Rolle's theorem

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 point, somewhere between them, at which the slope of the tangent line is zero.

His proof did not use the methods of differential calculus, which at that point in his life he considered to be fallacious.

[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.

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.

If differentiability fails at an interior point of the interval, the conclusion of Rolle's theorem may not hold.

The derivative of f changes its sign at x = 0, but without attaining the value 0.

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).

Rolle's theorem implies that a differentiable function whose derivative is ⁠

satisfies the hypotheses of Rolle's theorem on the interval ⁠

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).

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

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 left-hand 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,

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

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.)

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

Specifically, suppose that 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 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.

The case n = 1 is simply the standard version of Rolle's theorem.

We want to prove it for n. Assume the function f satisfies the hypotheses of the theorem.

Rolle's theorem is a property of differentiable functions over the real numbers, which are an ordered field.

Any algebraically closed field such as the complex numbers has Rolle's property.

The question of which fields satisfy Rolle's property was raised in Kaplansky 1972.

[4] For finite fields, the answer is that only F2 and F4 have Rolle's property.