update: add one condition according to answer below.

I post this question in MSE a week ago. I thought this should be an easy freshman exercise, but it turns out not easy...

The original question is very complicated, involving Bounded variation and other stuff. But I managed to simplify it into a simple algebra question, which stated below.

Define for $f(x)$, $x\geq 0$, (I don't care what happend for $x\leq 0$) $$ f(x):=\sum_{k=1}^\infty \frac{a_k^2\lambda_k^2x}{(1+x\lambda_k)^2} - \sum_{k=1}^\infty \frac{b_k^2\beta_k}{(1+x\beta_k)^3} $$ where $a_k\in \mathbb R$, $b_k\in\mathbb R$, $\lambda_k>0$, $\beta_k> 0$, and $$ \sum_{k=1}^\infty b_k^2< \sum_{k=1}^\infty a_k^2<\infty\,\text{ and }\sum_{k=1}^\infty a_k^2 \lambda_k< \sum_{k=1}^\infty b_k^2 \beta_k<\infty. $$ Additional assumption: we may think each $\beta_k$ is very large. You may take it as large as you want.

I am trying to prove that $f(x)$ has following graph. That is, prove that there exists $x_0>0$ such that $f(x_0)=0$, and for all $x<x_0$, $f(x)<0$, and for all $x>x_0$, $f(x)>0$.

**Also, feel free to add whatever assumption on $a_k$, $b_k$, $\lambda_k$ and $\beta_k$ which you think won't trivial the question but reasonable. I wish to have at least some of them can work**

**Moreover, I may assume that $k$ is finite. i.e., we only have finite sum in $f$, not infinite.**