MathDB
ugly but hard

Source: IMC 1999 day 1 problem 6

November 19, 2005
functionreal analysisreal analysis unsolved

Problem Statement

(a) Let p>1p>1 a real number. Find a real constant cpc_p for which the following statement holds: If f:[−1,1]→Rf: [-1,1]\rightarrow\mathbb{R} is a continuously differentiable function with f(1)>f(−1)f(1)>f(-1) and ∣f′(y)∣≤1∀y∈[−1,1]|f'(y)|\le1 \forall y\in[-1,1], then ∃x∈[−1,1]:f′(x)>0\exists x\in[-1,1]: f'(x)>0 so that ∀y∈[−1,1]:∣f(y)−f(x)∣≤cpf′(x)p∣y−x∣\forall y\in[-1,1]: |f(y)-f(x)|\le c_p\sqrt[p]{f'(x)}|y-x|. (b) What if p=1p=1?
Back to ProblemsView on AoPS