\section*{EXERCISE-A}
\textbf{Main topics covered: convexity, logarithm function}

\section*{Part I: graphical readings}
$f$ denotes a function defined and differentiable on $\mathbb{R}$.\\
We give below the representative curve of the derivative function $f'$.

\section*{With the precision allowed by the graph, answer the following questions}
\begin{enumerate}
  \item Determine the slope of the tangent line to the curve of function $f$ at 0.
  \item a. Give the variations of the derivative function $f'$.\\
b. Deduce an interval on which $f$ is convex.
\end{enumerate}

\section*{Part II: function study}
The function $f$ is defined on $\mathbb{R}$ by
$$f(x) = \ln\left(x^{2} + x + \frac{5}{2}\right)$$

\begin{enumerate}
  \item Calculate the limits of function $f$ at $+\infty$ and at $-\infty$.
  \item Determine an expression $f'(x)$ of the derivative function of $f$ for all $x \in \mathbb{R}$.
  \item Deduce the table of variations of $f$. Be sure to place the limits in this table.
  \item a. Justify that the equation $f(x) = 2$ has a unique solution $\alpha$ in the interval $\left[-\frac{1}{2}; +\infty\right[$.\\
b. Give an approximate value of $\alpha$ to $10^{-1}$ near.
  \item The function $f'$ is differentiable on $\mathbb{R}$. We admit that, for all $x \in \mathbb{R}$, $f''(x) = \frac{-2x^{2} - 2x + 4}{\left(x^{2} + x + \frac{5}{2}\right)^{2}}$. Determine the number of inflection points of the representative curve of $f$.
\end{enumerate}