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ÄÍÄÄìÑ À̶ó¸é Ç®ÀÌ °¡´É ÇÒµí?
Ãßõ : 44 À̸§ : ****** ÀÛ¼ºÀÏ : 2006-08-14 22:56:54 Á¶È¸¼ö : 290
[A] Suppose that a given positive number N is partitioned into two positive numbers a and b such that a+b = N. The product ab= a(N-a) has a maximum if a=b=N/2. To see this, let P = Na - a2. By dP/da = N - 2a = 0, we obtain a = N/2 and b = N - N/2 = N/2.

Let's look at this problem in another way. We first define the two numbers A and G such that A = (a+b)/2 and G=, and then prove that G achieves its maximum when a and b are equal. If a and b are different, A - G = (a+b)/2 - = ( - )2 / 2 > 0; if they are equal, A = G. Therefore, the maximum value G can take is A and the value is achieved when a=b.

In the case of n positive numbers a1, a2, ..., an with n being no smaller than 2 such that the sum of the numbers is equal to N, we define A=(a1+a2 + ... + an) / n and G = with respect to a1, a2, ..., an. The proof can be done by changing the numbers in such a way that A remains constant but G gets larger until all the numbers are equal, at which point A and G are equal.

Suppose that in the set { a1, a2, ..., an }, a1 is bigger than any other number in the set and a2 is smaller than any other number in the set so that we have a1 > A > a2. Now replace a1 by A and a2 by a2' where a2' = a1+a2-A which is > 0. The new set of numbers becomes {A, a2', a3, ..., an}. Notice that because A+a2' = a1+a2, the sum of the numbers in the new set is equal to that in the old set. Also, the value of A with respect to the numbers in the new set, (A+ a2' + ... + an) / n, is equal to the value of A with respect to the numbers in the old set. By some algebraic manipulation, we have that Aa2' - a1a2 = A(a1+a2-A) - a1a2 = (A-a2)(a1-A) > 0. So the product of the numbers in the new set is greater than that in the old set. This means that by the construction of the new set the value of G has increased.

Based on the above arguments, we can generate sets in sequence such that the numbers in the set generated at each stage will eventually all equal A, and A and G computed with respect to the numbers in the set will become equal. From this we can show that the product of the elements in a partition of N achieves its maximum when the elements are all equal.
Name Pass  
¹øÈ£ Á¦¸ñ ³¯Â¥ Á¶È¸
572  ÄɹÌÄÄÀº..   2002/04/18 569
571  ÄɹÌÄÄÀº..   2004/05/26 306
570  ÄɹÌÄÄÀº...   2004/05/23 451
569  ÄɹÌÄÄÀ» À§ÇÏ¿©~   2012/05/26 362
568  ÄɹÌÄÄÀÎÀÌ µÈÈÄ   2005/06/03 322
567  ÄÉÄÉÄÉÄÉÄÉÄÉÄÉ켘   2012/05/19 358
566  ÄÍ  [4] 2011/04/19 435
 ÄÍÄÄìÑ À̶ó¸é Ç®ÀÌ °¡´É ÇÒµí?   2006/08/14 290
564  ÄìÄìÄì   2005/05/24 332
563  Äí¿¡¿¢....½ºÃÅ  [1] 2003/04/18 279
562  ÄôǪ È­À̸µ   2008/06/27 327
561  ÄôǪÆÒ´õ Àç¹Õ¾î   2008/06/27 327
560  ÄûÁî   2018/04/02 95
559  Å©¸£··   2017/04/15 75
558  Å©Å©  [1] 2005/04/29 314
557  Åª; ¾Æ·¡ Áú¹®À» º¸°í³ª´Ï ¤»¤»   2003/04/27 398
556  Å°½ºÇÏ°í ½ÍÀº ÀÔ¼ú~  [3] 2002/04/29 436
555  Å°Å° ±³·É´Ô¹Ì¾È   2010/04/27 433
554  Å°Å°Å°Å°  [5] 2004/08/22 448
553  Å°Å°Å°Å°   2009/05/28 413
552  Å°Å°Å±È­ÀÌÆÃ!!!   2010/05/17 426
551  Å¸¶ó¶ù   2017/04/08 90
550  Å¸¹Ì5   2009/05/07 337
549  Å¸¹Ì6   2009/05/07 363
548  Å¸¹Ì7   2009/05/07 327
547  Å¸¹Ì8   2009/05/07 394
546  Å¸¹ÌŸ¹Ì   2010/04/29 388
545  Å¸¹ÌŸ¹Ì   2010/05/13 341
544  Å¸¹ÌŸ¹Ì   2010/05/13 378
543  Å¸¹ÌŸ¹Ì¾ð´ÉÇØÁÒ¤»¤»¤»¤»¤»¤»¤»¤»¤»   2010/05/10 411
542  Å¸¹ÌŸ¹ÌȸŸ¹Ì   2010/04/18 412
541  Å»¶ôÀÚ´Â ¶³¾îÁö°í 2Â÷ÀüÀ» °¡Áö°Ú½À´Ï´Ù  [4] 2005/04/15 342
540  Å½Å½   2011/04/07 454
539  Å±ØÀü»ç´Ôµé..........  [2] 2006/06/18 334
538  ÅÂdzÀÌ ¸ô¾Æ¿Â´Ù!!!   2002/07/05 307
537  ÅÂÈñ¾ß ȸ޾ðÁ¦ÇØ?  [4] 2011/09/06 469
536  ÅÂÈñ¾ß¾ó··Çऻ¤»¤»¤»¤»¤»¤»   2011/09/06 444
535  ÅÂÈñ¾ßȸްí°í   2011/09/04 405
534  ÅÂÈñ¾ßÈ¸Å½ÇØ!!!!!   2011/09/04 313
533  ÅÂÈ÷¾ßÈ­ÀÌÆÃ ! !   2011/09/06 372
532  Åä³Ê¸ÕÆ®   2009/04/17 336
531  Åä³Ê¸ÕÆ® ½ÃÀÛ~~  [1] 2009/04/03 311
530  Å丶Åä´Ô ¼ö°í ¸¹À¸½Ê´Ï´Ù...   2005/07/21 294
529  Åð¹° Áú¹® ¸ÞÀÌÄ¿ ¤Ð¤Ð  [6] 2010/05/24 371
528  Æ¯ÀÌÇÑ ¹é¹®¹é´ä   2003/03/31 385
527  Æ¼¿ëÀÌ   2009/06/26 359
526  ÆÒ´õ´ÔÀÇ È¸Å½   2008/06/18 361
525  ÆÛ´ã±â ÆÛ·¹À̵å..±× ù¹øÂ°   2004/06/06 372
524  ÆÛ¿À´Â °Íµµ Àº±ÙÈ÷ ±ÍÂú´Ù ¤»¤»   2004/08/18 375
523  ÆÛ¿Ã²¨´Ù   2008/04/17 339
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