# Math Help - Monotonically increasing function

1. ## Monotonically increasing function

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 $x$.

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