Visualizing 1 + 1/x Date: 10/10/2003 at 21:25:38 From: Mary
Subject: logic How can I show that the sum of a positive number
and its reciprocal is at least two? Date: 10/11/2003 at 06:06:27
From: Doctor Luis Subject: Re: logic Hi Mary, Adding a positive
number x > 0 and its reciprocal 1/x gives you the function f(x)
= x + 1/x If you're familiar with calculus, you can see that
solving for the extrema points gives you f'(x) = 1 - 1/x^2 = 0 1
= 1/x^2 x^2 = 1 x = 1 (reject negative root since x > 0) Since
f"(x) = 2/x^3 is positive for x>0, we know that f(x) is concave
upward. This means that the critical point x=1 gives you a
minimum. This minimum value is f(1) = 1 + 1/1 = 2. In the
following diagram, I've graphed the two functions y = x + 1/x
and y = 2. Even if you are not familiar with calculus maybe you
can follow the following chain of reasoning: The square of any
nonzero real number is positive. As an inspired guess, pick x-1
as the real number to be squared. Then, (x-1)^2 >= 0 (True for
all x. Equality holds