Inequality proof

  • Thread starter deancodemo
  • Start date
  • #1
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Homework Statement


If a, b, c > 0, prove [tex]\frac{a}{b} + \frac{b}{c} + \frac{c}{a} \ge 3[/tex]


Homework Equations





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.
 

Answers and Replies

  • #2
56
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Are a, b, and c integers?
 
  • #3
1,101
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.
 

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