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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

- Doctor Douglas, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Equations, Graphs, Translations
High School Polynomials

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