Exercise 4 (7 points)\\
Theme: numerical functions\\
This exercise is a multiple choice questionnaire. For each question, only one of the four proposed answers is correct. No justification is required. A wrong answer, multiple answers, or the absence of an answer to a question neither awards nor deducts points. The six questions are independent.

\begin{enumerate}
  \item The representative curve of the function $f$ defined on $\mathbb{R}$ by $f(x) = \frac{-2x^2 + 3x - 1}{x^2 + 1}$ admits as an asymptote the line with equation:\\
a. $x = -2$;\\
b. $y = -1$;\\
c. $y = -2$;\\
d. $y = 0$
  \item Let $f$ be the function defined on $\mathbb{R}$ by $f(x) = x\mathrm{e}^{x^2}$.\\
The antiderivative $F$ of $f$ on $\mathbb{R}$ which satisfies $F(0) = 1$ is defined by:\\
a. $F(x) = \frac{x^2}{2}\mathrm{e}^{x^2}$;\\
b. $F(x) = \frac{1}{2}\mathrm{e}^{x^2}$\\
c. $F(x) = \left(1 + 2x^2\right)\mathrm{e}^{x^2}$;\\
d. $F(x) = \frac{1}{2}\mathrm{e}^{x^2} + \frac{1}{2}$
  \item The representative graph $\mathscr{C}_{f'}$ of the derivative function $f'$ of a function $f$ defined on $\mathbb{R}$ is given below. We can affirm that the function $f$ is:\\
a. concave on $]0; +\infty[$;\\
b. convex on $]0; +\infty[$;\\
c. convex on $[0; 2]$;\\
d. convex on $[2; +\infty[$.
  \item Among the antiderivatives of the function $f$ defined on $\mathbb{R}$ by $f(x) = 3\mathrm{e}^{-x^2} + 2$:\\
a. all are increasing on $\mathbb{R}$;\\
b. all are decreasing on $\mathbb{R}$;\\
c. some are increasing on $\mathbb{R}$ and others decreasing on $\mathbb{R}$;\\
d. all are increasing on $]-\infty; 0]$ and decreasing on $[0; +\infty[$.
  \item The limit at $+\infty$ of the function $f$ defined on the interval $]0; +\infty[$ by $f(x) = \frac{2\ln x}{3x^2 + 1}$ is equal to:\\
a. $\frac{2}{3}$;\\
b. $+\infty$;\\
c. $-\infty$;\\
d. 0.
  \item The equation $\mathrm{e}^{2x} + \mathrm{e}^x - 12 = 0$ admits in $\mathbb{R}$:\\
a. three solutions;\\
b. two solutions;\\
c. only one solution;\\
d. no solution.
\end{enumerate}