Finding the conditions that $a,b,c$ should satisfy s.t. $f^{(2)}(x_{0})\approx0$, where $f(x)=ax^{4}+bx^{3}-c$ and $a,b,c$ are all positive.

Recently, I encountered a problem related to finding the roots of an equation, which has troubled me for quite some time. The problem can be described as follows:

Given that $a,b,c$ are all arbitrary positive real numbers, how can I determine the value of the second derivative function at the the root of its first derivative function?

The function is of the form $f(x) = ax^{4} + bx^{3} - cx$, and its first derivative is:$f'(x) = 4ax^{3} + 3bx^{2} - c$

Since $c$ is greater than $0$ and the above derivative is increasing and negative at $x=0$, this equation has only one positive root. Let's assume this positive root is $x_{0}$.

I want to obtain a result where $f^{(2)}(x_{0})$ is relatively small. What conditions should $a,b$ and $c$ satisfy to achieve this result?

Mathematica tells me that:

However, I am puzzled by the appearance of exponential functions in the final result. I would greatly appreciate any guidance or clarification from the experts. Moreover, the provided result does not help me identify the conditions that $a,b,c$ should meet in order to make $f^{(2)}(x_{0})\approx 0$.

Thank you to all the experts for your guidance!

Greatly appreciated!