# Lagrange's Theorem(?)

• January 10th 2010, 01:24 PM
bandedkrait
Lagrange's Theorem(?)
Given two positive real numbers http://latex.codecogs.com/gif.latex?a,b (a<b), prove that

$\sqrt{ab}<[(a-b)/(\ln(a)-\ln(b))]<(a+b)/2$

One thing that strikes me first is that the middle term is actually the point in (a,b) which satisfies Lagrange's Mean Value Theorem for the function $\ln(x)$

But I can't think of any way of proving that the point lies between the geometric mean and the arithmetic mean of the numbers a,b.

Any suggestions??
• January 14th 2010, 01:35 AM
Fresnel
maybe you can consider this
let $f(x)=\frac{x-a}{lnx-lna}-\sqrt{ax}$, then see $f'(x)$ and $f''(x)$.

another possible way: through Taylor's Mean Value Theorem. But I have not tried it.