I have the following function

$\displaystyle f=\frac{B}{y^{3}}+\frac{C}{y^{4}}\mid\frac{dy}{dx} \mid$

where $\displaystyle B$ and $\displaystyle C$ are constants and where $\displaystyle y$ is a monotonically

decreasing function of $\displaystyle x$ ($\displaystyle \mid\frac{dy}{dx}\mid$ stands for absolute value of derivative). According to my model, all

signs indicate that $\displaystyle f$ is a monotonically increasing function of

$\displaystyle x$. Numerical experiments and logical arguments confirm this but

I need a rigorous proof of this. If $\displaystyle f$ is not unconditionally monotonically

increasing function of $\displaystyle x$ I wish to know under what conditions it

will be monotonically increasing function of $\displaystyle x$.