This exercise is a multiple choice questionnaire. For each of the six 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.

\begin{enumerate}
  \item Consider the function $f$ defined and differentiable on $] 0 ; + \infty [$ by:
$$f ( x ) = x \ln ( x ) - x + 1 .$$
Among the four expressions below, which one is the derivative of $f$?
\begin{center}
\begin{tabular}{ | l | l | l | l | }
\hline
a. $\ln ( x )$ & b. $\frac { 1 } { x } - 1$ & c. $\ln ( x ) - 2$ & d. $\ln ( x ) - 1$ \\
\hline
\end{tabular}
\end{center}

  \item Consider the function $g$ defined on $] 0$; $+ \infty \left[ \text{ by } g ( x ) = x ^ { 2 } [ 1 - \ln ( x ) ] \right.$.
Among the four statements below, which one is correct?
\begin{center}
\begin{tabular}{ | l | l | l | l | }
\hline
a. $\lim _ { x \rightarrow 0 } g ( x ) = + \infty$ & b. $\lim _ { x \rightarrow 0 } g ( x ) = - \infty$ & c. $\lim _ { x \rightarrow 0 } g ( x ) = 0$ & \begin{tabular}{ l } d. The function $g$ \\ does not have a li- \\ mit at 0. \\ \end{tabular} \\
\hline
\end{tabular}
\end{center}

  \item Consider the function $f$ defined on $\mathbb { R }$ by $f ( x ) = x ^ { 3 } - 0,9 x ^ { 2 } - 0,1 x$. The number of solutions to the equation $f ( x ) = 0$ on $\mathbb { R }$ is:
\begin{center}
\begin{tabular}{ | l | l | l | l | }
\hline
a. 0 & b. 1 & c. 2 & d. 3 \\
\hline
\end{tabular}
\end{center}

  \item If $H$ is an antiderivative of a function $h$ defined and continuous on $\mathbb { R }$, and if $k$ is the function defined on $\mathbb { R }$ by $k ( x ) = h ( 2 x )$, then an antiderivative $K$ of $k$ is defined on $\mathbb { R }$ by:
\begin{center}
\begin{tabular}{ | l | l | l | l | }
\hline
a. $K ( x ) = H ( 2 x )$ & b. $K ( x ) = 2 H ( 2 x )$ & c. $K ( x ) = \frac { 1 } { 2 } H ( 2 x )$ & d. $K ( x ) = 2 H ( x )$ \\
\hline
\end{tabular}
\end{center}

  \item The equation of the tangent line at the point with abscissa 1 to the curve of the function $f$ defined on $\mathbb { R }$ by $f ( x ) = x \mathrm { e } ^ { x }$ is:
\begin{center}
\begin{tabular}{ | l | l | l | l | }
\hline
a. $y = \mathrm { e } x + \mathrm { e }$ & b. $y = 2 \mathrm { e } x - \mathrm { e }$ & c. $y = 2 \mathrm { e } x + \mathrm { e }$ & d. $y = \mathrm { e } x$ \\
\hline
\end{tabular}
\end{center}

  \item The integers $n$ that are solutions to the inequality $( 0,2 ) ^ { n } < 0,001$ are all integers $n$ such that:
\begin{center}
\begin{tabular}{ | l | l | l | l | }
\hline
a. $n \leqslant 4$ & b. $n \leqslant 5$ & c. $n \geqslant 4$ & d. $n \geqslant 5$ \\
\hline
\end{tabular}
\end{center}
\end{enumerate}