The Sum of a Number and Its ReciprocalDate: 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 http://mathforum.org/dr.math/ |
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