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
<br /> f = By^{-3} - Cy^{-4}y&#039;.<br />
Now differentiate, and find conditions necessary for f&#039; \geq 0.