Monotonically increasing function

[JulieK]

I have the following function

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

where $B$ and $C$ are constants and where $y$ is a monotonically
decreasing function of $x$ ($\mid\frac{dy}{dx}\mid$ stands for absolute value of derivative). According to my model, all
signs indicate that $f$ is a monotonically increasing function of
$x$. Numerical experiments and logical arguments confirm this but
I need a rigorous proof of this. If $f$ is not unconditionally monotonically
increasing function of $x$ I wish to know under what conditions it
will be monotonically increasing function of $x$.

 

[pasmith]

I have the following function

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

where $B$ and $C$ are constants and where $y$ is a monotonically
decreasing function of $x$ ($\mid\frac{dy}{dx}\mid$ stands for absolute value of derivative). According to my model, all
signs indicate that $f$ is a monotonically increasing function of
$x$. Numerical experiments and logical arguments confirm this but
I need a rigorous proof of this. If $f$ is not unconditionally monotonically
increasing function of $x$ I wish to know under what conditions it
will be monotonically increasing function of $x$.

If $y' \leq 0$ then $|y'| = -y'$, so you have
$$f = By^{-3} - Cy^{-4}y'.$$
Now differentiate, and find conditions necessary for $f' \geq 0$.