1. Monotonically increasing function

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 $\displaystyle x$.

2. Re: Monotonically increasing function

y=cosx is monotonically decreasing from 0 to 180.
y’=-sinx
Iy’I=sinx which is increasing from 0 to 90 and decreasing from 90 to 180.

dIy’I/dx needs to be positive. For above example,
dIy’I/dx=cosx which is positive from 0 to 90.

a,b positive: b<a -> b3<a3 & b4<a4
a,b negative: b<a -> b3<a3 & b4>a4