We consider the function $f$ defined on $\mathbb{R}$ by:
$$f(x) = \mathrm{e}^{3x} - (2x+1)\mathrm{e}^{x}$$

The purpose of this exercise is to study the function $f$ on $\mathbb{R}$.

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

We define the function $g$ on $\mathbb{R}$ by:
$$g(x) = 3\mathrm{e}^{2x} - 2x - 3$$

\begin{enumerate}
  \item a. Determine the limit of function $g$ at $-\infty$.\\
b. Determine the limit of function $g$ at $+\infty$.
  \item a. We admit that function $g$ is differentiable on $\mathbb{R}$, and we denote by $g'$ its derivative function. Prove that for every real number $x$, we have $g'(x) = 6\mathrm{e}^{2x} - 2$.\\
b. Study the sign of the derivative function $g'$ on $\mathbb{R}$.\\
c. Deduce the table of variations of function $g$ on $\mathbb{R}$. Verify that function $g$ has a minimum equal to $\ln(3) - 2$.
  \item a. Show that $x = 0$ is a solution of the equation $g(x) = 0$.\\
b. Show that the equation $g(x) = 0$ has a second non-zero solution, denoted $\alpha$, for which you will give an interval of amplitude $10^{-1}$.
  \item Deduce from the previous questions the sign of function $g$ on $\mathbb{R}$.
\end{enumerate}

\textbf{Part B - Study of function $f$}

\begin{enumerate}
  \item Function $f$ is differentiable on $\mathbb{R}$, and we denote by $f'$ its derivative function. Prove that for every real number $x$, we have $f'(x) = \mathrm{e}^{x} g(x)$, where $g$ is the function defined in Part A.
  \item Deduce the sign of the derivative function $f'$ and then the variations of function $f$ on $\mathbb{R}$.
  \item Why is function $f$ not convex on $\mathbb{R}$? Explain.
\end{enumerate}