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The Sum of a Number and Its Reciprocal

Date: 11/04/2003 at 11:58:02
From: Amy
Subject: sum of a number and its reciprocal

I need a formal proof showing that the sum of a positive number and
its reciprocal is at least 2. I can prove it algebraically, but I need
a visual justification.

Date: 11/04/2003 at 14:22:50
From: Doctor Douglas
Subject: Re: sum of a number and its reciprocal

Hi Amy -

For a geometric demonstration, you can do the following:

  1.  Graph y = x and y = 1/x on the same coordinate plane.
      Obviously, both curves go through the point (1,1).
      Also add the horizontal lines y = 2 and y = 1 and the line
      y = 2 - x to the same graph.

  2.  Note that for both regions 0 < x < 1 and x > 1, the reciprocal
      curve y = 1/x lies *above* the "deficit" curve y = 2 - x.  This
      means that when you add the reciprocal 1/x to x, you must 
      obtain a number that exceeds 2.0.

  3.  At x = 1, both the graphs of y = x and y = 1/x have values of 1,
      so their sum is 2.  Thus, for all x > 0, the sum of x and 1/x is
      at least 2.

Of course, these results can be adapted into an algebraic proof as
well.  We want to show that x + 1/x >= 2 for all x > 0.

  x + 1/x >= 2

We can multiply through by x without flipping the inequality since we 
know x > 0, or positive:

  x^2 + 1      >= 2x
  x^2 - 2x + 1 >= 0
  (x - 1)^2    >= 0

Clearly this last statement is true for all x since for x = 1 we get 0 
= 0 and for any x other than 1 the result of (x - 1)^2 is positive or 
greater than 0.  Keeping in mind, though, that one stage in solving
was predicated on x being greater than 0, we can now only safely say
that for all x > 0 the sum of x and 1/x must be > or = to 2.

I hope this helps.  Please write back if you have more questions
about this.

- Doctor Douglas, The Math Forum 
Associated Topics:
High School Equations, Graphs, Translations
High School Polynomials

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