How do you Use the Mean Value Theorem to prove this

• Feb 17th 2011, 05:26 AM
maximus101
How do you Use the Mean Value Theorem to prove this
Use the Mean Value Theorem to prove that if p > 1 then $(1+x)^p$ $>$ $1+px$ for
$x$ $\in$ (−1, 0) U (0, $\infty$).
• Feb 17th 2011, 07:52 AM
Sudharaka
Quote:

Originally Posted by maximus101
Use the Mean Value Theorem to prove that if p > 1 then $(1+x)^p$ $>$ $1+px$ for
$x$ $\in$ (−1, 0) U (0, $\infty$).

Dear maximus101,

Using the Taylor's theorem you could write,

$(1+x)^p=1+px+p(p-1)(1+c)^{p-2}x^2~\text{where}~0------(A)

$p>1\Rightarrow{p(p-1)>0}$-----(1)

If $0-----(2)

If $x----(3)

$x>-1\Rightarrow{1+x>0}$------(4)

By, (3) and (4); $If~x----(5)

By (2) and (5); for both cases, $0------(6)

Therefore, by (1) and (6); $p(p-1)(1+c)^{p-2}x^2>0~if~x\neq{0}~and~x>-1$-----(6)

By (A);

$(1+x)^p-1-px=p(p-1)(1+c)^{p-2}x^2>0~if~x\neq{0}~and~x>-1$

$(1+x)^p>1+px~if~x>-1~and~x\neq{0}$
• Feb 19th 2011, 06:24 AM
maximus101
Hey, thank you for this it was very helpful could you explain how I could use the mean value theorem to prove it?
• Feb 19th 2011, 06:50 AM
Sudharaka
Quote:

Originally Posted by maximus101
Hey, thank you for this it was very helpful could you explain how I could use the mean value theorem to prove it?

There are several mean value theorems; Rolle's mean value theorem, Cauchy's mean value theorem and the Taylor's mean value theorem. Please refer Mean value theorem - Wikipedia, the free encyclopedia. I have used the Taylor's mean value theorem in the above answer.
• Feb 19th 2011, 07:08 AM
maximus101
Quote:

Originally Posted by Sudharaka
There are several mean value theorems; Rolle's mean value theorem, Cauchy's mean value theorem and the Taylor's mean value theorem. Please refer Mean value theorem - Wikipedia, the free encyclopedia. I have used the Taylor's mean value theorem in the above answer.

Hey, sorry for my mistake, I meant the first one showing

$f'(c)$= $\frac{f(b)-f(a)}{b-a}$

kind thanks
• Feb 19th 2011, 07:12 AM
Sudharaka
Quote:

Originally Posted by maximus101
Hey, sorry for my mistake, I meant the first one showing

$f'(c)$= $\frac{f(b)-f(a)}{b-a}$

kind thanks

That is also know as Lagrange's mean value theorem. It is used to prove Taylor's theorem. Therefore it had been used in the answer.
• Feb 19th 2011, 07:34 AM
maximus101
ok I worked it out :) thank you