Consider a function $f$ defined and twice differentiable on $]-2; +\infty[$. We denote by $\mathscr{C}_f$ its representative curve in an orthogonal coordinate system of the plane, $f'$ its derivative and $f''$ its second derivative. The curve $\mathscr{C}_f$ and its tangent $T$ at point B with abscissa $-1$ are drawn. It is specified that the line $T$ passes through the point $\mathrm{A}(0; -1)$.

\textbf{Part A: exploitation of the graph.}

Using the graph, answer the questions below.

\begin{enumerate}
  \item Specify $f(-1)$ and $f'(-1)$.
  \item Is the function $f$ convex on its domain of definition? Justify.
  \item Conjecture the number of solutions of the equation $f(x) = 0$ and give a value rounded to $10^{-1}$ near a solution.
\end{enumerate}

\textbf{Part B: study of the function $f$}

Consider that the function $f$ is defined on $]-2; +\infty[$ by:
$$f(x) = x^2 + 2x - 1 + \ln(x+2),$$
where ln denotes the natural logarithm function.

\begin{enumerate}
  \item Determine by calculation the limit of the function $f$ at $-2$. Interpret this result graphically.
\end{enumerate}

It is admitted that $\lim_{x \rightarrow +\infty} f(x) = +\infty$.

\begin{enumerate}
  \setcounter{enumi}{1}
  \item Show that for all $x > -2$, $\quad f'(x) = \frac{2x^2 + 6x + 5}{x+2}$.
  \item Study the variations of the function $f$ on $]-2; +\infty[$ then draw up its complete variation table.
  \item Show that the equation $f(x) = 0$ admits a unique solution $\alpha$ on $]-2; +\infty[$ and give a value of $\alpha$ rounded to $10^{-2}$ near.
  \item Deduce the sign of $f(x)$ on $]-2; +\infty[$.
  \item Show that $\mathscr{C}_f$ admits a unique inflection point and determine its abscissa.
\end{enumerate}

\textbf{Part C: a minimum distance.}

Let $g$ be the function defined on $]-2; +\infty[$ by $\quad g(x) = \ln(x+2)$. We denote by $\mathscr{C}_g$ its representative curve in an orthonormal coordinate system $(O; I, J)$. Let $M$ be a point of $\mathscr{C}_g$ with abscissa $x$. The purpose of this part is to determine for which value of $x$ the distance $JM$ is minimal. Consider the function $h$ defined on $]-2; +\infty[$ by $\quad h(x) = JM^2$.

\begin{enumerate}
  \item Justify that for all $x > -2$, we have: $\quad h(x) = x^2 + [\ln(x+2) - 1]^2$.
  \item It is admitted that the function $h$ is differentiable on $]-2; +\infty[$ and we denote by $h'$ its derivative function. It is also admitted that for all real $x > -2$,
$$h'(x) = \frac{2f(x)}{x+2}$$
where $f$ is the function studied in part B.\\
a. Draw up the variation table of $h$ on $]-2; +\infty[$. The limits are not required.\\
b. Deduce that the value of $x$ for which the distance $JM$ is minimal is $\alpha$ where $\alpha$ is the real number defined in question 4 of part B.
  \item We will denote by $M_\alpha$ the point of $\mathscr{C}_g$ with abscissa $\alpha$.\\
a. Show that $\ln(\alpha + 2) = 1 - 2\alpha - \alpha^2$.\\
b. Deduce that the tangent to $\mathscr{C}_g$ at point $M_\alpha$ and the line $(JM_\alpha)$ are perpendicular.\\
One may use the fact that, in an orthonormal coordinate system, two lines are perpendicular when the product of their slopes is equal to $-1$.
\end{enumerate}