លំហាត់ 47: van khea

គេអោយ a, b, c>0 ; abc=1 ។ ស្រាយបញ្ជាក់ថាៈ
\displaystyle \frac{1}{a^3(\sqrt[3]{b}+\sqrt[3]{c})}+\frac{1}{b^3(\sqrt[3]{c}+\sqrt[3]{a})}+\frac{1}{c^3(\sqrt[3]{a}+\sqrt[3]{b})}\geq \frac{3}{2}
\displaystyle \sqrt[3]{a}+\sqrt[3]{b}\leq 2\sqrt[3]{\frac{a+b}{2}}=\sqrt[3]{4}\sqrt[3]{a+b}
\displaystyle \frac{1}{a^3(\sqrt[3]{b}+\sqrt[3]{c})}+\frac{1}{b^3(\sqrt[3]{c}+\sqrt[3]{a})}+\frac{1}{c^3(\sqrt[3]{a}+\sqrt[3]{b})}\displaystyle \geq \frac{1}{\sqrt[3]{4}a^3\sqrt[3]{b+c}}+\frac{1}{\sqrt[3]{4}b^3\sqrt[3]{c+a}}+\frac{1}{\sqrt[3]{4}c^3\sqrt[3]{a+b}}
\displaystyle \frac{1}{\sqrt[3]{4}a^3\sqrt[3]{b+c}}+\frac{1}{\sqrt[3]{4}b^3\sqrt[3]{c+a}}+\frac{1}{\sqrt[3]{4}c^3\sqrt[3]{a+b}}\geq \frac{3}{2}
\displaystyle \Leftrightarrow  \frac{1}{a^3\sqrt[3]{b+c}}+\frac{1}{b^3\sqrt[3]{c+a}}+\frac{1}{c^3\sqrt[3]{a+b}}\geq \frac{3}{\sqrt[3]{2}}
\displaystyle \frac{1}{a^3\sqrt[3]{b+c}}+\frac{1}{b^3\sqrt[3]{c+a}}+\frac{1}{c^3\sqrt[3]{a+b}}\displaystyle =\frac{\biggl(\frac{1}{a}\biggl)^3}{1^{\frac{5}{3}}(b+c)^{\frac{1}{3}}}+\frac{\biggl(\frac{1}{b}\biggl)^3}{1^{\frac{5}{3}}(c+a)^{\frac{1}{3}}}+\frac{\biggl(\frac{1}{c}\biggl)^3}{1^{\frac{5}{3}}(a+b)^{\frac{1}{3}}}
ដោយ \displaystyle 3-\frac{5}{3}-\frac{1}{3}=1 នោះតាមវិសមភាព V - K:07 គេបានៈ
\displaystyle =\frac{\biggl(\frac{1}{a}\biggl)^3}{1^{\frac{5}{3}}(b+c)^{\frac{1}{3}}}+\frac{\biggl(\frac{1}{b}\biggl)^3}{1^{\frac{5}{3}}(c+a)^{\frac{1}{3}}}+\frac{\biggl(\frac{1}{c}\biggl)^3}{1^{\frac{5}{3}}(a+b)^{\frac{1}{3}}}\displaystyle \geq \frac{(\frac{1}{a}+\frac{1}{b}+\frac{1}{c})^3}{3^{\frac{5}{3}}(2(a+b+c))^{\frac{1}{3}}}=\frac{(\frac{1}{a}+\frac{1}{b}+\frac{1}{c})^2(\frac{1}{a}+\frac{1}{b}+\frac{1}{c})}{3^{\frac{5}{3}}(2(a+b+c))^{\frac{1}{3}}}
យើងមានៈ \displaystyle (\frac{1}{a}+\frac{1}{b}+\frac{1}{c})^2\geq 3\biggl(\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ca}\biggl)=3(a+b+c)
តាមវិសមភាព AM - GM យើងមានៈ
\displaystyle \frac{1}{a}+\frac{1}{b}+\frac{1}{c}\geq \frac{3}{\sqrt[3]{abc}}=3
\displaystyle \Rightarrow \frac{1}{a^3\sqrt[3]{b+c}}+\frac{1}{b^3\sqrt[3]{c+a}}+\frac{1}{c^3\sqrt[3]{a+b}}\displaystyle \geq \frac{9(a+b+c)}{3^{\frac{5}{3}}\sqrt[3]{2}(a+b+c)^{\frac{1}{3}}}=\frac{\sqrt[3]{3}(a+b+c)^{\frac{2}{3}}}{\sqrt[3]{2}}
ដោយ a+b+c\geq 3\sqrt[3]{abc}=3
\displaystyle \frac{1}{a^3\sqrt[3]{b+c}}+\frac{1}{b^3\sqrt[3]{c+a}}+\frac{1}{c^3\sqrt[3]{a+b}}\geq \frac{\sqrt[3]{3}.3^{\frac{2}{3}}}{\sqrt[3]{2}}=\frac{3}{\sqrt[3]{2}} ពិត ។
ដូចនេះវិសមភាពត្រូវបានស្រាយបញ្ជាក់ ។ សមភាពកើតមានពលេ a=b=c=1


38 Responses to លំហាត់ 47: van khea

  1. Pingback: មេរៀនទី 4 « ចំណេះដឹងនិងចំណេះធ្វើ !!!

  2. Pingback: មេរៀនទី 4 « 2010inequality

  3. Many people don’t understand the effects misleading effects. I knew what you meant. It can be more effective to give people little opinion with facts here. I recognize this problem but with little clue, so we must face the topic together. With new assurance, we hope that the skill of other categories can be found together. Nice people can work with great people, and great resource is important for us in analyzing the case. I’ve studied for more than 4 years to analyze them, but the fact still can’t be discovered. In order to do that, we must deal with the changes of management resources. With so many resources here, we can develop new ideas for solving to problem. Let’s give opinion, let’s discuss them, we can take charge with our resources. The last one but not least, I support your analyzes to face new one. With many resources that we can collect, all of them can be handled well.

  4. Surprising time in here; I forget to do my other activity because of your wonderful site. It doesn’t matter with me, because it is worthed and I will learn new knowledge, hope progressively I can meet with your speech. Linguists and educationalists (in my school) had conflicted and debated in several subject, I got the correction when I read the full article here. The positive effects of debatable discussion in my school are great brain for future time (for me and for my friends). Many subjects and topics with great confusing material in my school, but I have initiation step that your site has better correct conclusion. The above discussion in Linguists and educationalists is great. I’ve used several techniques for my research, for example online comparison. Would you mind if I make citation for my future project? (Of course I will tell you later, once I got the project plan in my hand). Thanks for your attention in reading my comments; you can shot me in my comment details to execute this project, so you can to be as a great part in my project.

    • kheavan says:

      thanks for visiting here. I know English a little, so i don’t understand your idea much.
      I hope to talk to you too but maybe you can’t understand me, just your comments i spent
      so much time to read , to translate. I don’t know your projects are, and what can i help you??

  5. Jene Tencer says:

    By visiting your webpage, the first impression for me is strong. I can’t imagine when and why you share this great topic but don’t spread it with social bookmarking. This information can be published as reference in online journal, or even in press release site. An early improvement in your site is great, can give us more time in your website. Would you mind if I capture several screenshot as my collection, because I’ve joined several researches? General purpose for me is to tell you about this discussion. My critical question for us is the resource that you have used to manage this site. In order to make great discussion, you are great because you post new topic in several areas. But, I suggest in giving personal opinion, please refer to big or authority sites, I am sure you will be fine in giving past or future experiences. In my environment, I am sure your capability to enrich people can be strong advantage for your future.

    • khea says:

      Thanks for visiting here. I am happy to hear that, it’s not problem for your research here. I also want to publish my resource more, but i don’t know how to show my topics and i have never though that i could create a personal site. Can you show me any way how could i do??

  6. ឌីយ៉ា says:

    ម្ចាស់ថ្លៃ ទោះបីជាញ៉ុមអ្នកបកប្រែ ក៏ខ្ចិលដែរ ហិហិ!

  7. khea says:

    ធ្វើកុំខ្ជិលមើល ជួយផងមើល ចាំប៉ាវមួយកេស 😀

  8. Lupe Aycock says:

    When I firstly saw your web, I’ve been around in many pages that I’ll use to get my final assignment. I googling with my short idea in my brain, and then I met with useful article. I hope you can give several advices to me, because your knowledge is great, it can help many students to get great score in every task or project. If you don’t mind, I will contact you to make short discussion for my project. I write my short comment for you to give support for your site; my friends had used your article as reference in specific term and condition. After that, my friends are satisfied with your article (I’ve given reference to read many pages in your site). We can make other discussion later, because it will be worth to me. Thanks in advance, hope you contact me, so we can broaden our knowledge together.

  9. Laila Ullom says:

    Before I meet your pilot website, I incredible didn’t know something about this subject that I will use in my school task. After I read page by page, I found the significant thing for my initial idea, great article in your webpage, I discover many missing file and data in my book guidance, but you gave me the several options. I am postgraduate college and want to finish my subject; it is hard task (for me). I spent over 2 weeks, just browsing and finding this nice article.

  10. hioptecas says:

    I simply had to appreciate you once again. I’m not certain the things that I would have tried without the actual aspects shown by you over such field. This has been a real distressing matter in my position, nevertheless witnessing a new skilled manner you solved the issue forced me to jump with joy. I will be thankful for this support and hope that you recognize what a powerful job you have been undertaking educating the others with the aid of your webblog. Probably you have never come across any of us.

  11. Rashida Can says:

    I just wish emphasize the excellent work on this website, has excellent views and in addition a clear vision of what you are seeking.

  12. Pleasant a person, there is actually some really good things about this web site many of my visitors might find the following valuable, I need to send out a web link, with thanks.

  13. Cinda Cason says:

    It’s the best time to make some plans for the future and it’s time to be happy. I have read this post and if I could I want to suggest you some interesting things or advice. Perhaps you could write next articles referring to this article. Pleas check articles in online poker minimum deposit www. I wish to read more things about it!

  14. i had been reading throught a few of the posts and that i find them to become very interesting. abject my english isn’t exaclty the very best. would there be anyway to transalte this into my argot, spanish. it might actually usurp us a lot. since i could compare the english terminology towards the spanish language.

  15. Elly Ollis says:

    Enjoyed the read, went well with my night cap.

  16. What i find difficult is to find a weblog that can capture me for a minute however your weblog is different. Bravo.

  17. Very efficiently written information. It is going to be helpful to anybody who usess it, including myself. Keep up the great work – for positive i will check out extra posts.

  18. Great blog here! Also your website loads up very fast! What host are you using? Can I get your affiliate link to your host? I wish my website loaded up as fast as yours lol

  19. I used to be more than happy to search out this internet-site.I wanted to thanks on your time for this glorious read!! I definitely having fun with every little little bit of it and I have you bookmarked to check out new stuff you weblog post.

  20. If you guys have absolutely nothing to say about the article topic then do not say anything at all. You guys are obviously just spamming. Please avoid such indecent behavior .

  21. I’ve been absent for a while, but now I remember why I used to love this website. Thanks , I’ll try and check back more frequently. How frequently you update your site?

  22. Many thanks for your efforts to have decided to put these things together on this weblog. Janet and I very much valued your ideas through the articles on certain things. I understand that you have many demands with your timetable and so the fact that you actually took as much time as you did to help people really like us through this article is even highly appreciated.

  23. 4400 says:

    I want to tell you that the article is not good, but the comments are 100 x better

  24. Josiah says:

    I actually couldn’t agree more with your insight on this. You did a fantastic job addressing every thing.

  25. An Shutty says:

    I love the layout of your site, what theme do you use?

  26. Pikolinos says:

    Very interesting topic , regards for posting .


Fill in your details below or click an icon to log in:

ឡូហ្កូ WordPress.com

អ្នក​កំពុង​បញ្ចេញ​មតិ​ដោយ​ប្រើ​គណនី WordPress.com របស់​អ្នក​។ Log Out / ផ្លាស់ប្តូរ )

រូប Twitter

អ្នក​កំពុង​បញ្ចេញ​មតិ​ដោយ​ប្រើ​គណនី Twitter របស់​អ្នក​។ Log Out / ផ្លាស់ប្តូរ )

រូបថត Facebook

អ្នក​កំពុង​បញ្ចេញ​មតិ​ដោយ​ប្រើ​គណនី Facebook របស់​អ្នក​។ Log Out / ផ្លាស់ប្តូរ )

Google+ photo

អ្នក​កំពុង​បញ្ចេញ​មតិ​ដោយ​ប្រើ​គណនី Google+ របស់​អ្នក​។ Log Out / ផ្លាស់ប្តូរ )

កំពុង​ភ្ជាប់​ទៅ​កាន់ %s

%d bloggers like this: