We admit that the function $f$ from part $\mathbf{A}$ is defined on $\mathbb{R}$ by

$$f(x) = \left(x^{2} - 5x + 6\right)\mathrm{e}^{x}$$

We denote $\mathscr{C}$ the representative curve of the function $f$ in a coordinate system.

\begin{enumerate}
  \item a. Determine the limit of the function $f$ at $+\infty$.\\
b. Determine the limit of the function $f$ at $-\infty$.
  \item Show that, for all real $x$, we have $f^{\prime}(x) = \left(x^{2} - 3x + 1\right)\mathrm{e}^{x}$.
  \item Deduce the direction of variation of the function $f$.
  \item Determine the reduced equation of the tangent line $(\mathscr{T})$ to the curve $\mathscr{C}$ at the point with abscissa 0.
\end{enumerate}

We admit that the function $f$ is twice differentiable on $\mathbb{R}$. We denote $f^{\prime\prime}$ the second derivative function of $f$. We admit that, for all real $x$, we have $f^{\prime\prime}(x) = (x+1)(x-2)\mathrm{e}^{x}$.\\
5. a. Study the convexity of the function $f$ on $\mathbb{R}$.\\
b. Show that, for all $x$ belonging to the interval $[-1; 2]$, we have $f(x) \leqslant x + 6$.