\section*{Part A}

In the orthonormal coordinate system above, the representative curves of a function $f$ and its derivative function, denoted $f ^ { \prime }$, are drawn, both defined on $] 3 ; + \infty [$.

\begin{enumerate}
  \item Associate each curve with the function it represents. Justify.
  \item Determine graphically the possible solution(s) of the equation $f ( x ) = 3$.
  \item Indicate, by graphical reading, the convexity of the function $f$.
\end{enumerate}

\section*{Part B}
\begin{enumerate}
  \item Justify that the quantity $\ln \left( x ^ { 2 } - x - 6 \right)$ is well defined for values $x$ in the interval ]3; $+ \infty$ [, which we will call $I$ in the following.
  \item We admit that the function $f$ from Part A is defined by $f ( x ) = \ln \left( x ^ { 2 } - x - 6 \right)$ on $I$. Calculate the limits of the function $f$ at the two endpoints of the interval $I$.\\
Deduce an equation of an asymptote to the representative curve of the function $f$ on $I$.
  \item a. Calculate $f ^ { \prime } ( x )$ for all $x$ belonging to $I$.\\
b. Study the direction of variation of the function $f$ on $I$.
\end{enumerate}

Draw the variation table of the function $f$ showing the limits at the endpoints of $I$.\\
4. a. Justify that the equation $f ( x ) = 3$ admits a unique solution $\alpha$ on the interval ]5; 6[.\\
b. Determine, using a calculator, an approximation of $\alpha$ to within $10 ^ { - 2 }$.\\
5. a. Justify that $f ^ { \prime \prime } ( x ) = \frac { - 2 x ^ { 2 } + 2 x - 13 } { \left( x ^ { 2 } - x - 6 \right) ^ { 2 } }$.\\
b. Study the convexity of the function $f$ on $I$.