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.
Standard version of the theorem
If a real-valued function is continuous on a proper closed interval , differentiable on the open interval , 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, 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. The name "Rolle's theorem" was first used by Moritz Wilhelm Drobisch of Germany in 1834 and by Giusto Bellavitis of Italy in 1846.
Examples
First example
For a radius , consider the function
Its graph is the upper semicircle 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.
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 , 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).
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 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 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