Determine all possible value(s) of a positive integer constant c that satisfy this relationship:

√(y-1) + √(y-2) + .....+ √(y-c) ≤ y

whenever y is a positive real number ≥ c

Trying c=1 and c=2 for graphing f(y) = sqrt(y-1)+...+sqrt(y-c), as well as g(y) = y, results in a curve that stays below the straight line and therefore fills the conditions of the problem.

Starting at c=3, the curve of the sum of the square roots starts to go above the g(y)=y line. At c=3, for example, the violations occur between somewhat before y=4 and somewhat after y=5, including all point in between.

Therefore it would seem that 1 and 2 are the only values of c that work, though I haven't proved that at no larger c does the curve cease intersecting the straight line.

Posted by Charlie on 2010-04-10 15:56:00