送交者: k2d 于 2005-7-25, 17:37:03:

X = Sum(Xi),
Y = Sum(Yi),
where i = 1, 2, ..., n

Z = Sum(Xi * log(Yi))

To maximize Z, Yi should be (Y/X) * Xi.

请问这个怎么证明呢？提示里说要用到 convexity of the logarithm 。
我想所谓的 "convexity of the logarithm" ，就是指算术平均大于等于
几何平均吧？可是我还是不知道该怎么证。

请高手指教！

• Re: 一道数学题请教大家 - k2d (25 bytes) 2005-7-26, 10:58:59
• A better proof using only high school knowledge - Yush (845 bytes) 2005-7-26, 01:14:39
  • 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
• try google "lagrange multiplier" - Enlighten (0 bytes) 2005-7-25, 23:14:45
• 拉 格 郎 日 方 程 - HunHunSheng (228 bytes) 2005-7-25, 22:13:28
• Create Lagrange equation and find partial derivatives - HunHunSheng (86 bytes) 2005-7-25, 17:44:41
  • Re: Create Lagrange equation and find partial derivatives - k2d (105 bytes) 2005-7-25, 18:01:06

加跟贴