\section*{Exercise 2}

During a laboratory experiment, a projectile is launched into a fluid medium. The objective is to determine for which firing angle $\theta$ with respect to the horizontal the height of the projectile does not exceed 1.6 meters.\\
Since the projectile does not move through air but through a fluid, the usual parabolic model is not adopted.\\
Here we model the projectile as a point that moves, in a vertical plane, on the curve representing the function $f$ defined on the interval $[0; 1[$ by:
$$f(x) = bx + 2\ln(1-x)$$
where $b$ is a real parameter greater than or equal to 2, $x$ is the abscissa of the projectile, $f(x)$ its ordinate, both expressed in meters.

\begin{enumerate}
  \item The function $f$ is differentiable on the interval $[0; 1[$. We denote $f'$ its derivative function.

We admit that the function $f$ has a maximum on the interval $[0; 1[$ and that, for every real $x$ in the interval $[0; 1[$:
$$f'(x) = \frac{-bx + b - 2}{1 - x}$$
Show that the maximum of the function $f$ is equal to $b - 2 + 2\ln\left(\frac{2}{b}\right)$.
  \item Determine for which values of the parameter $b$ the maximum height of the projectile does not exceed 1.6 meters.
  \item In this question, we choose $b = 5.69$.

The firing angle $\theta$ corresponds to the angle between the abscissa axis and the tangent to the curve of the function $f$ at the point with abscissa 0.\\
Determine an approximate value of the angle $\theta$ to the nearest tenth of a degree.
\end{enumerate}