Exercise 3 — Theme: Functions; 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 no points and loses no points. To answer, indicate on your paper the question number and the letter of the chosen answer. No justification is required.
Let $g$ be the function defined on $\mathbb{R}$ by $g(x) = x^{1000} + x$. We can affirm that: a. the function $g$ is concave on $\mathbb{R}$. b. the function $g$ is convex on $\mathbb{R}$. c. the function $g$ has exactly one inflection point. d. the function $g$ has exactly two inflection points.
Consider a function $f$ defined and differentiable on $\mathbb{R}$. Let $f'$ denote its derivative function. Let $\mathscr{C}$ denote the representative curve of $f$. Let $\Gamma$ denote the representative curve of $f'$. The curve $\Gamma$ is plotted below. Let $T$ denote the tangent to the curve $\mathscr{C}$ at the point with abscissa 0. We can affirm that the tangent $T$ is parallel to the line with equation: a. $y = x$ b. $y = 0$ c. $y = 1$ d. $x = 0$
Consider the sequence $(u_n)$ defined for every natural number $n$ by $u_n = \frac{(-1)^n}{n+1}$. We can affirm that the sequence $(u_n)$ is: a. bounded above and not bounded below. b. bounded below and not bounded above. c. bounded. d. not bounded above and not bounded below.
Let $k$ be a non-zero real number. Let $(v_n)$ be a sequence defined for every natural number $n$. Suppose that $v_0 = k$ and that for all $n$, we have $v_n \times v_{n+1} < 0$. We can affirm that $v_{10}$ is: a. positive. b. negative. c. of the same sign as $k$. d. of the same sign as $-k$.
Consider the sequence $(w_n)$ defined for every natural number $n$ by: $$w_{n+1} = 2w_n - 4 \quad \text{and} \quad w_2 = 8.$$ We can affirm that: a. $w_0 = 0$ b. $w_0 = 5$. c. $w_0 = 10$. d. It is not possible to calculate $w_0$.
Consider the sequence $(a_n)$ defined for every natural number $n$ by: $$a_{n+1} = \frac{\mathrm{e}^n}{\mathrm{e}^n + 1} a_n \quad \text{and} \quad a_0 = 1.$$ We can affirm that: a. the sequence $(a_n)$ is strictly increasing. b. the sequence $(a_n)$ is strictly decreasing. c. the sequence $(a_n)$ is not monotone. d. the sequence $(a_n)$ is constant.
A cell reproduces by dividing into two identical cells, which divide in turn, and so on. The generation time is defined as the time required for a given cell to divide into two cells. 1 cell was placed in culture. After 4 hours, there are approximately 4000 cells. We can affirm that the generation time is approximately equal to: a. less than one minute. b. 12 minutes. c. 20 minutes. d. 1 hour.
\textbf{Exercise 3 — Theme: Functions; 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 no points and loses no points. To answer, indicate on your paper the question number and the letter of the chosen answer. No justification is required.
\begin{enumerate}
\item Let $g$ be the function defined on $\mathbb{R}$ by $g(x) = x^{1000} + x$.\\
We can affirm that:\\
a. the function $g$ is concave on $\mathbb{R}$.\\
b. the function $g$ is convex on $\mathbb{R}$.\\
c. the function $g$ has exactly one inflection point.\\
d. the function $g$ has exactly two inflection points.
\item Consider a function $f$ defined and differentiable on $\mathbb{R}$. Let $f'$ denote its derivative function. Let $\mathscr{C}$ denote the representative curve of $f$. Let $\Gamma$ denote the representative curve of $f'$. The curve $\Gamma$ is plotted below.\\
Let $T$ denote the tangent to the curve $\mathscr{C}$ at the point with abscissa 0.\\
We can affirm that the tangent $T$ is parallel to the line with equation:\\
a. $y = x$\\
b. $y = 0$\\
c. $y = 1$\\
d. $x = 0$
\item Consider the sequence $(u_n)$ defined for every natural number $n$ by $u_n = \frac{(-1)^n}{n+1}$.\\
We can affirm that the sequence $(u_n)$ is:\\
a. bounded above and not bounded below.\\
b. bounded below and not bounded above.\\
c. bounded.\\
d. not bounded above and not bounded below.
\item Let $k$ be a non-zero real number. Let $(v_n)$ be a sequence defined for every natural number $n$. Suppose that $v_0 = k$ and that for all $n$, we have $v_n \times v_{n+1} < 0$.\\
We can affirm that $v_{10}$ is:\\
a. positive.\\
b. negative.\\
c. of the same sign as $k$.\\
d. of the same sign as $-k$.
\item Consider the sequence $(w_n)$ defined for every natural number $n$ by:
$$w_{n+1} = 2w_n - 4 \quad \text{and} \quad w_2 = 8.$$
We can affirm that:\\
a. $w_0 = 0$\\
b. $w_0 = 5$.\\
c. $w_0 = 10$.\\
d. It is not possible to calculate $w_0$.
\item Consider the sequence $(a_n)$ defined for every natural number $n$ by:
$$a_{n+1} = \frac{\mathrm{e}^n}{\mathrm{e}^n + 1} a_n \quad \text{and} \quad a_0 = 1.$$
We can affirm that:\\
a. the sequence $(a_n)$ is strictly increasing.\\
b. the sequence $(a_n)$ is strictly decreasing.\\
c. the sequence $(a_n)$ is not monotone.\\
d. the sequence $(a_n)$ is constant.
\item A cell reproduces by dividing into two identical cells, which divide in turn, and so on. The generation time is defined as the time required for a given cell to divide into two cells. 1 cell was placed in culture. After 4 hours, there are approximately 4000 cells.\\
We can affirm that the generation time is approximately equal to:\\
a. less than one minute.\\
b. 12 minutes.\\
c. 20 minutes.\\
d. 1 hour.
\end{enumerate}