200 3 Real Analysis
571.Determine allnth-degree polynomialsP(x), with real zeros, for which the equality
∑ni= 11
P(x)−xi=
n^2
xP′(x)holds for all nonzero real numbersxfor whichP′(x) =0, wherexi,i= 1 , 2 ,...,n,
are the zeros ofP(x).572.LetCbe the class of all real-valued continuously differentiable functionsfon the
interval[ 0 , 1 ]withf( 0 )=0 andf( 1 )=1. Determine
u=inf
f∈C∫ 1
0|f′(x)−f(x)|dx.