Exercise 1 — Multiple Choice Questionnaire (Logarithmic function) For each of the following questions, only one of the four proposed answers is correct. The six questions are independent.
Consider the function $f$ defined for all real $x$ by $f(x) = \ln\left(1 + x^2\right)$. On $\mathbb{R}$, the equation $f(x) = 2022$ a. has no solution. b. has exactly one solution. c. has exactly two solutions. d. has infinitely many solutions.
Let the function $g$ defined for all strictly positive real $x$ by: $$g(x) = x\ln(x) - x^2$$ We denote $\mathscr{C}_g$ its representative curve in a coordinate system of the plane. a. The function $g$ is convex on $]0; +\infty[$. b. The function $g$ is concave on $]0; +\infty[$. c. The curve $\mathscr{C}_g$ has exactly one inflection point on $]0; +\infty[$. d. The curve $\mathscr{C}_g$ has exactly two inflection points on $]0; +\infty[$.
Consider the function $f$ defined on $]-1; 1[$ by $$f(x) = \frac{x}{1 - x^2}$$ An antiderivative of the function $f$ is the function $g$ defined on the interval $]-1; 1[$ by: a. $g(x) = -\frac{1}{2}\ln\left(1 - x^2\right)$ b. $g(x) = \frac{1 + x^2}{\left(1 - x^2\right)^2}$ c. $g(x) = \frac{x^2}{2\left(x - \frac{x^3}{3}\right)}$ d. $g(x) = \frac{x^2}{2}\ln\left(1 - x^2\right)$
The function $x \longmapsto \ln\left(-x^2 - x + 6\right)$ is defined on a. $]-3; 2[$ b. $]-\infty; 6]$ c. $]0; +\infty[$ d. $]2; +\infty[$
Consider the function $f$ defined on $]0.5; +\infty[$ by $$f(x) = x^2 - 4x + 3\ln(2x - 1)$$ An equation of the tangent line to the representative curve of $f$ at the point with abscissa 1 is: a. $y = 4x - 7$ b. $y = 2x - 4$ c. $y = -3(x - 1) + 4$ d. $y = 2x - 1$
The set $S$ of solutions in $\mathbb{R}$ of the inequality $\ln(x + 3) < 2\ln(x + 1)$ is: a. $S = ]-\infty; -2[ \cup ]1; +\infty[$ b. $S = ]1; +\infty[$ c. $S = \varnothing$ d. $S = ]-1; 1[$
\textbf{Exercise 1 — Multiple Choice Questionnaire (Logarithmic function)}
For each of the following questions, only one of the four proposed answers is correct. The six questions are independent.
\begin{enumerate}
\item Consider the function $f$ defined for all real $x$ by $f(x) = \ln\left(1 + x^2\right)$.
On $\mathbb{R}$, the equation $f(x) = 2022$\\
a. has no solution.\\
b. has exactly one solution.\\
c. has exactly two solutions.\\
d. has infinitely many solutions.
\item Let the function $g$ defined for all strictly positive real $x$ by:
$$g(x) = x\ln(x) - x^2$$
We denote $\mathscr{C}_g$ its representative curve in a coordinate system of the plane.\\
a. The function $g$ is convex on $]0; +\infty[$.\\
b. The function $g$ is concave on $]0; +\infty[$.\\
c. The curve $\mathscr{C}_g$ has exactly one inflection point on $]0; +\infty[$.\\
d. The curve $\mathscr{C}_g$ has exactly two inflection points on $]0; +\infty[$.
\item Consider the function $f$ defined on $]-1; 1[$ by
$$f(x) = \frac{x}{1 - x^2}$$
An antiderivative of the function $f$ is the function $g$ defined on the interval $]-1; 1[$ by:\\
a. $g(x) = -\frac{1}{2}\ln\left(1 - x^2\right)$\\
b. $g(x) = \frac{1 + x^2}{\left(1 - x^2\right)^2}$\\
c. $g(x) = \frac{x^2}{2\left(x - \frac{x^3}{3}\right)}$\\
d. $g(x) = \frac{x^2}{2}\ln\left(1 - x^2\right)$
\item The function $x \longmapsto \ln\left(-x^2 - x + 6\right)$ is defined on\\
a. $]-3; 2[$\\
b. $]-\infty; 6]$\\
c. $]0; +\infty[$\\
d. $]2; +\infty[$
\item Consider the function $f$ defined on $]0.5; +\infty[$ by
$$f(x) = x^2 - 4x + 3\ln(2x - 1)$$
An equation of the tangent line to the representative curve of $f$ at the point with abscissa 1 is:\\
a. $y = 4x - 7$\\
b. $y = 2x - 4$\\
c. $y = -3(x - 1) + 4$\\
d. $y = 2x - 1$
\item The set $S$ of solutions in $\mathbb{R}$ of the inequality $\ln(x + 3) < 2\ln(x + 1)$ is:\\
a. $S = ]-\infty; -2[ \cup ]1; +\infty[$\\
b. $S = ]1; +\infty[$\\
c. $S = \varnothing$\\
d. $S = ]-1; 1[$
\end{enumerate}