This exercise is a multiple choice questionnaire. For each of the following questions, only one of the four proposed answers is correct. No justification is required. A wrong answer, no answer, or multiple answers, neither gives nor takes away points.

\begin{enumerate}
  \item We consider the function $f$ defined on $\mathbb{R}$ by $f(x) = 2x\mathrm{e}^x$.

The number of solutions on $\mathbb{R}$ of the equation $f(x) = -\dfrac{73}{100}$ is equal to:

a. 0

b. 1

c. 2

d. infinitely many.

  \item We consider the function $g$ defined on $\mathbb{R}$ by:
$$g(x) = \frac{x+1}{\mathrm{e}^x}.$$
The limit of the function $g$ at $-\infty$ is equal to:

a. $-\infty$

b. $+\infty$

c. $0$

d. it does not exist.

  \item We consider the function $h$ defined on $\mathbb{R}$ by:
$$h(x) = (4x - 16)\mathrm{e}^{2x}.$$
We denote $\mathscr{C}_h$ the representative curve of $h$ in an orthogonal coordinate system. We can affirm that:

a. $h$ is convex on $\mathbb{R}$.

b. $\mathscr{C}_h$ has an inflection point at $x = 3$.

c. $h$ is concave on $\mathbb{R}$.

d. $\mathscr{C}_h$ has an inflection point at $x = 3.5$.

  \item We consider the function $k$ defined on the interval $]0; +\infty[$ by:
$$k(x) = 3\ln(x) - x.$$
We denote $\mathscr{C}$ the representative curve of the function $k$ in an orthonormal coordinate system. We denote $T$ the tangent line to the curve $\mathscr{C}$ at the point with abscissa $x = \mathrm{e}$. An equation of $T$ is:

a. $y = (3 - \mathrm{e})x$

b. $y = \left(\dfrac{3 - \mathrm{e}}{\mathrm{e}}\right)x$

c. $y = \left(\dfrac{3}{\mathrm{e}} - 1\right)x + 1$

d. $y = (\mathrm{e} - 1)x + 1$

  \item We consider the equation $[\ln(x)]^2 + 10\ln(x) + 21 = 0$, with $x \in ]0; +\infty[$.

The number of solutions of this equation is equal to:

a. 0

b. 1

c. 2

d. infinitely many.
\end{enumerate}