Consider the function $f$ defined and differentiable on the interval $[0 ; +\infty[$ by

$$f ( x ) = x \mathrm { e } ^ { - x } - 0,1$$

\begin{enumerate}
  \item Determine the limit of $f$ as $x \to +\infty$.
  \item Study the variations of $f$ on $[0 ; +\infty[$ and draw the variation table.
  \item Prove that the equation $f ( x ) = 0$ has a unique solution denoted $\alpha$ on the interval $[0 ; 1]$.
\end{enumerate}

We admit the existence of a strictly positive real number $\beta$ such that $\alpha < \beta$ and $f ( \beta ) = 0$.\\
We denote by $\mathscr { C }$ the representative curve of the function $f$ on the interval $[\alpha ; \beta]$ in an orthogonal coordinate system and $\mathscr { C } ^ { \prime }$ the curve symmetric to $\mathscr { C }$ with respect to the $x$-axis.

The unit on each axis represents 5 meters. These curves are used to delimit a floral bed in the shape of a candle flame on which tulips will be planted.

\begin{enumerate}
  \setcounter{enumi}{3}
  \item Prove that the function $F$, defined on the interval $[\alpha ; \beta]$ by
$$F ( x ) = - ( x + 1 ) \mathrm { e } ^ { - x } - 0,1 x$$
is an antiderivative of the function $f$ on the interval $[\alpha ; \beta]$.
  \item Calculate, in square units, a value rounded to 0.01 of the area of the region between the curves $\mathscr { C }$ and $\mathscr { C } ^ { \prime }$.\\
Use the following values rounded to 0.001: $\alpha \approx 0.112$ and $\beta \approx 3.577$.
  \item Knowing that 36 tulip plants can be placed per square meter, calculate the number of tulip plants needed for this floral bed.
\end{enumerate}