Exercise 1 — Multiple Choice (Exponential function) For each of the following questions, only one of the four proposed answers is correct.
Let $f$ be the function defined on $\mathbb{R}$ by $$f(x) = \frac{x}{\mathrm{e}^{x}}$$ We assume that $f$ is differentiable on $\mathbb{R}$ and we denote $f'$ its derivative function. a. $f'(x) = \mathrm{e}^{-x}$ b. $f'(x) = x\mathrm{e}^{-x}$ c. $f'(x) = (1-x)\mathrm{e}^{-x}$ d. $f'(x) = (1+x)\mathrm{e}^{-x}$
Let $f$ be a function twice differentiable on the interval $[-3;1]$. The graphical representation of its second derivative function $f''$ is given. We can then affirm that: a. The function $f$ is convex on the interval $[-2;0]$ b. The function $f$ is concave on the interval $[-1;1]$ c. The function $f'$ is decreasing on the interval $[-2;0]$ d. The function $f'$ admits a maximum at $x = -1$
We consider the function $f$ defined on $\mathbb{R}$ by: $$f(x) = x^3 \mathrm{e}^{-x^2}$$ If $F$ is an antiderivative of $f$ on $\mathbb{R}$, a. $F(x) = -\frac{1}{6}\left(x^3+1\right)\mathrm{e}^{-x^2}$ b. $F(x) = -\frac{1}{4}x^4 \mathrm{e}^{-x^2}$ c. $F(x) = -\frac{1}{2}\left(x^2+1\right)\mathrm{e}^{-x^2}$ d. $F(x) = x^2\left(3-2x^2\right)\mathrm{e}^{-x^2}$
What is the value of: $$\lim_{x \rightarrow +\infty} \frac{\mathrm{e}^x + 1}{\mathrm{e}^x - 1}$$ a. $-1$ b. $1$ c. $+\infty$ d. does not exist
We consider the function $f$ defined on $\mathbb{R}$ by $f(x) = \mathrm{e}^{2x+1}$. The only antiderivative $F$ on $\mathbb{R}$ of the function $f$ such that $F(0) = 1$ is the function: a. $x \longmapsto 2\mathrm{e}^{2x+1} - 2\mathrm{e} + 1$ b. $x \longmapsto 2\mathrm{e}^{2x+1} - \mathrm{e}$ c. $x \longmapsto \frac{1}{2}\mathrm{e}^{2x+1} - \frac{1}{2}\mathrm{e} + 1$ d. $x \longmapsto \mathrm{e}^{x^2+x}$
In a coordinate system, the representative curve of a function $f$ defined and twice differentiable on $[-2;4]$ is drawn. Among the following curves (a, b, c, d), which one represents the function $f''$, the second derivative of $f$?
\textbf{Exercise 1 — Multiple Choice (Exponential function)}
For each of the following questions, only one of the four proposed answers is correct.
\begin{enumerate}
\item Let $f$ be the function defined on $\mathbb{R}$ by
$$f(x) = \frac{x}{\mathrm{e}^{x}}$$
We assume that $f$ is differentiable on $\mathbb{R}$ and we denote $f'$ its derivative function.\\
a. $f'(x) = \mathrm{e}^{-x}$\\
b. $f'(x) = x\mathrm{e}^{-x}$\\
c. $f'(x) = (1-x)\mathrm{e}^{-x}$\\
d. $f'(x) = (1+x)\mathrm{e}^{-x}$
\item Let $f$ be a function twice differentiable on the interval $[-3;1]$. The graphical representation of its second derivative function $f''$ is given. We can then affirm that:\\
a. The function $f$ is convex on the interval $[-2;0]$\\
b. The function $f$ is concave on the interval $[-1;1]$\\
c. The function $f'$ is decreasing on the interval $[-2;0]$\\
d. The function $f'$ admits a maximum at $x = -1$
\item We consider the function $f$ defined on $\mathbb{R}$ by:
$$f(x) = x^3 \mathrm{e}^{-x^2}$$
If $F$ is an antiderivative of $f$ on $\mathbb{R}$,\\
a. $F(x) = -\frac{1}{6}\left(x^3+1\right)\mathrm{e}^{-x^2}$\\
b. $F(x) = -\frac{1}{4}x^4 \mathrm{e}^{-x^2}$\\
c. $F(x) = -\frac{1}{2}\left(x^2+1\right)\mathrm{e}^{-x^2}$\\
d. $F(x) = x^2\left(3-2x^2\right)\mathrm{e}^{-x^2}$
\item What is the value of:
$$\lim_{x \rightarrow +\infty} \frac{\mathrm{e}^x + 1}{\mathrm{e}^x - 1}$$
a. $-1$\\
b. $1$\\
c. $+\infty$\\
d. does not exist
\item We consider the function $f$ defined on $\mathbb{R}$ by $f(x) = \mathrm{e}^{2x+1}$. The only antiderivative $F$ on $\mathbb{R}$ of the function $f$ such that $F(0) = 1$ is the function:\\
a. $x \longmapsto 2\mathrm{e}^{2x+1} - 2\mathrm{e} + 1$\\
b. $x \longmapsto 2\mathrm{e}^{2x+1} - \mathrm{e}$\\
c. $x \longmapsto \frac{1}{2}\mathrm{e}^{2x+1} - \frac{1}{2}\mathrm{e} + 1$\\
d. $x \longmapsto \mathrm{e}^{x^2+x}$
\item In a coordinate system, the representative curve of a function $f$ defined and twice differentiable on $[-2;4]$ is drawn. Among the following curves (a, b, c, d), which one represents the function $f''$, the second derivative of $f$?
\end{enumerate}