1. ## analysis help

for positive real numbers x how to show that

1/x+1 < or = ln(x+1)/x < or = to 1

2. ## Re: analysis help

Originally Posted by sbacon1991
for positive real numbers x how to show that
1/x+1 < or = ln(x+1)/x < or = to 1
Napier"s inequality is:
If $\displaystyle 0<a<b$ then $\displaystyle \frac{1}{b}\le\frac{\ln(b)-\ln(a)}{b-a}\le\frac{1}{a}$.

That is proved using the mean value theorem.

Let $\displaystyle b=x+1~\&~a=1$.

3. ## Re: analysis help

Show it using our de nition of ln(x) as intergral between 1 and x of 1/t dt. do not use any other defination. and how could you draw a diagram of this

4. ## Re: analysis help

Originally Posted by sbacon1991
Show it using our de nition of ln(x) as intergral between 1 and x of 1/t dt. do not use any other defination. and how could you draw a diagram of this
Why did you not say that in the first place.
Let $\displaystyle x>0$ and $\displaystyle \ln (x) = \int_1^x {\frac{1}{t}dt}$
Now clearly $\displaystyle \frac{d\ln(x)}{dx}=\frac{1}{x}.$
Use the mean value theorem to prove the Napier inequality.

5. ## Re: analysis help

what do you mean by uthe sing mean value theorem to prove the Napier inequality?

6. ## Re: analysis help

Originally Posted by sbacon1991
what do you mean by uthe sing mean value theorem to prove the Napier inequality?
Do you know what the mean value theorem says?
You posted this in the analysis forum.
Therefore, we can safely assume that something as basic as that theorem can be used. If not, what level is this question on?

7. ## Re: analysis help

its a university question and is there any way to draw a graph for the answer?