MathDB

5

Part of 2005 IMC

Problems(2)

IMC 2005 day1 pb 5

Source: Peter

7/26/2005
5) f twice cont diff, ∣f′′(x)+2xf′(x)+(x2+1)f(x)∣≤1|f''(x)+2xf'(x)+(x^{2}+1)f(x)|\leq 1. prove lim⁡x→+∞f(x)=0\lim_{x\rightarrow +\infty} f(x) = 0
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IMC 2005 day 2 pb 5

Source: Peter

7/26/2005
Find all r>0 r > 0 such that when f:R2→R f: \mathbb R^{2}\to \mathbb R is differentiable, \|\textrm{grad} \; f(0,0)\| \equal{} 1, \|\textrm{grad} \; f(u) \minus{} \textrm{grad} \; f(v)\| \leq \| u \minus{} v\|, then the max of f f on the disk ∥u∥≤r \|u\|\leq r, is attained at exactly one point.
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