Нақты дифференциалдауға ие функциялар үшін Роль теоремасының тұрақтанған нүктелері

On stationary points between two equal values of a real differentiable function

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.

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.

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 1=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 1=x = 0. The derivative of f changes its sign at 1=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 , but it may not yield a horizontal tangent (as in the case of the absolute value represented in the graph).

The proof uses mathematical induction. The case 1=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 such that 1=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.

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. 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, 1=x^(3) − x = x(x − 1)(x + 1) factors over the rationals, but its derivative,

does not. The question of which fields satisfy Rolle's property was raised in For finite fields, the answer is that only 'F'^(2) and 'F'^(4) have Rolle's property. For a complex version, see Voorhoeve index.