\textbf{Exercise 2 (7 points)}\\
Theme: functions, exponential function

\section*{Part A}
Let $p$ be the function defined on the interval $[ - 3 ; 4 ]$ by:
$$p ( x ) = x ^ { 3 } - 3 x ^ { 2 } + 5 x + 1$$

\begin{enumerate}
  \item Determine the variations of the function $p$ on the interval $[ - 3 ; 4 ]$.
  \item Justify that the equation $p ( x ) = 0$ admits in the interval $[-3;4]$ a unique solution which will be denoted $\alpha$.
  \item Determine an approximate value of the real number $\alpha$ to the nearest tenth.
  \item Give the sign table of the function $p$ on the interval $[ - 3 ; 4 ]$.
\end{enumerate}

\section*{Part B}
Let $f$ be the function defined on the interval $[ - 3 ; 4 ]$ by:
$$f ( x ) = \frac { \mathrm { e } ^ { x } } { 1 + x ^ { 2 } }$$
We denote by $\mathscr { C } _ { f }$ its representative curve in an orthogonal coordinate system.

\begin{enumerate}
  \item a. Determine the derivative of the function $f$ on the interval $[ - 3 ; 4 ]$.\\
b. Justify that the curve $\mathscr { C } _ { f }$ admits a horizontal tangent at the point with abscissa 1.
  \item The designers of a water slide use the curve $\mathscr { C } _ { f }$ as the profile of a water slide. They estimate that the water slide provides good sensations if the profile has at least two inflection points.\\
a. Based on the graph, does the water slide seem to provide good sensations? Argue.\\
b. It is admitted that the function $f ^ { \prime \prime }$, the second derivative of the function $f$, has the following expression for every real $x$ in the interval $[ - 3 ; 4 ]$:
$$f ^ { \prime \prime } ( x ) = \frac { p ( x ) ( x - 1 ) \mathrm { e } ^ { x } } { \left( 1 + x ^ { 2 } \right) ^ { 3 } }$$
where $p$ is the function defined in Part A.\\
Using the above expression for $f ^ { \prime \prime }$, answer the question: ``does the water slide provide good sensations?''. Justify.
\end{enumerate}