Question 4: Consider the function $f$ defined on $\mathbb{R}$ by $f(x) = 3\mathrm{e}^x - x$.

a. $\lim_{x\rightarrow+\infty} f(x) = 3$	b. $\lim_{x\rightarrow+\infty} f(x) = +\infty$	c. $\lim_{x\rightarrow+\infty} f(x) = -\infty$	\begin{tabular}{l} d. We cannot
determine the limit
of the function $f$
as $x$ tends to
$+\infty$

\hline \end{tabular}

Exercise 1 — Multiple Choice (Exponential function)
For each of the following questions, only one of the four proposed answers is correct.

1. 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. $f'(x) = (1+x)\mathrm{e}^{-x}$
   
2. 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 $x = -1$
   
3. 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. $F(x) = x^2\left(3-2x^2\right)\mathrm{e}^{-x^2}$
   
4. 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
   
5. 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. $x \longmapsto \mathrm{e}^{x^2+x}$
   
6. 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 $f$?

Consider the function $g$ defined on $\mathbb { R }$ by: $$g ( x ) = \frac { 2 \mathrm { e } ^ { x } } { \mathrm { e } ^ { x } + 1 } .$$ The representative curve of the function $g$ admits as an asymptote at $+ \infty$ the line with equation: a. $x = 2$; b. $y = 2$; c. $y = 0$; d. $x = - 1$

There are electric showers on the market with different power ratings, which represent different consumptions and costs. The power (P) of an electric shower is given by the product between its electrical resistance (R) and the square of the electrical current (i) flowing through it. The consumption of electrical energy (E), in turn, is directly proportional to the power of the device.
Considering the characteristics presented, which of the following graphs represents the relationship between the energy consumed (E) by an electric shower and the electrical current (i) flowing through it?
(A) [graph A]
(B) [graph B]
(C) [graph C]
(D) [graph D]
(E) [graph E]

QUESTION 173
The function $f(x) = 3^x$ passes through the point
(A) $(0, 0)$
(B) $(0, 1)$
(C) $(1, 0)$
(D) $(1, 3)$
(E) $(3, 1)$

7. D

Solution: By the concavity of the function, the point $(a, b)$ cannot be above the curve $y = e ^ { x }$. Since $y = 0$ is an asymptote, the point lies between the curve and the asymptote, so $0 < b < e ^ { a }$. The answer is $D$.