1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Inequality proof

  1. Jul 29, 2009 #1
    1. The problem statement, all variables and given/known data
    If a, b, c > 0, prove [tex]\frac{a}{b} + \frac{b}{c} + \frac{c}{a} \ge 3[/tex]

    2. Relevant equations

    3. The attempt at a solution
    I'm not so sure how to do this. Usually I would try to prove that [tex]\frac{a}{b} + \frac{b}{c} + \frac{c}{a} - 3 \ge 0[/tex] but this gets me nowhere: [tex]\frac{a^2 c + b^2 a + c^2 b - 3abc}{abc}[/tex]. I can't factorise the numerator.

    I know of a similar inequality that I can prove easily using this method, which is [tex]\frac{a}{b} + \frac{b}{a} \ge 2[/tex] but the inequality above is harder for me. Please help.
  2. jcsd
  3. Jul 29, 2009 #2
    Are a, b, and c integers?
  4. Jul 29, 2009 #3
    It doesn't matter if a, b, c are integers or not. The inequality holds for real a,b,c > 0.

    Clear denominators, divide by 3, and apply the AM-GM inequality.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook