Comparing two functions: Proving f(x) > g(x) for all x > 0

[the_kid]

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]

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?

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.

 

[the_kid]

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.

Right. That was the other method I tried. I just can't figure out a way to do it without some "hand-waving."

 

[jbunniii]

Not sure if this simplifies anything, but if 0 < x < 1, then 1/x^2 > 1/x so clearly f(x) > g(x). So you only have to worry about x > 1.

 

[HallsofIvy]

Using that, a fourth method: it is easy to show this is true for x= 1 so if you can prove that f'(x)> g'(x) for all x>1 you are done.

 

[SammyS]

Right. That was the other method I tried. I just can't figure out a way to do it without some "hand-waving."

How about comparing f(1/x) and g(1/x) ?

 

[the_kid]

How about comparing f(1/x) and g(1/x) ?

Hmmm, OK, how does this help me?

 

[Ray Vickson]

Well, derivatives are a bit simpler, etc.

RGV

 

[the_kid]

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)?

 

[SammyS]

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)?

That shows that f(x)>g(x) for 0 < x < 1, which isn't a big help.