Results 1 to 4 of 4

Math Help - Inequality

  1. #1
    Newbie
    Joined
    Aug 2008
    Posts
    9

    Inequality

    Prove that for all positive real numbers x,y,z,

    2+\frac{1}{xyz}\ \ge\ \frac{9}{x+y+z}.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7
    Quote Originally Posted by Sigyn3 View Post
    Prove that for all positive real numbers x,y,z,


    2+\frac{1}{xyz}\ \ge\ \frac{9}{x+y+z}.
    by AM-GM we have x+y+z \geq 3 \sqrt[3]{xyz}. thus: \frac{9}{x+y+z} \leq \frac{3}{\sqrt[3]{xyz}}. so we only need to prove that \frac{3}{\sqrt[3]{xyz}} \leq 2 + \frac{1}{xyz}. \ \ \ \ \ \ (1)

    let xyz=\frac{1}{a^3}, \ a > 0. then (1) becomes a^3 - 3a + 2 \geq 0, which is obviously true because a^3 - 3a + 2 =(a-1)^2(a+2). \ \ \ \square
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member JaneBennet's Avatar
    Joined
    Dec 2007
    Posts
    293
    Another solution.

    By AM–GM, x+y+\frac{1}{xy}\ \ge3\ \sqrt[3]{(x)(y)\left(\frac{1}{xy}\right)}\ =\ 3.

    Similarly y+z+\frac{1}{yz}\ \ge\ 3.

    And z+x+\frac{1}{zx}\ \ge\ 3.

    Adding up,

    2(x+y+z)+\frac{1}{xy}+\frac{1}{yz}+\frac{1}{zx}\ \ge\ 9

    \Rightarrow\ (x+y+z)\left(2+\frac{1}{xyz}\right)\ \ge\ 9
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7
    here's a general form of the inequality that i just made it up:

    suppose \alpha, \beta, \gamma are positive real numbers with \frac{\gamma^3}{\alpha^2 \beta} \leq \frac{729}{4}. prove that for all positive real numbers x,y,z: \ \ \alpha + \frac{\beta}{xyz} \geq \frac{\gamma}{x+y+z}.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 2
    Last Post: January 11th 2011, 09:20 PM
  2. Replies: 3
    Last Post: December 12th 2010, 02:16 PM
  3. inequality
    Posted in the Math Challenge Problems Forum
    Replies: 7
    Last Post: July 25th 2010, 07:11 PM
  4. Inequality help
    Posted in the Pre-Calculus Forum
    Replies: 5
    Last Post: July 8th 2010, 07:24 AM
  5. Inequality :\
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: October 12th 2009, 02:57 PM

Search Tags


/mathhelpforum @mathhelpforum