發表回覆

主題 通關密語 訪客發文, 請參考 這裡 輸入通關密語.

顯示表情符號

站內上傳圖檔     Upload.cc免費圖片上傳

數學塗鴉工具     常用數學符號表    

用Latex打數學方程式

 


 

+ / -檢視主題

發表 宇智波鼬 於 星期一 三月 19, 2007 9:51 pm

To:aaaaa
那部份我以用函數圖形確定過了...(也就是證明過了)
因為懶的打所以沒說明....若是競賽時我一定會補上...請放心^^

發表 galaxylee 於 星期一 三月 19, 2007 6:43 pm

#ed_op#DIV#ed_cl##ed_op#P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"#ed_cl##ed_op#SPAN lang=EN-US#ed_cl#a^3+b^3+(-1)^3-3ab(-1)=0#ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"#ed_cl##ed_op#SPAN lang=EN-US#ed_cl#(a+b-1)(a^2+b^2+1-ab+a+b)=0#ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"#ed_cl##ed_op#SPAN lang=EN-US#ed_cl#a+b=1#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 新細明體; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'"#ed_cl#或#ed_op#/SPAN#ed_cl##ed_op#SPAN lang=EN-US#ed_cl#a^2+b^2+1-ab+a+b=0(#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 新細明體; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'"#ed_cl#不合#ed_op#/SPAN#ed_cl##ed_op#SPAN lang=EN-US#ed_cl#)#ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"#ed_cl##ed_op#SPAN lang=EN-US#ed_cl#(#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 新細明體; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'"#ed_cl#因為#ed_op#/SPAN#ed_cl##ed_op#SPAN lang=EN-US#ed_cl#a^2+b^2+1-ab+a+b=(a-b)^2+(a+1)(b+1)#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 新細明體; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'"#ed_cl#≠#ed_op#/SPAN#ed_cl##ed_op#SPAN lang=EN-US#ed_cl#0)#ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"#ed_cl##ed_op#SPAN style="FONT-FAMILY: 新細明體; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl# #ed_op#/P#ed_cl##ed_op#P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"#ed_cl##ed_op#SPAN style="FONT-FAMILY: 新細明體; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'"#ed_cl#考慮函數#ed_op#/SPAN#ed_cl##ed_op#SPAN lang=EN-US#ed_cl#y=f(x)=(x+1/x)^3#ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"#ed_cl##ed_op#SPAN style="FONT-FAMILY: 新細明體; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'"#ed_cl#因為#ed_op#/SPAN#ed_cl##ed_op#SPAN lang=EN-US#ed_cl#y=f(x)#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 新細明體; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'"#ed_cl#在區間#ed_op#/SPAN#ed_cl##ed_op#SPAN lang=EN-US#ed_cl#(0,1)#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 新細明體; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'"#ed_cl#是凸函數#ed_op#FONT face="Times New Roman"#ed_cl#(凹向上)#ed_op#/FONT#ed_cl##ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"#ed_cl##ed_op#SPAN style="FONT-FAMILY: 新細明體; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'"#ed_cl#所以由Jensen不等式#ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"#ed_cl##ed_op#SPAN style="FONT-FAMILY: 新細明體; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'"#ed_cl#若#ed_op#/SPAN#ed_cl##ed_op#SPAN lang=EN-US#ed_cl#0#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 新細明體; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'"#ed_cl#<#ed_op#/SPAN#ed_cl##ed_op#SPAN lang=EN-US#ed_cl#a,b#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 新細明體; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'"#ed_cl#<#ed_op#/SPAN#ed_cl##ed_op#SPAN lang=EN-US#ed_cl#1#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 新細明體; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'"#ed_cl#,#ed_op#/SPAN#ed_cl##ed_op#SPAN lang=EN-US#ed_cl#a+b=1#ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"#ed_cl##ed_op#SPAN lang=EN-US#ed_cl#則有[f(a)+f(b)]/2#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 新細明體; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'"#ed_cl#≧#ed_op#/SPAN#ed_cl##ed_op#SPAN lang=EN-US#ed_cl#f[(a+b)/2]#ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"#ed_cl##ed_op#SPAN lang=EN-US#ed_cl#(1/2)*[(a+1/a)^3+(b+1/b)^3]#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 新細明體; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'"#ed_cl#≧#ed_op#/SPAN#ed_cl##ed_op#SPAN lang=EN-US#ed_cl#(2+1/2)^3#ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"#ed_cl##ed_op#SPAN lang=EN-US#ed_cl#[(a+1/a)^3+(b+1/b)^3]#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 新細明體; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'"#ed_cl#≧#ed_op#/SPAN#ed_cl##ed_op#SPAN lang=EN-US#ed_cl#125/4#ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"#ed_cl##ed_op#SPAN style="FONT-FAMILY: 新細明體; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'"#ed_cl#等號成立時,取#ed_op#/SPAN#ed_cl##ed_op#SPAN lang=EN-US#ed_cl#a=b=1/2#ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"#ed_cl##ed_op#SPAN lang=EN-US#ed_cl#最小值125/4#ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"#ed_cl##ed_op#SPAN style="FONT-FAMILY: 新細明體; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl# #ed_op#/P#ed_cl##ed_op#/DIV#ed_cl#

發表 aaaaa 於 星期日 三月 18, 2007 10:28 pm

#ed_op#DIV#ed_cl##ed_op#FONT size=2#ed_cl#雖然大部分的情形,不等式等號成立的條件#ed_op#/FONT#ed_cl##ed_op#FONT size=2#ed_cl#都是a=b時#ed_op#/FONT#ed_cl##ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl##ed_op#FONT size=2#ed_cl#但也不能由a=b來反推a,b的值。#ed_op#/FONT#ed_cl##ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl##ed_op#FONT size=2#ed_cl##ed_op#/FONT#ed_cl# #ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl##ed_op#FONT size=2#ed_cl#若是參加數學競試,這樣寫肯定會被扣分。#ed_op#/FONT#ed_cl##ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl##ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl##ed_op#/DIV#ed_cl#

發表 宇智波鼬 於 星期日 三月 18, 2007 5:48 pm

Let me put it this way...
a+(1/a)>=2>1. So, when a or 1/a increases, a+(1/a) will also increase.
That means the min value of a+(1/a) appears when a=1/a.
The situation b is the same.
a^3+b^3+3ab=1 tells us a is not equal to 1.
But obviously, we'll get the min when a=b.
Solve the equation: 2a^3+3a^2=1.
We get a=-1 or 1/2.
a is postive real, so we choose 1/2.
That's how we get the answer.

發表 aaaaa 於 星期日 三月 18, 2007 2:27 pm

#ed_op#DIV#ed_cl##ed_op#SPAN class=postbody#ed_cl##ed_op#FONT size=2#ed_cl# #ed_op#/FONT#ed_cl##ed_op#DIV#ed_cl#若正實數a,b滿足a^3+b^3+3ab=1#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#則 [a+(1/a)]^3+[b+(1/b)]^3≧125/4#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#要如何證明???#ed_op#/DIV#ed_cl##ed_op#/SPAN#ed_cl##ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl##ed_op#/DIV#ed_cl#

發表 宇智波鼬 於 星期日 三月 18, 2007 10:23 am

很明顯[a+(1/a)]^3+[b+(1/b)]^3的min決定於ab的值.
而a^3+b^3+3ab=1
當a=b=1/2時會使ab有最大值1/4.
如此會使得原式越小.
因此原式最小值=2*(5/2)^3=250/8.

[數學]不等式題

發表 skywalker 於 星期六 三月 17, 2007 9:54 pm

#ed_op#DIV#ed_cl#設正實數a,b滿足a^3+b^3+3ab=1#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl# #ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#試求[a+(1/a)]^3+[b+(1/b)]^3之最小值#ed_op#/DIV#ed_cl#