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How do I prove that x + 1/x is greater than or equal to 2 if x > 0

  1. Mar 4, 2006 #1
    How to prove this???

    How do I prove that x + 1/x is greater than or equal to 2 if x > 0

    i'm not allowed to use calculus either.

    i got that x + 1/x is greater than zero, but i can't get greater than or equal to 2.
     
  2. jcsd
  3. Mar 4, 2006 #2

    0rthodontist

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    Assume for contradiction that x > 0 yet x + 1/x < 2. Transform it into a quadratic and show that this is impossible.
     
  4. Mar 4, 2006 #3

    George Jones

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    Hint: (x - 1)^2 > 0.

    Regards,
    George
     
  5. Mar 4, 2006 #4
    First make the equation into x^2-2x+1 >= 0 by rearranging and multiplying by x.

    You should find it is (x-1)^2 so for any x (not just x>0) this function is greater than or equal to 0 because of ^2.

    For your situation it is obvious that x>0 which is certainly true for the factorised function. Therefore you have proved that x+1/x is greater than or equal to 2.
     
  6. Mar 4, 2006 #5
    before i posted this, i did get the algebraic manipulation of (x-1)^2, but i thought that only proved it for x greater than or equal to 1. but i guess since it's an equivalent statement, it's the same thing.

    thanks everyone
     
  7. Mar 5, 2006 #6
    Another method is arithmetis-geometric mean inequality.
     
  8. Mar 5, 2006 #7

    HallsofIvy

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    Strictly speaking a proof would work the other way:

    for any x, [itex](x-1)^2\ge 0[/itex] so [itex]x^2- 2x+ 1\ge 0[/itex].
    Adding 2x to both sides, [itex]x^2+ 1\ge 2x[/itex]. Finally, dividing both sides by the positive number x, [itex]x+ \frac{1}{x}\ge 2[/itex]
     
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