We propose to study in this part the function $f$ encountered in Part B question 2.\\
We recall that, for every real $x$, $f(x) = \left(6x^2 + 2x - 2\right)\mathrm{e}^{-5x+1}$.\\
We denote $f'$ the derivative function of the function $f$. We call $\mathscr{C}_f$ the representative curve of $f$ in a coordinate system of the plane.

\begin{enumerate}
  \item We admit that $\lim_{x \rightarrow +\infty} f(x) = 0$. Determine the limit of the function $f$ at $-\infty$.
  \item Using Part A (Exercise 4), show that $\mathscr{C}_f$ intersects the $x$-axis at two points (the coordinates of these points are not expected).
  \item Using Parts A and B (Exercise 4), show that $\mathscr{C}_f$ has two horizontal tangent lines.
  \item Draw the complete variation table of the function $f$.
  \item Determine by justifying the number of solution(s) of the equation $f(x) = 1$.
  \item For every real $m$ strictly greater than 0.2, we define $I_m$ by $I_m = \int_{0.2}^{m} f(x)\,\mathrm{d}x$.\\
a. Verify that the function $F$ defined on $\mathbb{R}$ by
$$F(x) = \left(-\frac{6}{5}x^2 - \frac{22}{25}x + \frac{28}{125}\right)\mathrm{e}^{-5x+1}$$
is a primitive of the function $f$ on $\mathbb{R}$.\\
b. Does there exist a value of $m$ for which $I_m = 0$? Interpret this result graphically.
\end{enumerate}