Does the function $y(x)$ increasing /decreasing?

Question

Given two decreasing functions $H(y), B(y)$, And the implicit function $y(x)$:

$3H(y)-4y^3+x^5+1=-B(y)-e^{5x+1}$.

What can I say about the function $y(x)$ in terms of increasing /decreasing?

I know that it's an odd polynomial, but I can't see how it has to do anything about increasing /decreasing.

Thank you.

Answer 1

Yuo have that $-4y^3=-B(y)-e^{5x+1}-x^5-1-3H(y)$, so you have $y^3=\frac{1}{4} \cdot(B(y)+e^{5x+1}+x^5+1+3H(y))$.

If $B(y)=H(y)=-y^3$ then $y$ is increasing.

If $B(y)=H(y)={2y^3}$ then $y$ is decreasing.

So, both can happen.

Another approach (more general approach):

You have $3H(y)-4y^3+x^5+1=-B(y)-e^{5x+1}$,

so you have $-B(y)-3H(y)+4y^3=x^5+1+e^{5x+1}$

Write $G(y)=-B(y)-3H(y)$ and $f(x)=x^5+1+e^{5x+1}$.

Then $4y^3=f(x)+(-G(y))$, so $y^3$ is a sum of increasing $f$ and decreasing $-G(y)$, so you see that you cannot say anything in this generality about $y(x)$ because you do not know anything about a manner in which $-G(y)$ decreases.