Given $r_1, r_2 \in \Bbb R^+$, let the function $f : \Bbb R^+ \to \Bbb R^+$ be defined by $$f(x) = \frac{1- \text{exp}(-(r_1 +r_2)x)}{1-\text{exp}(-r_1x)}$$ I need to show that $f$ is increasing in $x$ for all $x > 0$.
I've tried obviously to differentiate, without success. Can someone help me? I basically get that this is increasing if and only if:
$\frac{1- \text{exp}(-(r_1 +r_2)\Lambda)}{1-\text{exp}(-r_2\Lambda)} > 1 + r_1/r_2$
I differentiate again the LHS and and I get that the LHS is increasing in $x$ iff (I did this to try to find some contradiction)
$\frac{1- \text{exp}(-(r_1 +r_2)\Lambda)}{1-\text{exp}(-r_1\Lambda)} > 1 + r_2/r_1$
And so on... What should I conclude!?
Edit: the domain and the codomain are $R_{+}$
