The discussion revolves around demonstrating that the function f(x)=1/x² is greater than g(x)=1/x - ln(1+1/x) for all x > 0. Participants are exploring various methods to establish this inequality rigorously.
There are multiple lines of reasoning being explored, including derivative comparisons and transformations of the functions. Some participants have noted specific cases where the inequality holds, but there is no explicit consensus on a complete method yet.
Participants mention the challenge of rigorously proving the inequality without resorting to informal reasoning. There is a focus on the behavior of the functions in different ranges of x, particularly around the point x = 1.
Well... then the function (f-g)(x) must be positive ...the_kid said:Homework Statement
I'm trying to show that the function f(x)=\frac{1}{x^{2}} is greater than g(x)=\frac{1}{x}-ln(1+\frac{1}{x}) for all x>0.
Homework Equations
The Attempt at a Solution
This is really simple, but I'm having some trouble showing it rigorously. I thought about trying to compare the derivatives, but it's not that simple. Are there any good ways to do this?
SammyS said:Well... then the function (f-g)(x) must be positive ...
the function (f/g)(x) must be greater than 1, or be negative ...
the function (g/f)(x) must be less than 1, or be negative.
How about comparing f(1/x) and g(1/x) ?the_kid said:Right. That was the other method I tried. I just can't figure out a way to do it without some "hand-waving."
SammyS said:How about comparing f(1/x) and g(1/x) ?
That shows that f(x)>g(x) for 0 < x < 1, which isn't a big help.the_kid said:I've been able to show that f(1/x)>g(1/x) for all x>1. From this, how exactly can I conclude that f(x)>g(x)?