We consider the function $f$ defined for all real $x$ by $f(x) = x\mathrm{e}^{1-x^{2}}$.

\begin{enumerate}
  \item Calculate the limit of the function $f$ at $+\infty$.\\
Hint: you may use the fact that for all real $x$ different from 0,\\
$f(x) = \frac{\mathrm{e}}{x} \times \frac{x^{2}}{\mathrm{e}^{x^{2}}}$.\\
It is admitted that the limit of the function $f$ at $-\infty$ is equal to 0.
  \item a. It is admitted that $f$ is differentiable on $\mathbb{R}$ and we denote by $f'$ its derivative. Prove that for all real $x$,
$$f'(x) = \left(1 - 2x^{2}\right)\mathrm{e}^{1-x^{2}}$$
b. Deduce the table of variations of the function $f$.
\end{enumerate}