Determine all possible value(s) of a positive integer constant c that satisfy this relationship:
√(y1) + √(y2) + .....+ √(yc) ≤ y
whenever y is a positive real number ≥ c
let f(c,y) = sum( sqrt(yt), t=1 to c)
then we want to know for what integers c does
y>=c imply f(c,y)<=y
now f(c,c)>=integral sqrt(x) x=0 to c1
f(c,c) >= (2/3)*(c1)^(3/2)
now we want to know when (2/3)*(c1)^(3/2)<=c
4*(c1)^3<=9c^2
4c^34c^2+4c4<=9c^2
4c^313c^2+4c4<=0
which implies c<=3
so now we need to check if c=1,2 or 3 work
c=1
this is simply sqrt(y1)<=y when y>=1 which is true
c=2
sqrt(y1)+sqrt(y2)<=y
y1+2*sqrt((y1)(y2))+y2<=y^2
2*sqrt((y1)(y2))<=y^22y=y(y2)
4*(y1)(y2)<=y^2(y2)^2
4*(y1)<=y^2(y2)
4y1<=y^32y^2
y^32y^24y+1>=0
the polynomial has no real solutions and thus never crosses the xaxis since it is positive for y=0 then it is positive for all y thus it works for c=2
c=3
sqrt(y1)+sqrt(y2)+sqrt(y3)<=y
looking at y=4 we get
sqrt(3)+sqrt(2)+1<=4
which is not true, thus it does not hold for c=3
thus it only holds for c=1 and c=2
Edited on April 11, 2010, 7:08 am

Posted by Daniel
on 20100410 19:01:31 