Set of role

If there is a real-valued function with continuously on and differentiable on , then there is a such that it holds. ${\displaystyle f}$ ${\displaystyle f(a)=f(b)}$ ${\displaystyle [a,b]}$ ${\displaystyle (a,b)}$ ${\displaystyle x_{0}\in (a,b)}$ ${\displaystyle f'(x_{0})=0}$

The von Rolle theorem (named after the French mathematician Michel Rolle ) is a central theorem of differential calculus .

history

Rolle formulated the theorem named after him in 1691 (in his work Démonstration d'une méthode pour résoudre les égalitéz de tous les dégrez ), but only for polynomials and purely algebraic. The sentence was named after Rolle in 1834 by Moritz Wilhelm Drobisch , in 1860 by Giusto Bellavitis and in 1868 in the German edition of Serret 's lectures on calculus (Volume 1, p. 216). The mean value theorem of differential calculus was proven by Joseph Louis Lagrange (1797) and by Augustin Louis Cauchy , published in 1823 in his lectures on calculus (Calcul infinitésimal, Lecture 7). An explicit connection with the theorem of Rolle was first drawn by Pierre Ossian Bonnet (presented in the lectures on calculus by Joseph Serret in 1868 (he did not mention Rolle). A forerunner of the theorem of Rolle was in the astronomical work of Bhaskara II in the 12th century formulated.

statement

Let and be a continuous function that is differentiable in the open interval . If it fulfills , there is a place with ${\displaystyle a<b}$ ${\displaystyle f\colon [a,b]\to \mathbb {R} }$ ${\displaystyle (a,b)}$ ${\displaystyle f(a)=f(b)}$ ${\displaystyle x_{0}\in (a,b)}$

${\displaystyle f'(x_{0})=0}$.

interpretation

This clearly means: On the graph of the function there is at least one point between two curve points with matching function values ​​at which the slope is equal to zero. At this point the tangent is horizontal and thus parallel to the x-axis. The theorem means in particular that there is a zero of the derivative between two zeros of a differentiable function. The von Rolle theorem is a special case of the mean value theorem in differential calculus , which, conversely, can easily be proven from the von Rolle theorem. ${\displaystyle f}$

Visualizations

• 

  Function that has a minimum within the definition range. There the derivative is zero.

• 

  The function has only a maximum and no minimum inside the domain. At this point the derivative is zero. ${\displaystyle f}$

• 

  The function has only a minimum and no maximum inside the domain. At this point the derivative is zero. ${\displaystyle f}$

• 

  If the function is constant, then the derivative is zero everywhere.

proof

Since is continuous over the compact interval , it assumes a minimum at one point and a maximum at one point (according to Weierstrasse's theorem ) . If not constant, then at least or must apply. This extreme point is denoted by. If constant, there is an extreme point inside the interval . ${\displaystyle f}$ ${\displaystyle [a,b]}$ ${\displaystyle m\in [a,b]}$ ${\displaystyle M\in [a,b]}$ ${\displaystyle f}$ ${\displaystyle f(a)=f(b)}$ ${\displaystyle m\in (a,b)}$ ${\displaystyle M\in (a,b)}$ ${\displaystyle x_{0}}$ ${\displaystyle f}$ ${\displaystyle x_{0}={\frac {a+b}{2}}}$ ${\displaystyle (a,b)}$

If the inner extremal point is a maximum point , it follows from the differentiability of at the point that ${\displaystyle x_{0}}$ ${\displaystyle f}$ ${\displaystyle x_{0}}$

${\displaystyle f'(x_{0})=\lim _{h\searrow 0}{\frac {f(x_{0}+h)-f(x_{0})}{h}}\leq 0}$

${\displaystyle f'(x_{0})=\lim _{h\nearrow 0}{\frac {f(x_{0}+h)-f(x_{0})}{h}}\geq 0}$

So is . ${\displaystyle f'(x_{0})=0}$

If there is a minimum of , then a maximum of and is obtained and thus . ${\displaystyle x_{0}}$ ${\displaystyle f}$ ${\displaystyle x_{0}}$ ${\displaystyle -f}$ ${\displaystyle -f'(x_{0})=0}$ ${\displaystyle f'(x_{0})=0}$

literature

• Konrad Königsberger : Analysis 1 . Springer, Berlin 2004, ISBN 3-540-41282-4
• Otto Forster : Analysis 1. Differential and integral calculus of a variable. Vieweg-Verlag, 8th edition 2006, ISBN 3-528-67224-2

Web links

Wikibooks: Math for Non-Freaks: Set of Role  - Learning and Teaching Materials

Wikibooks: Evidence Archive: Set of Role  - Learning and Teaching Materials

Commons : set of role  - collection of images, videos and audio files

Individual evidence

1. ↑ ^{a b c} Florian Cajori: On Michel Rolle's book "Méthode pour resoudre les égalitez" and the history of Rolle's theorem, Bibliotheca Mathematica, 1911, pp. 300-313
2. ^ Drobisch, Fundamentals of the Doctrine of Higher Numerical Equations, Leipzig 1834, p. 179