠ $f {\displaystyle f}$ ⁠ is constant.

Generalization

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The second example illustrates the following generalization of Rolle's theorem:

Consider a real-valued, continuous functionf on a closed interval[a,b] withf(a) =f(b). If for everyx in the open interval(a,b) theright-hand limit $f'(x^{+}):=\lim _{h\to 0^{+}}{\frac {f(x+h)-f(x)}{h}}$ and the left-hand limit $f'(x^{-}):=\lim _{h\to 0^{-}}{\frac {f(x+h)-f(x)}{h}}$

exist in theextended real line[−∞, ∞], then there is some numberc in the open interval(a,b) such that one of the two limits $f'(c^{+})\quad {\text{and}}\quad f'(c^{-})$ is≥ 0 and the other one is≤ 0 (in the extended real line). If the right- and left-hand limits agree for everyx, then they agree in particular forc, hence the derivative off exists atc and is equal to zero.

Remarks

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Iff 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 proveconvexity when the one-sided derivatives aremonotonically increasing:^[3] $f'(x^{-})\leq f'(x^{+})\leq f'(y^{-}),\quad x<y.$

Proof of the generalized version

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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 iff(a) =f(b), thenf must attain eithera maximum or a minimum somewhere betweena andb, say atc, and the function must change from increasing to decreasing (or the other way around) atc. In particular, if the derivative exists, it must be zero atc.

By assumption,f is continuous on[a,b], and by theextreme value theorem attains both its maximum and its minimum in[a,b]. If these are both attained at the endpoints of[a,b], thenf isconstant on[a,b] and so the derivative off is zero at every point in(a,b).

Suppose then that the maximum is obtained at aninterior pointc of(a,b) (the argument for the minimum is very similar, just consider−f). We shall examine the above right- and left-hand limits separately.

For a realh such thatc +h is in[a,b], the valuef(c +h) is smaller or equal tof(c) becausef attains its maximum atc. Therefore, for everyh > 0, ${\frac {f(c+h)-f(c)}{h}}\leq 0,$ hence $f'(c^{+}):=\lim _{h\to 0^{+}}{\frac {f(c+h)-f(c)}{h}}\leq 0,$ where the limit exists by assumption; it may be minus infinity.

Similarly, for everyh < 0, the inequality turns around because the denominator is now negative and we get ${\frac {f(c+h)-f(c)}{h}}\geq 0,$ hence $f'(c^{-}):=\lim _{h\to 0^{-}}{\frac {f(c+h)-f(c)}{h}}\geq 0,$ where the limit might be plus infinity.

Finally, when the above right- and left-hand limits agree (in particular whenf is differentiable), then the derivative off atc must be zero.

(Alternatively, we can applyFermat's stationary point theorem directly.)

Generalization to higher derivatives

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We can also generalize Rolle's theorem by requiring thatf has more points with equal values and greater regularity. Specifically, suppose that

the functionf isn − 1 timescontinuously differentiable on the closed interval[a,b] and thenth derivative exists on the open interval(a,b), and
there aren intervals given bya₁ <b₁ ≤a₂ <b₂ ≤ ⋯ ≤a_n <b_n in[a,b] such thatf(a_k) =f(b_k) for everyk from 1 ton.

Then there is a numberc in(a,b) such that thenth derivative off atc 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 thenth derivative off can be weakened as in the generalization above, giving the corresponding (possibly weaker) assertions for the right- and left-hand limits defined above withf^{(n − 1)} in place off.

Particularly, this version of the theorem asserts that if a function differentiable enough times hasn roots (so they have the same value, that is 0), then there is an internal point wheref^{(n − 1)} vanishes.

Proof

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The proof usesmathematical induction. The casen = 1 is simply the standard version of Rolle's theorem. Forn > 1, take as the induction hypothesis that the generalization is true forn − 1. We want to prove it forn. Assume the functionf satisfies the hypotheses of the theorem. By the standard version of Rolle's theorem, for every integerk from 1 ton, there exists ac_k in the open interval(a_k,b_k) such thatf′(c_k) = 0. Hence, the first derivative satisfies the assumptions on then − 1 closed intervals[c₁,c₂], …, [c_{n − 1},c_n]. By the induction hypothesis, there is ac such that the(n − 1)st derivative off′ atc is zero.

Generalizations to other fields

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Rolle's theorem is a property of differentiable functions over the real numbers, which are anordered field. As such, it does not generalize to otherfields, 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 fieldRolle's property.^[4] 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 thecomplex numbers has Rolle's property. However, the rational numbers do not – for example,x³ −x =x(x − 1)(x + 1) factors over therationals, but its derivative, $3x^{2}-1=3\left(x-{\tfrac {1}{\sqrt {3}}}\right)\left(x+{\tfrac {1}{\sqrt {3}}}\right),$ does not. The question of which fields satisfy Rolle's property was raised inKaplansky 1972.^[5] Forfinite fields, the answer is that onlyF² andF⁴ have Rolle's property.^[6]^[7]

For a complex version, seeVoorhoeve index.

References

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^Besenyei, A. (September 17, 2012)."A brief history of the mean value theorem"(PDF).
^SeeCajori, Florian (1999).A History of Mathematics. American Mathematical Soc. p. 224.ISBN 9780821821022.
^Artin, Emil (1964) [1931],The Gamma Function, translated by Butler, Michael,Holt, Rinehart and Winston, pp. 3–4.
^Ron Brown; Thomas C. Craven; M. J. Pelling."ORDERED FIELDS SATISFYING ROLLE'S THEOREM"(PDF). University of Hawaiʻi, Honolulu. Retrieved2025-12-08.
^Kaplansky, Irving (1972),Fields and Rings.^{[full citation needed]}
^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,109 (1), Mathematical Association of America:72–74,doi:10.2307/2695770,JSTOR 2695770.