3. We admit that for all $x$ belonging to $] 0$; $+ \infty \left[ , f ^ { \prime } ( x ) = 2 ( \ln x ) ^ { 2 } + \ln x - 1 \right.$. a. Show that for all $x$ belonging to $] 0 ; + \infty \left[ , f ^ { \prime \prime } ( x ) = \frac { 1 } { x } ( 4 \ln x + 1 ) \right.$. b. Study the convexity of the function $f$ on the interval $] 0 ; + \infty [$ and specify the exact value of the abscissa of the inflection point. c. Show that the curve $C _ { f }$ is above the tangent $T _ { B }$ on the interval $[ 1 ; + \infty [$.
Part C: Area calculation
- Justify that the tangent $T _ { B }$ has the reduced equation $y = 2 x - \mathrm { e }$.
- Using integration by parts, show that $\int _ { 1 } ^ { \mathrm { e } } x \ln x d x = \frac { \mathrm { e } ^ { 2 } + 1 } { 4 }$.
- We denote by $\mathcal { A }$ the area of the shaded region in the figure, bounded by the curve $C _ { f }$, the tangent $T _ { B }$, and the lines with equations $x = 1$ and $x = \mathrm { e }$. We admit that $\int _ { 1 } ^ { \mathrm { e } } x ( \ln x ) ^ { 2 } d x = \frac { \mathrm { e } ^ { 2 } - 1 } { 4 }$. Deduce the exact value of $\mathcal { A }$ in square units.
Exercise 3 (4 points)
For each of the following statements, indicate whether it is true or false. Justify each answer. An unjustified answer earns no points.
We equip space with an orthonormal coordinate system ( $O ; \vec { \imath } , \vec { \jmath } , \vec { k }$ ).
- We consider the points $A ( - 1 ; 0 ; 5 )$ and $B ( 3 ; 2 ; - 1 )$.
Statement 1: A parametric representation of the line ( $A B$ ) is
$$\left\{ \begin{array} { l }
x = 3 - 2 t \\
y = 2 - t \\
z = - 1 + 3 t
\end{array} \quad \text { with } t \in \mathbb { R } \right.$$
Statement 2: The vector $\vec { n } \left( \begin{array} { c } 5 \\ - 2 \\ 1 \end{array} \right)$ is normal to the plane $( O A B )$.
2. We consider:
- the line $d$ with parametric representation $\left\{ \begin{array} { l } x = 15 + k \\ y = 8 - k \\ z = - 6 + 2 k \end{array} \right.$ with $k \in \mathbb { R }$;
- the line $d ^ { \prime }$ with parametric representation $\left\{ \begin{array} { l } x = 1 + 4 s \\ y = 2 + 4 s \\ z = 1 - 6 s \end{array} \right.$ with $s \in \mathbb { R }$.
Statement 3: The lines $d$ and $d ^ { \prime }$ are not coplanar.
3. We consider the plane $\mathcal { P }$ with equation $x - y + z + 1 = 0$.
Statement 4: The distance from point $C ( 2 ; - 1 ; 2 )$ to the plane $\mathcal { P }$ is equal to $2 \sqrt { 3 }$.
Exercise 4 (5 points)
A team of biologists is studying the evolution of the area covered by a marine algae called seagrass, on the bottom of Alycastre Bay, near the island of Porquerolles. The studied area has a total area of 20 hectares (ha), and on July 1, 2024, seagrass covered 1 ha of this area.
Part A: study of a discrete model
For any natural integer $n$, we denote by $u _ { n }$ the area of the zone, in hectares, covered by seagrass on July 1 of the year $2024 + n$. Thus, $u _ { 0 } = 1$.
A study conducted on this area made it possible to establish that for any natural integer $n$:
$$u _ { n + 1 } = - 0,02 u _ { n } ^ { 2 } + 1,3 u _ { n }$$
- Calculate the area that seagrass should cover on July 1, 2025 according to this model.
- We denote by $h$ the function defined on [ 0 ; 20] by $h ( x ) = - 0,02 x ^ { 2 } + 1,3 x$. We admit that $h$ is increasing on [0;20]. a. Prove that for any natural integer $n , 1 \leq u _ { n } \leq u _ { n + 1 } \leq 20$. b. Deduce that the sequence ( $u _ { n }$ ) converges. We denote its limit by $L$. c. Justify that $\mathrm { L } = 15$.
- The biologists wish to know after how long the area covered by seagrass will exceed 14 hectares. a. Without any calculation, justify that, according to this model, this will occur. b. Copy and complete the following algorithm so that at the end of execution, it displays the answer to the biologists' question.
\begin{verbatim} def seuil(): n=0 u=1 while ..................... : n= ............ u= ............ return n \end{verbatim}
Part B: study of a continuous model
We wish to describe the area of the studied zone covered by seagrass over time with a continuous model.
In this model, for a duration $t$, in years, elapsed from July 1, 2024, the area of the studied zone covered by seagrass is given by $f ( t )$, where $f$ is a function defined on [ $0 ; + \infty [$ satisfying:
- $f ( 0 ) = 1$;
- $f$ does not vanish on [ 0 ; $+ \infty [$;
- $f$ is differentiable on $[ 0 ; + \infty [$;
- $f$ is a solution on $\left[ 0 ; + \infty \left[ \