We consider the function $f$ defined on $\mathbb{R}$ by
$$f(x) = \ln\left(\mathrm{e}^{2x} - \mathrm{e}^{x} + 1\right).$$
We denote $\mathscr{C}_f$ its representative curve.

A student formulates the following conjectures based on this graphical representation:
\begin{enumerate}
  \item The equation $f(x) = 2$ seems to admit at least one solution.
  \item The largest interval on which the function $f$ seems to be increasing is $[-0{,}5; +\infty[$.
  \item The equation of the tangent line at the point with abscissa $x = 0$ seems to be: $y = 1{,}5x$.
\end{enumerate}

\textbf{Part A: Study of an auxiliary function}

We define on $\mathbb{R}$ the function $g$ defined by
$$g(x) = \mathrm{e}^{2x} - \mathrm{e}^{x} + 1.$$

\begin{enumerate}
  \item Determine $\lim_{x \rightarrow -\infty} g(x)$.
  \item Show that $\lim_{x \rightarrow +\infty} g(x) = +\infty$.
  \item Show that $g'(x) = \mathrm{e}^{x}\left(2\mathrm{e}^{x} - 1\right)$ for all $x \in \mathbb{R}$.
  \item Study the monotonicity of the function $g$ on $\mathbb{R}$. Draw up the variation table of the function $g$ showing the exact value of the extrema if any, as well as the limits of $g$ at $-\infty$ and $+\infty$.
  \item Deduce the sign of $g$ on $\mathbb{R}$.
  \item Without necessarily carrying out the calculations, explain how one could establish the result of question 5 by setting $X = \mathrm{e}^{x}$.
\end{enumerate}

\textbf{Part B}

\begin{enumerate}
  \item Justify that the function $f$ is well defined on $\mathbb{R}$.
  \item The derivative function of the function $f$ is denoted $f'$. Justify that $f'(x) = \frac{g'(x)}{g(x)}$ for all $x \in \mathbb{R}$.
  \item Determine an equation of the tangent line to the curve at the point with abscissa 0.
  \item Show that the function $f$ is strictly increasing on $[-\ln(2); +\infty[$.
  \item Show that the equation $f(x) = 2$ admits a unique solution $\alpha$ on $[-\ln(2); +\infty[$ and determine an approximate value of $\alpha$ to $10^{-2}$ near.
\end{enumerate}

\textbf{Part C}

Using the results of Part B, indicate, for each conjecture of the student, whether it is true or false. Justify.