Results 1 to 8 of 8

Math Help - inequality

  1. #1
    Junior Member
    Joined
    Jan 2009
    Posts
    42

    inequality

    Prove that the positive numbers a, b and c satisfy the inequality

    \frac{1}{a^{3}+b^{3}+abc }+\frac{1}{b^{3}+c^{3}+abc}+\frac{1}{c^{3}+a^{3}+a  bc}\leq\frac{1}{abc}
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Junior Member
    Joined
    Oct 2008
    From
    Guernsey
    Posts
    69
    x^3 + y^3 \ge xy(x + y)

    Therefore

    \sum \frac {1}{a^3 + b^3 + abc} \leq \sum \frac {1}{ab(a + b + c)} = \sum \frac {c}{abc(a + b + c)} = \frac {1}{abc}
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Jan 2009
    Posts
    42
    can you tell me a little bit more why it is so please
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Junior Member
    Joined
    Oct 2008
    From
    Guernsey
    Posts
    69
    Which step didn't you understand, I'll go into that in more detail
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Junior Member
    Joined
    Jan 2009
    Posts
    42
    i didn't understand this step :
    Quote Originally Posted by SimonM View Post

    \sum \frac {1}{a^3 + b^3 + abc} \leq \sum \frac {1}{ab(a + b + c)} = \sum \frac {c}{abc(a + b + c)} = \frac {1}{abc}
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Junior Member
    Joined
    Oct 2008
    From
    Guernsey
    Posts
    69
    a^3+b^3+abc \ge ab(a+b)+abc = ab(a+b+c)

    Now take the reciprocal
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Junior Member
    Joined
    Jan 2009
    Posts
    42
    and why this :  \sum \frac{c}{abc(a + b + c)} = \frac {1}{abc}<br />
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Junior Member
    Joined
    Oct 2008
    From
    Guernsey
    Posts
    69
    \sum \frac{c}{abc(a+b+c)} = \frac{c}{abc(a+b+c)} + \frac{a}{abc(a+b+c)} + \frac{b}{abc(a+b+c)}  =\frac{c+a+b}{abc(a+b+c)}
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 2
    Last Post: January 11th 2011, 08:20 PM
  2. Replies: 3
    Last Post: December 12th 2010, 01:16 PM
  3. inequality
    Posted in the Math Challenge Problems Forum
    Replies: 7
    Last Post: July 25th 2010, 06:11 PM
  4. Inequality help
    Posted in the Pre-Calculus Forum
    Replies: 5
    Last Post: July 8th 2010, 06:24 AM
  5. Inequality :\
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: October 12th 2009, 01:57 PM

Search Tags


/mathhelpforum @mathhelpforum