# Thread: Prove that this function is increasing

1. ## Prove that this function is increasing

We are given, for a,b>0

$f(a,b)=\frac{a-b}{ln a-ln b} \ \ \ if a \neq b$
and
$f(a,b)=a$ if a=b

prove that f(a,b) is strictly increasing in both a and b

I take the derivative, but I can't prove that it is positive. Help is much appreciated

2. Originally Posted by altave86
We are given, for a,b>0

$f(a,b)=\frac{a-b}{ln a-ln b} \ \ \ if a \neq b$
and
$f(a,b)=a$ if a=b

prove that f(a,b) is strictly increasing in both a and b

I take the derivative, but I can't prove that it is positive. Help is much appreciated
$b<\frac{a-b}{\ln(a)-\ln(b)}$, see this using the MVT on $f(x)=\ln(x)$.

3. Originally Posted by altave86
We are given, for a,b>0

$f(a,b)=\frac{a-b}{ln a-ln b} \ \ \ if a \neq b$
and
$f(a,b)=a$ if a=b

prove that f(a,b) is strictly increasing in both a and b

I take the derivative, but I can't prove that it is positive. Help is much appreciated

$\frac{df(a,b)}{da}=\frac{\ln a - \ln b -1 +\frac{b}{a}}{(\ln a-\ln b)^2}=\frac{\ln(a\slash b)+b\slash a -1}{(\ln a-\ln b)^2}$

Since $\ln x+x^{-1}>=1\,\,\,\forall\,x>0$ (check the minimal point of $f(x):=\ln x+\frac{1}{x}-1$) , we're done.