\section*{Exercise 2 — 7 points}
Topics: Sequences, Functions\\
This exercise is a multiple choice questionnaire. For each of the following questions, only one of the four proposed answers is correct.\\
A wrong answer, multiple answers, or no answer to a question earns neither points nor deducts points.\\
To answer, indicate on your answer sheet the question number and the letter of the chosen answer. No justification is required.
\begin{enumerate}
  \item We consider the sequences $\left( a _ { n } \right)$ and $\left( b _ { n } \right)$ defined by $a _ { 0 } = 1$ and, for every natural number $n$, $a _ { n + 1 } = 0.5 a _ { n } + 1$ and $b _ { n } = a _ { n } - 2$.\\
We can affirm that:\\
a. $\left( a _ { n } \right)$ is arithmetic;\\
b. $\left( b _ { n } \right)$ is geometric;\\
c. $\left( a _ { n } \right)$ is geometric;\\
d. $\left( b _ { n } \right)$ is arithmetic.
  \item In questions 2. and 3., we consider the sequences $\left( u _ { n } \right)$ and $\left( v _ { n } \right)$ defined by:
$$u _ { 0 } = 2 , v _ { 0 } = 1 \mathrm { and } , \text { for every natural number } n : \left\{ \begin{array} { l } 
u _ { n + 1 } = u _ { n } + 3 v _ { n } \\
v _ { n + 1 } = u _ { n } + v _ { n } .
\end{array} \right.$$
We can affirm that:\\
a. $\left\{ \begin{array} { l } u _ { 2 } = 5 \\ v _ { 2 } = 3 \end{array} \right.$\\
b. $u _ { 2 } ^ { 2 } - 3 v _ { 2 } ^ { 2 } = - 2 ^ { 2 }$\\
c. $\frac { u _ { 2 } } { v _ { 2 } } = 1.75$\\
d. $5 u _ { 1 } = 3 v _ { 1 }$.
  \item We consider the sequences $\left( u _ { n } \right)$ and $\left( v _ { n } \right)$ defined by:
$$u _ { 0 } = 2 , v _ { 0 } = 1 \mathrm { and } , \text { for every natural number } n : \left\{ \begin{array} { l } 
u _ { n + 1 } = u _ { n } + 3 v _ { n } \\
v _ { n + 1 } = u _ { n } + v _ { n } .
\end{array} \right.$$
We consider the program below written in Python language:
\begin{verbatim}
def valeurs() :
    u = 2
    v = 1
    for k in range(1,11)
        c = u
        u = u + 3*v
        v = c + v
    return (u, v)
\end{verbatim}
This program returns:\\
a. $u _ { 11 }$ and $v _ { 11 }$;\\
b. $u _ { 10 }$ and $v _ { 11 }$;\\
c. the values of $u _ { n }$ and $v _ { n }$ for $n$ ranging from 1 to 10;\\
d. $u _ { 10 }$ and $v _ { 10 }$.
  \item For questions 4. and 5., we consider a function $f$ twice differentiable on the interval $[-4; 2]$. We denote by $f ^ { \prime }$ the derivative function of $f$ and $f ^ { \prime \prime }$ the second derivative of $f$.\\
We are given below the representative curve $\mathscr { C } ^ { \prime }$ of the derivative function $f ^ { \prime }$ in a coordinate system of the plane. We are also given the points $\mathrm { A } ( - 2 ; 0 ) , \mathrm { B } ( 1 ; 0 )$ and $\mathrm { C } ( 0 ; 5 )$.\\
The function $f$ is:\\
a. concave on $[-2; 1]$;\\
b. convex on $[-4; 0]$;\\
c. convex on $[ - 2 ; 1 ]$;\\
d. convex on $[ 0 ; 2 ]$.
  \item For questions 4. and 5., we consider a function $f$ twice differentiable on the interval $[-4; 2]$. We denote by $f ^ { \prime }$ the derivative function of $f$ and $f ^ { \prime \prime }$ the second derivative of $f$.\\
We are given below the representative curve $\mathscr { C } ^ { \prime }$ of the derivative function $f ^ { \prime }$ in a coordinate system of the plane. We are also given the points $\mathrm { A } ( - 2 ; 0 ) , \mathrm { B } ( 1 ; 0 )$ and $\mathrm { C } ( 0 ; 5 )$.\\
We admit that the line (BC) is tangent to the curve $\mathscr { C } ^ { \prime }$ at point B. We have:\\
a. $f ^ { \prime } ( 1 ) < 0$;\\
b. $f ^ { \prime } ( 1 ) = 5$;\\
c. $f ^ { \prime \prime } ( 1 ) > 0$;\\
d. $f ^ { \prime \prime } ( 1 ) = - 5$.
  \item Let $f$ be the function defined on $\mathbb { R }$ by $f ( x ) = \left( x ^ { 2 } + 1 \right) \mathrm { e } ^ { x }$.\\
The antiderivative $F$ of $f$ on $\mathbb { R }$ such that $F ( 0 ) = 1$ is defined by:\\
a. $F ( x ) = \left( x ^ { 2 } - 2 x + 3 \right) \mathrm { e } ^ { x }$;\\
b. $F ( x ) = \left( x ^ { 2 } - 2 x + 3 \right) \mathrm { e } ^ { x } - 2$;\\
c. $F ( x ) = \left( \frac { 1 } { 3 } x ^ { 3 } + x \right) \mathrm { e } ^ { x } + 1$;\\
d. $F ( x ) = \left( \frac { 1 } { 3 } x ^ { 3 } + x \right) \mathrm { e } ^ { x }$.
\end{enumerate}