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

(In reply to by graphing by Charlie)

While increasing c, as soon as you have violations, at particular c and y, if you, say double both c and y, there will be twice as many terms on the left and the terms will be approximately multiplied by sqrt(2) so the LHS increases as the 1.5 power of the common factor of increase of y and c, while the RHS increases as only the 1st power, so the violations will be getting worse as you increase c. This is evidence that indeed 1 and 2 are the only values of c that work.

Posted by Charlie on 2010-04-10 16:04:00