送交者: Yush 于 2005-7-26, 01:14:39:

回答: 一道数学题请教大家 由 k2d 于 2005-7-25, 17:37:03:

A better proof using only high school knowledge

I think the convexity of the logarithm could mean the inequality we learned in high school:
(0) log(x) =0 and Yi>=0.

(1) Z - Sum( Xi*log(Xi*Y/X) )
(2) = Sum( Xi*log(Yi) ) - Sum( Xi*log(Xi*Y/X) )
(3) = Sum( Xi*log( X*Yi/(Xi*Y) ) )
(4) <=Sum( Xi*( X*Yi/(Xi*Y) - 1 ) )
(5) = X/Y*Sum(Yi) - Sum(Xi)
(6) = 0

In step (4), we use inequality (0). Note, Xi's have to be positive for this step. The equality of this step holds iif X*Yi/(Xi*Y) = 1, as what inequality (0) states.

This actually completes the proof. You may need further explaination. From the above proof, we have
Z - Sum( Xi*log(Xi*Y/X) ) <= 0, where the equality holds iif X*Yi/(Xi*Y) = 1
In other words, Z reaches its its maximum, Sum( Xi*log(Xi*Y/X) ), iif Yi = (Y/X) * Xi.

• xys does not like 'less than' and 'greater than'. It eat those. - Yush (1213 bytes) 2005-7-26, 01:53:59
  • the logarithm curve y = log(x) is always BELOW the line y = x-1 - Yush (0 bytes) 2005-7-26, 01:56:45

加跟贴