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Math question

  1. Sep 15, 2003 #1
    ok i have a really stupid problem ....but i can't seem to get it...would appreciate any help...
    how do u prove that x+1/x >= 2 for all x>0.....
    is the above state ment true in the first place..i think it is...but can't seem to prove it...i keep coming to the fact that it holds for x>1 and equality holds for x=1 but can't seem to prove it...obviously for 0<x<1.........
     
  2. jcsd
  3. Sep 15, 2003 #2

    HallsofIvy

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    Since you are specifying that x> 0, you can multiply through the inequality by x without any problem:

    You want to get x+ 1/x>= 2 and, multiplying by x, that is the same as x2+ 1>= 2x.

    That, of course, is the same as x2- 2x+ 1>= 0 and you KNOW that's true.

    Well, HOW do you know it's true? Also remember you have to ARRIVE at "x+ 1/x>= 2", not start there. Start with what you KNOW is true and work backwards.
     
  4. Sep 15, 2003 #3
    Another method is to use the fact that A.M. > = G.M. as x > 0

    Where AM stands for arithmatic mean and GM stands for Geometric mean.
     
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