Theorem of Poker

See also: Poker (homonymy)

Statement

In analyzes, the Théorème of Poker , in reference to Michel Rolle, states themselves in the following way:

For two real numbers has and B such as and $f\; \backslash ;$ a function with real values continues on $\backslash ;$ and derivable on $]\; has,\; B\; \backslash ;$ such as: : $f\; (a)=f\; (b)\; \backslash ,$ then there exists (at least) an element C of $]\; has,\; B\; \backslash ;$ such as: : $f\text{'}\; (c)=0\; \backslash ;$.

Remarks

The theorem of Poker is intuitively obvious. To say that there exists at least an element where the derivative of F is null, it is to say that there exists a point where the tangent with the graph is horizontal . To say that the function is derivable, it is to say that its graph does not have angular points. The assumptions guarantee to us by the Théorème of the terminals that the function has a minimum and a maximum. There is thus well a point C between has and B such as F (c) is a maximum or a minimum. In such a point, the tangent with the graph is horizontal .

This theorem makes it possible to integrate the necessary topological properties of the real numbers in the analysis of the real functions of a real variable. The topological properties are integrated into the demonstration through the Théorème of the terminals.

The theorem of Poker does not extend to the functions from a complex variable or vectorial values . For example, the function

$f:\; 2\; \backslash \; pi\; \backslash \; to\; \backslash \; mathbb\; \{C\},\; T\; \backslash \; mapsto\; e^\; \{I\; \backslash ,\; T\}$ is continuous and derivable on the closed interval, and takes the same value at the two boundaries of the interval; but its derivative is not cancelled in any point: for all $T\; \backslash \; in\; 2\; \backslash \; pi,\; F\; \backslash ,\; \text{'}(T)\; =\; I\; \backslash ,\; e^\; \{I\; \backslash ,\; T\}$.

Applications

This theorem is used for the demonstration of the Théorème of the finished increases which is used with the analysis with the Développement limited of a function and Théorème as Taylor. This is why this theorem is impossible to circumvent in the construction of the analysis.

If P is a real polynomial having at least p distinct real roots ( p ≥ 2 ), then its derived polynomial has at least p - 1 distinct real roots.

History

Michel Rolle proposed a first demonstration of this theorem at the end of the 17th century. However, at that time, this theorem was algebraic and of anything the did not relate to analyzes. Indeed, it was not formulated that within the framework of the polynomials. Two contemporaries of Michel Poker, Isaac Newton and Gottfried Wilhelm von Leibniz, found the Infinitesimal calculus. And nobody is able to understand that this theorem will become later one of the pillars of the analysis of the functions of a real variable with actual values. More serious still, Michel Rolle judged work of infinitesimal calculus like vague and defective. It is true that according to our current criteria, or even according to the criteria which the algebra had reached at that time, logic formalization, Michel Rolle was right. On the other hand, the results of Newton and Leibniz were primarily right and especially founders of an new approach of most fertile in the history of mathematics. It did not realize from there that at the end of its life, but without imagining the destiny which the theorem could have which bears from now on its name.

This time was not ripe to see appearing a rigorous formalization of the analysis. The absence of a major comprehension of the nature of the real numbers and, consequently, concept of continuity did not allow such a fast progress. It was thus necessary to await a century and half so that the statement of this theorem is subtly modified. In 1860 Pierre-Ossian Bonnet generalizes the theorem with the derivable functions. This generalization changes whole with the whole the statute of this theorem. Of a a little anecdotic result on the theory of the real polynomials, this theorem becomes an impossible to circumvent base of the real analysis.

Demonstration

If F is constant, it is immediate. In the contrary case, as F is continuous on the closed interval limited '' B '', she admits (according to the Théorème of the terminals) a total minimum and a total maximum; taking into account what F (a) = F (b) and of what F is not constant, one at least from these two extrema is reached in a point C pertaining to the open interval ] has , B .

It is supposed here that F (c) is maximum total. The rates of increase in the function F between C and a second point have a known sign then.

For H strictly positive and such as c+h belongs to the interval '' B ''

$\backslash \; frac\; \{F\; (c+h)\; -\; F\; (c)\}\; H\; \backslash \; Leq\; 0$. By considering the limit when H tends towards 0, the derived number $f\text{'}\; (c)$ is negative.

For H strictly negative and such as c+h belongs to the interval '' B ''

$\backslash \; frac\; \{F\; (c+h)\; -\; F\; (c)\}\; H\; \backslash \; geq\; 0$. By considering the limit when H tends towards 0, the derived number $f\text{'}\; (c)$ is positive.

At the end of the day, the derivative of F is null at the point C .

The demonstration is similar if F (c) is a total minimum, with the signs of the rates of variation which are the opposites.

It is because C belongs to the open interval ] has, B which one could consider these rates of increase: if C were equal to B the first would not have direction and if it were equal to it has is the second who would not be defined.

See too

• Theorem of the finished increases

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