
In
calculus
Calculus is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations.
Originally called infinitesimal calculus or "the ...
, Rolle's theorem or Rolle's lemma essentially states that any real-valued
differentiable function
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...
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 a
stationary point
In mathematics, particularly in calculus, a stationary point of a differentiable function of one variable is a point on the graph of a function, graph of the function where the function's derivative is zero. Informally, it is a point where the ...
. It is a point at which the first derivative of the function is zero. The theorem is named after
Michel Rolle.
Standard version of the theorem
If a
real-valued
function is
continuous on a proper
closed interval
In mathematics, a real interval is the set of all real numbers lying between two fixed endpoints with no "gaps". Each endpoint is either a real number or positive or negative infinity, indicating the interval extends without a bound. A real in ...
,
differentiable
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...
on the
open interval
In mathematics, a real interval is the set (mathematics), set of all real numbers lying between two fixed endpoints with no "gaps". Each endpoint is either a real number or positive or negative infinity, indicating the interval extends without ...
, and , then there exists at least one in the open interval such that
This version of Rolle's theorem is used to prove the
mean value theorem
In mathematics, the mean value theorem (or Lagrange's mean value theorem) states, roughly, that for a given planar arc (geometry), arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant lin ...
, of which Rolle's theorem is indeed a special case. It is also the basis for the proof of
Taylor's theorem
In calculus, Taylor's theorem gives an approximation of a k-times differentiable function around a given point by a polynomial of degree k, called the k-th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the truncation a ...
.
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
In mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus—the study of the area beneath a curve. ...
, 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
In mathematics, the mean value theorem (or Lagrange's mean value theorem) states, roughly, that for a given planar arc (geometry), arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant lin ...
. The name "Rolle's theorem" was first used by
Moritz Wilhelm Drobisch of Germany in 1834 and by
Giusto Bellavitis
Giusto Bellavitis (22 November 1803 – 6 November 1880) was an Italian mathematician, senator, and municipal councilor. Charles Laisant (1880) "Giusto Bellavitis. Nécrologie", ''Bulletin des sciences mathématiques et astronomiques'', 2nd ...
of Italy in 1846.
Examples
Half circle
For a radius , consider the function
Its
graph
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discret ...
is the upper
semicircle
In mathematics (and more specifically geometry), a semicircle is a one-dimensional locus of points that forms half of a circle. It is a circular arc that measures 180° (equivalently, radians, or a half-turn). It only has one line of symmetr ...
centered at the origin. This function is continuous on the closed interval and differentiable in the open interval , but not differentiable at the endpoints and . Since , Rolle's theorem applies, and indeed, there is a point where the derivative of 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.
Absolute value

If differentiability fails at an interior point of the interval, the conclusion of Rolle's theorem may not hold. Consider the
absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if x is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...
function
Then , but there is no between −1 and 1 for which the is zero. This is because that function, although continuous, is not differentiable at . The derivative of changes its sign at , 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 in the open interval. However, when the differentiability requirement is dropped from Rolle's theorem, will still have a
critical number in the open interval , but it may not yield a horizontal tangent (as in the case of the absolute value represented in the graph).
Functions with zero derivative
Rolle's theorem implies that a
differentiable function
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...
whose derivative is in an interval is constant in this interval.
Indeed, if and are two points in an interval where a function is differentiable, then the function
satisfies the hypotheses of Rolle's theorem on the interval .
If the derivative of is zero everywhere, the derivative of is
and Rolle's theorem implies that there is such that
Hence, for every and , and the function is constant.
Generalization
The second example illustrates the following generalization of Rolle's theorem:
Consider a real-valued, continuous function on a closed interval with . If for every in the open interval the
right-hand limit
and the left-hand limit
exist in the
extended real line , then there is some number in the open interval such that one of the two limits
is and the other one is (in the extended real line). If the right- and left-hand limits agree for every , then they agree in particular for , hence the derivative of exists at and is equal to zero.
Remarks
* If 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:
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 , then must attain either
a maximum or a minimum somewhere between and , say at , and the function must change from increasing to decreasing (or the other way around) at . In particular, if the derivative exists, it must be zero at .
By assumption, is continuous on , and by the
extreme value theorem
In calculus, the extreme value theorem states that if a real-valued function f is continuous on the closed and bounded interval ,b/math>, then f must attain a maximum and a minimum, each at least once.
That is, there exist numbers c and ...
attains both its maximum and its minimum in . If these are both attained at the endpoints of , then is
constant on and so the derivative of is zero at every point in .
Suppose then that the maximum is obtained at an
interior point
In mathematics, specifically in topology,
the interior of a subset of a topological space is the union of all subsets of that are open in .
A point that is in the interior of is an interior point of .
The interior of is the complement of t ...
of (the argument for the minimum is very similar, just consider ). We shall examine the above right- and left-hand limits separately.
For a real such that is in , the value is smaller or equal to because attains its maximum at . Therefore, for every ,
hence
where the limit exists by assumption, it may be minus infinity.
Similarly, for every , 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 left-hand limits agree (in particular when is differentiable), then the derivative of at 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 has more points with equal values and greater regularity. Specifically, suppose that
* the function is times
continuously differentiable
In mathematics, a differentiable function of one Real number, real variable is a Function (mathematics), function whose derivative exists at each point in its Domain of a function, domain. In other words, the Graph of a function, graph of a differ ...
on the closed interval and the th derivative exists on the open interval , and
* there are intervals given by in such that for every from 1 to .
Then there is a number in such that the th derivative of at is zero.

The requirements concerning the th derivative of can be weakened as in the generalization above, giving the corresponding (possibly weaker) assertions for the right- and left-hand limits defined above with in place of .
Particularly, this version of the theorem asserts that if 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
Mathematical induction is a method for mathematical proof, proving that a statement P(n) is true for every natural number n, that is, that the infinitely many cases P(0), P(1), P(2), P(3), \dots all hold. This is done by first proving a ...
. The case is simply the standard version of Rolle's theorem. For , take as the induction hypothesis that the generalization is true for . We want to prove it for . Assume the function satisfies the hypotheses of the theorem. By the standard version of Rolle's theorem, for every integer from 1 to , there exists a in the open interval such that . Hence, the first derivative satisfies the assumptions on the closed intervals . By the induction hypothesis, there is a such that the st derivative of at is zero.
Generalizations to other fields
Rolle's theorem is a property of differentiable functions over the real numbers, which are an
ordered field
In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. Basic examples of ordered fields are the rational numbers and the real numbers, both with their standard ord ...
. As such, it does not generalize to other
fields
Fields may refer to:
Music
*Fields (band), an indie rock band formed in 2006
* Fields (progressive rock band), a progressive rock band formed in 1971
* ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010)
* "Fields", a song by ...
, 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. 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
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...
has Rolle's property. However, the rational numbers do not for example, factors over the
rationals, but its derivative,
does not. The question of which fields satisfy Rolle's property was raised in . For
finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), s ...
s, the answer is that only and have Rolle's property.
[.]
For a complex version, see
Voorhoeve index.
See also
*
Mean value theorem
In mathematics, the mean value theorem (or Lagrange's mean value theorem) states, roughly, that for a given planar arc (geometry), arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant lin ...
*
Intermediate value theorem
In mathematical analysis, the intermediate value theorem states that if f is a continuous function whose domain contains the interval , then it takes on any given value between f(a) and f(b) at some point within the interval.
This has two imp ...
*
Linear interpolation
In mathematics, linear interpolation is a method of curve fitting using linear polynomials to construct new data points within the range of a discrete set of known data points.
Linear interpolation between two known points
If the two known po ...
*
Gauss–Lucas theorem
References
Further reading
*
*
External links
*
Rolle's and Mean Value Theoremsat
cut-the-knot
Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet Union, Soviet-born Israeli Americans, Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow ...
.
*
Mizar system proof: http://mizar.org/version/current/html/rolle.html#T2
{{DEFAULTSORT:Rolle's Theorem
Theorems in real analysis
Articles containing proofs
Theorems in calculus