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IMC 2005 day 2 pb 5

Source: Peter

July 26, 2005
IMCcollege contests

Problem Statement

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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