I want to find the upper bound value of $x$ in $$ax^2+bx+c=x^{ax^2+bx+c}, \tag{1}$$ where $a,b,c\in \mathbb{R}$.
This is how I've approached the problem:
Let $q=ax^2+bx+c$. Then, $(1)$ becomes $$\begin{align} q&=x^q \\ \log_x({q})&=q \\ \frac{\ln({q})}{\ln({x})}&=q \\ \tag{2}x&=e^{\frac{\ln({q})}{q}}.\end{align}$$
Let $f(x)=\frac{\ln({x})}{x}$. The maximum of $f$ would occur at an $x=a$ such that $f'(a)=0$.
We find that
$$f'(x)=\frac{1-\ln({x})}{x^2},$$
and so
$$\frac{1-\ln({x})}{x^2}=0 \implies x=e$$
With reference to $(2)$, the largest value possible for $x$ is thus $$x=e^{\frac{\ln({e})}{e}}=\sqrt[e]{e}.$$
I have a few questions about this:
Is this correct?
I am unsure of the validity of $(2)$ and its use since $q$ is expressed in terms of $x$.
Does the relationship between $q$ and $x$ affect the logic of the argument?
Furthermore, is my derivation of the maximum of $f$ correct? I made an assumption that the maximum occurs at $x=a$ such that $f'(a)=0$, but can't really back it up.
Overall, how was the wording of the work? I'm currently at the high school level, so I'm not too familiar with this type of math formatting.
Thanks in advance for your suggestions!
Note: this question is a "byproduct" of this one.