Exercise 2 (7 points)\\
Themes: numerical functions and sequences\\
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 paper the question number and the letter of the chosen answer. No justification is required.

For questions 1 to 3 below, consider a function $f$ defined and twice differentiable on $\mathbb{R}$. The curve of its derivative function $f'$ is given. We admit that $f'$ has a maximum at $-\frac{3}{2}$ and that its curve intersects the x-axis at the point with coordinates $\left(-\frac{1}{2}; 0\right)$.

\textbf{Question 1:}\\
a. The function $f$ has a maximum at $-\frac{3}{2}$;\\
b. The function $f$ has a maximum at $-\frac{1}{2}$;\\
c. The function $f$ has a minimum at $-\frac{1}{2}$;\\
d. At the point with abscissa $-1$, the curve of function $f$ has a horizontal tangent.

\textbf{Question 2:}\\
a. The function $f$ is convex on $]-\infty; -\frac{3}{2}[$;\\
c. The curve $\mathscr{C}_f$ representing function $f$ does not have an inflection point;

\textbf{Question 3:}\\
The second derivative $f''$ of function $f$ satisfies:\\
a. $f''(x) \geqslant 0$ for $x \in ]-\infty; -\frac{1}{2}[$;\\
b. $f''(x) \geqslant 0$ for $x \in [-2; -1]$;\\
c. $f''\left(-\frac{3}{2}\right) = 0$;\\
d. $f''(-3) = 0$.

\textbf{Question 4:}\\
Consider three sequences $\left(u_n\right)$, $\left(v_n\right)$ and $\left(w_n\right)$. We know that, for every natural number $n$, we have: $u_n \leqslant v_n \leqslant w_n$ and furthermore: $\lim_{n \rightarrow +\infty} u_n = 1$ and $\lim_{n \rightarrow +\infty} w_n = 3$.\\
We can then affirm that:\\
a. the sequence $\left(v_n\right)$ converges;\\
b. If the sequence $(u_n)$ is increasing then the sequence $(v_n)$ is bounded below by $u_0$;\\
c. $1 \leqslant v_0 \leqslant 3$;\\
d. the sequence $(v_n)$ diverges.

\textbf{Question 5:}\\
Consider a sequence $(u_n)$ such that, for every non-zero natural number $n$: $u_n \leqslant u_{n+1} \leqslant \frac{1}{n}$. We can then affirm that:\\
a. the sequence $(u_n)$ diverges;\\
b. the sequence $(u_n)$ converges;\\
c. $\lim_{n \rightarrow +\infty} u_n = 0$;\\
d. $\lim_{n \rightarrow +\infty} u_n = 1$.

\textbf{Question 6:}\\
Consider $(u_n)$ a real sequence such that for every natural number $n$, we have: $n < u_n < n+1$. We can affirm that:\\
a. There exists a natural number $N$ such that $u_N$ is an integer;\\
b. the sequence $(u_n)$ is increasing;\\
c. the sequence $(u_n)$ is convergent;\\
d. The sequence $(u_n)$ has no limit.