Ivan Shishkin, Rye (1878)

Problems/OtherNeeds work

Wirtinger’s inequality

by Sequoia·
67
Difficulty scaleÉchelle de difficulté

This score reflects both the level of the required concepts and the difficulty of the solution.Ce score tient compte à la fois du niveau des notions nécessaires et de la difficulté de la résolution.

  1. 1–10First steps / middle schoolPremiers pas / collège
  2. 11–25Beginner / high schoolDébutant / lycée
  3. 26–50Intermediate / undergraduateIntermédiaire / licence
  4. 51–70Advanced / graduateAvancé / master
  5. 71–90Expert / specializedExpert / spécialisé
  6. 91–100Research levelNiveau recherche
These levels are approximate guides.Ces niveaux sont des repères approximatifs.
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Needs work. This problem has been marked as needing work.

Let f ⁣:[0,π]→Rf\colon [0,\pi]\to \R be a C1\mathcal{C}^{1} function such that f(0)=f(π)=0f(0)=f(\pi)=0.

Show the inequality
∫0πf2⩽∫0πf′2,\int_0^{\pi} f^2\leqslant \int_0^{\pi} f'^2,and that f(x):=csin⁡(x),c∈Rf(x):=c\sin(x), c\in\R are the only such functions that satisfy the equality case.

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