Analysis Basics

Take 10 minutes to prepare this exercise.

Then, if you lack ideas to begin, look at the given clue and start searching for the solution.

A detailed solution is then proposed to you.

If you have more questions, feel to ask then on the forum.

Question

Let be a differentiable application from to such that and

Let be a non zero natural integer, prove that there exists real numbers belonging to such that

Indice

Slice the segment in segments of equal length and apply the Rolle's theorem to each of them.

Solution

Let us consider the steady subdivision of step meaning:

From the formula of mean values:

By summing the previous equalities for from to we obtain:

The terms of the sum of the left member of the previous equality are zero when taken two by two, meaning there remains which gives us the expected result using the hypotheses on and

PrécédentPrécédentSuivantSuivant
AccueilAccueilImprimerImprimerRéalisé avec Scenari (nouvelle fenêtre)