Find the smallest possible value of
in terms of the constant k, given that xy=k, and x>y>1.
(In reply to
Correct solution (with hand-waving) by Steve Herman)
This is the same thing I got with one exception. This fails for very small k (<2) because y ends up needing to be smaller than 1.
Note that a = (x^2 - y^2).
Take k = 3/2
then a = 2*sqrt(9/2 - 3/2) = sqrt(3)
but the largest possible a given x>y>1 is < a when
x=3/2 y=1 which gives (9/4 - 1) = 5/4 (squared is less than 2)
I didn't calculate the exact cut-off but if we relax the constraints to simply x>|y| the spirit of the problem is preserved and we get the interesting answer that when k is between 0 and 1/2 there is no minimum, when it is exactly 0 or 1/2 f can be abritrarily close to (above) but not equal to 0, and when x is outside of [0,1/2] the answer you give is correct.
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Posted by Joel
on 2007-03-29 19:55:41 |
re: Correct solution (with hand-waving)
