We consider the function $f$ defined on $] 0 ; + \infty [$ by
$$f ( x ) = \frac { 1 } { x } ( 1 + \ln x )$$

1. In the three situations below, we have drawn, in an orthonormal coordinate system, the representative curve $\mathscr { C } _ { f }$ of the function $f$ and a curve $\mathscr { C } _ { F }$. In only one situation, the curve $\mathscr { C } _ { F }$ is the representative curve of a primitive $F$ of the function $f$. Which one? Justify the answer.

Exercise 1 -- Part B
The proportion of individuals who possess a certain type of equipment in a population is modelled by the function $p$ defined on $[ 0 ; + \infty [$ by $$p ( x ) = \frac { 1 } { 1 + \mathrm { e } ^ { - 0,2 x } } .$$ The real number $x$ represents the time elapsed, in years, since January 1st, 2000. The number $p ( x )$ models the proportion of equipped individuals after $x$ years. Thus, for this model, $p ( 0 )$ is the proportion of equipped individuals on January 1st, 2000 and $p ( 3.5 )$ is the proportion of equipped individuals in the middle of 2003.

1. What is, for this model, the proportion of equipped individuals on January 1st, 2010? Give a value rounded to the nearest hundredth.
2. 1. [a.] Determine the direction of variation of the function $p$ on $[ 0 ; + \infty [$.
   2. [b.] Calculate the limit of the function $p$ as $x \to + \infty$.
   3. [c.] Interpret this limit in the context of the exercise.
3. It is considered that, when the proportion of equipped individuals exceeds $95\%$, the market is saturated. Determine, by explaining the approach, the year in which this occurs.
4. The average proportion of equipped individuals between 2008 and 2010 is defined by $$m = \frac { 1 } { 2 } \int _ { 8 } ^ { 10 } p ( x ) \mathrm { d } x$$
   1. [a.] Verify that, for all real $x \geqslant 0$, $$p ( x ) = \frac { \mathrm { e } ^ { 0,2 x } } { 1 + \mathrm { e } ^ { 0,2 x } }$$
   2. [b.] Deduce an antiderivative of the function $p$ on $[ 0 ; + \infty [$.
   3. [c.] Determine the exact value of $m$ and its approximation to the nearest hundredth.

Let $f : (0,2) \rightarrow R$ be a twice differentiable function such that $f''(x) > 0$, for all $x \in (0,2)$. If $\phi(x) = f(x) + f(2-x)$, then $\phi$ is
(1) decreasing on $(0,2)$
(2) increasing on $(0,2)$
(3) increasing on $(0,1)$ and decreasing on $(1,2)$
(4) decreasing on $(0,1)$ and increasing on $(1,2)$