Lagrange's Theorem(?)

Given two positive real numbers http://latex.codecogs.com/gif.latex?a,b (a<b), prove that



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

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??

let , then see and .

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