Results 1 to 5 of 5

Thread: inequalities

  1. #1
    Junior Member
    Joined
    Sep 2009
    Posts
    40

    inequalities

    If$\displaystyle x+y = 3z$ prove that $\displaystyle x^2 + y^2 >= 3z^2$

    Thanks
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Oct 2009
    Posts
    5,577
    Thanks
    790
    Note that $\displaystyle 3z^2=3(3z/3)^2$, so one has to show that $\displaystyle x^2+y^2\ge3((x+y)/3)^2$. After multiplying both sides by 3 and moving everything to the left, that expression can be represented as a sum of two squares.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member
    Joined
    Jun 2009
    Posts
    671
    Thanks
    136
    Quote Originally Posted by emakarov View Post
    Note that $\displaystyle 3z^2=3(3z/3)^2$, so one has to show that $\displaystyle x^2+y^2\ge3((x+y)/3)^2$. After multiplying both sides by 3 and moving everything to the left, that expression can be represented as a sum of two squares.
    Maybe I have a blind spot, but I don't see that last step of getting to the sum of two squares. Is there some identity that I've forgotten ? .

    I can get to the result using the AM >= GM inequality, (so $\displaystyle x^{2}+y^{2}\geq 2xy).$
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Joined
    Oct 2009
    Posts
    5,577
    Thanks
    790
    $\displaystyle \begin{aligned}
    x^2+y^2&\ge3((x+y)/3)^2\Leftrightarrow{}\\
    3x^2+3y^2-x^2-2xy-y^2&\ge0\Leftrightarrow{}\\
    x^2-xy+y^2&\ge0\Leftrightarrow{}\\
    (x-y/2)^2+3y^2/4&\ge0
    \end{aligned}
    $
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Super Member
    Joined
    Jun 2009
    Posts
    671
    Thanks
    136
    Nice one. Thanks, I didn't see that .
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. inequalities
    Posted in the Algebra Forum
    Replies: 7
    Last Post: Aug 20th 2011, 04:56 AM
  2. inequalities
    Posted in the Algebra Forum
    Replies: 0
    Last Post: Mar 20th 2011, 02:42 AM
  3. Inequalities
    Posted in the Pre-Calculus Forum
    Replies: 1
    Last Post: Jul 1st 2009, 04:35 AM
  4. Inequalities
    Posted in the Algebra Forum
    Replies: 4
    Last Post: Jun 15th 2009, 11:26 AM
  5. inequalities
    Posted in the Pre-Calculus Forum
    Replies: 1
    Last Post: Oct 14th 2008, 09:57 PM

Search Tags


/mathhelpforum @mathhelpforum