The graph of the function $f$ is shown in the figure above. The value of $\lim _ { x \rightarrow 1 ^ { + } } f ( x )$ is
(A) - 2
(B) - 1
(C) 2
(D) nonexistent

Exercise 3 — 5 points Theme: function study Parts A and B can be treated independently
Part A
The plane is equipped with an orthogonal coordinate system. Below is represented the curve of a function $f$ defined and twice differentiable on $\mathbb{R}$, as well as that of its derivative $f'$ and its second derivative $f''$.

1. Determine, by justifying your choice, which curve corresponds to which function.
2. Determine, with the precision allowed by the graph, the slope of the tangent line to curve $\mathscr{C}_{2}$ at the point with abscissa 4.
3. Give, with the precision allowed by the graph, the abscissa of each inflection point of curve $\mathscr{C}_{1}$.

Part B
Let $k$ be a strictly positive real number. We consider the function $g$ defined on $\mathbb{R}$ by: $$g(x) = \frac{4}{1 + \mathrm{e}^{-kx}}$$

1. Determine the limits of $g$ at $+\infty$ and at $-\infty$.
2. Prove that $g'(0) = k$.
3. By admitting the result below obtained with computer algebra software, prove that the curve of $g$ has an inflection point at the point with abscissa 0.

$\triangleright$	Computer algebra
	$g(x) = 4 / (1 + \mathrm{e}^{\wedge}(-kx))$
1
	$\rightarrow g(x) = \frac{4}{\mathrm{e}^{-kx} + 1}$
	Simplify $(g''(x))$
2
	$\rightarrow g''(x) = -4\mathrm{e}^{kx}(\mathrm{e}^{kx} - 1)\frac{k^{2}}{(\mathrm{e}^{kx} + 1)^{3}}$

We consider a function $f$ defined on $[0; +\infty[$, represented by the curve $\mathscr{C}$ below. The line $T$ is tangent to the curve $\mathscr{C}$ at point A with abscissa $\frac{5}{2}$.

1. Draw up, by graphical reading, the table of variations of the function $f$ on the interval $[0;5]$.
2. What does the curve $\mathscr{C}$ appear to present at point A?
3. The derivative $f'$ and the second derivative $f''$ of the function $f$ are represented by the curves $\mathscr{C}_1$ and $\mathscr{C}_2$. Associate with each of these two functions the curve that represents it. This choice will be justified.
4. Can the curve $\mathscr{C}_3$ be the graphical representation on $[0; +\infty[$ of a primitive of the function $f$? Justify.

Question 144
A figura mostra o gráfico de uma função $f$.
[Figure]
Com base no gráfico, é correto afirmar que
(A) $f(-2) = 0$ (B) $f(0) = -2$ (C) $f(1) = 3$ (D) $f(2) = 0$ (E) $f(3) = 1$

Some electronic equipment can ``burn out'' during operation when its internal temperature reaches a maximum value $\mathrm{T}_{\mathrm{M}}$. For greater durability of its products, the electronics industry connects temperature sensors to this equipment, which activate an internal cooling system, turning it on when the temperature of the electronic device exceeds a critical level $\mathrm{T}_{\mathrm{c}}$, and turning it off only when the temperature drops to values below $\mathrm{T}_{\mathrm{m}}$. The graph illustrates the oscillation of the internal temperature of an electronic device during the first six hours of operation, showing that its internal cooling system was activated several times.
How many times did the temperature sensor activate the system, turning it on or off?
(A) 2
(B) 3
(C) 4
(D) 5
(E) 9

Research in the area of neurobiology confirms that meditative practice is responsible for considerably reducing respiratory frequency for advanced practitioners, who, after initiating meditation, have their respiratory frequencies reduced until they stabilize at a lower level. The graph presents the relationship of respiratory frequency, in breaths per minute (rpm), in relation to time, in minutes, of an advanced practitioner, in which $(\mathrm{f}_1)$ represents the frequency at instant $\mathrm{t}_1$, when meditative practice begins; and $(\mathrm{f}_2)$, the frequency at instant $t_2$, from which it stabilizes during meditation.
From the instant $\mathrm{t}_1$, when the meditative practice begins, the behavior of respiratory frequency, in relation to time,
(A) remains constant.
(B) is directly proportional to time.
(C) is inversely proportional to time.
(D) decreases until the instant $\mathrm{t}_2$, after which it becomes constant.
(E) decreases proportionally to time, both between $\mathrm{t}_1$ and $\mathrm{t}_2$ and after $t_2$.

The graph of the function $y = f(x)$ is shown in the figure. What is the value of $\lim _ { x \rightarrow - 1 - 0 } f ( x ) + \lim _ { x \rightarrow + 0 } f ( x )$? [3 points]
(1) $-2$
(2) $-1$
(3) 0
(4) 1
(5) 2

The graph of the function $y = f ( x )$ is shown in the figure.
What is the value of $\lim _ { x \rightarrow - 1 - 0 } f ( x ) + \lim _ { x \rightarrow + 0 } f ( x )$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

The graph of the function $y = f ( x )$ is shown in the figure. What is the value of $\lim _ { x \rightarrow - 0 } f ( x ) + \lim _ { x \rightarrow 1 + 0 } f ( x )$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

The graph of the function $y = f ( x )$ is shown in the figure. What is the value of $\lim _ { x \rightarrow - 1 - 0 } f ( x ) + \lim _ { x \rightarrow + 0 } f ( x )$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

The graph of the function $y = f ( x )$ is shown in the figure. What is the value of $\lim _ { x \rightarrow 0 - } f ( x ) + \lim _ { x \rightarrow 1 + } f ( x )$? [3 points]
(1) - 1
(2) - 2
(3) - 3
(4) - 4
(5) - 5

The graph of the function $y = f ( x )$ is shown in the figure. Find the value of $\lim _ { x \rightarrow 0 - } f ( x ) + \lim _ { x \rightarrow 1 + } f ( x )$. [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

The graph of the function $y = f ( x )$ is shown in the figure. [Figure] What is the value of $\lim _ { x \rightarrow - 1 - } f ( x ) - \lim _ { x \rightarrow 1 + } f ( x )$? [3 points]
(1) - 2
(2) - 1
(3) 0
(4) 1
(5) 2

The graph of the function $y = f ( x )$ is shown in the figure. What is the value of $\lim _ { x \rightarrow 0 + } f ( x ) - \lim _ { x \rightarrow 1 - } f ( x )$? [3 points]
(1) $-2$
(2) $-1$
(3) 0
(4) 1
(5) 2

The graph of the function $y = f ( x )$ is shown in the figure.
What is the value of $\lim _ { x \rightarrow - 1 - } f ( x ) + \lim _ { x \rightarrow 2 } f ( x )$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

Which of the following is the domain of the function $f$ whose graph is given above?
A) $[-3,0) \cup [4,7)$
B) $(-3,0) \cup (3,7]$
C) $[-3,2] \cup (3,7)$
D) $(-3,3) \cup (3,7]$
E) $[-3,2) \cup (4,7]$

The graph of the function $f: \mathbb{R}\setminus\{-1\} \rightarrow \mathbb{R}\setminus\{2\}$ is shown in the figure above.
Accordingly, $$\lim_{x \rightarrow -\infty} f(x) + \lim_{x \rightarrow 0} f(x)$$ What is the sum of these limits?
A) $-2$
B) $-1$
C) $0$
D) $1$
E) $3$

When 130 liters of milk in a dairy is used to make cheese, the graph of the linear relationship between the remaining milk and the amount of cheese produced is given.
According to this, when 10 kg of cheese is produced in this dairy, how many liters of milk remain?
A) 50
B) 60
C) 65
D) 75
E) 80

In the rectangular coordinate plane, the graphs of functions $f$, $g$, and $h$ are given in the figure.
Accordingly, for a real number $a$ satisfying the condition $0 < a < 2$
I. When $f(a) < g(a)$, then $g(a) < h(a)$ holds. II. When $g(a) < h(a)$, then $h(a) < f(a)$ holds. III. When $h(a) < f(a)$, then $f(a) < g(a)$ holds.
Which of the following statements are true?
A) Only I
B) Only II
C) Only III
D) I and II
E) I and III

In the rectangular coordinate plane, the graph of a function $f$ defined on the closed interval $[-5,5]$ is given in the figure.
For distinct numbers $a, b, c$ and $d$ in the domain of this function
$$\begin{aligned} & f(a) = f(b) = 1 \\ & f(c) = f(d) = 3 \end{aligned}$$
the equalities are satisfied. Accordingly, regarding the ordering of $a, b, c$ and $d$ numbers I. $a < b < c < d$ II. $c < a < b < d$ III. $c < d < a < b$ Which of the following inequalities can be true?
A) Only I
B) Only II
C) I and II
D) II and III
E) I, II and III

In the rectangular coordinate plane, the graphs of functions $f$ and $g$ defined on the closed interval $[0,1]$ and the line $y = x$ are given below.
For real numbers $a, b$ and $c$ in the open interval $(0,1)$
$$\begin{aligned} & a < f(a) < g(a) \\ & g(b) < b < f(b) \\ & c < g(c) < f(c) \end{aligned}$$
If these inequalities are satisfied, which of the following orderings is correct?
A) $a < b < c$ B) $a < c < b$ C) $b < a < c$ D) $c < a < b$ E) $c < b < a$

In the rectangular coordinate plane, the graphs of linear functions $f$, $g$ and $h$ are shown in the figure.
Regarding these functions, the following equalities are given:
$$\begin{aligned} & f(x-5) = g(x) \\ & h(x) = -f(x) \end{aligned}$$
Which of the following orderings is correct for the values $f(0)$, $g(0)$ and $h(0)$?
A) $g(0) < f(0) < h(0)$ B) $f(0) < h(0) < g(0)$ C) $f(0) < g(0) < h(0)$ D) $g(0) < h(0) < f(0)$ E) $h(0) < g(0) < f(0)$

In the rectangular coordinate plane, the graphs of functions $f$, $g$ and $h$ are given in the figure.
For functions $f$, $g$ and $h$
$$(f - g)(1) \cdot (f - h)(1) < 0$$
$$(g - h)(2) \cdot (g - f)(2) > 0$$
Given that the inequalities are satisfied, which of the following orderings is correct?
A) $f(3) < g(3) < h(3)$
B) $f(3) < h(3) < g(3)$
C) $g(3) < f(3) < h(3)$
D) $h(3) < f(3) < g(3)$
E) $h(3) < g(3) < f(3)$

In the rectangular coordinate plane, the graphs of the functions $f + g$ and $f \cdot g$ defined on the closed interval $[0, 10]$ are shown below.
For the real numbers $a, b$ and $c$ in the closed interval $[0, 10]$,

• $f(a), f(b)$ and $g(b)$ values are positive,
• $g(a), f(c)$ and $g(c)$ values are negative.

Accordingly, which of the following is the correct ordering of $a, b$ and $c$?
A) $a < c < b$ B) $b < a < c$ C) $b < c < a$ D) $c < a < b$ E) $c < b < a$

In the rectangular coordinate plane, the graphs of $f$, $g$, and $h$ linear functions defined on the closed interval $[0,1]$ are given in the figure.
Let $a$ and $b$ be real numbers in the open interval $(0,1)$ and
$$f(1) = g(1)$$
$$f(a) = g(b) = h(b)$$
equalities are satisfied.
Accordingly, which of the following orderings is correct?
A) $f(a) < h(a) < g(a)$ B) $g(a) < f(a) < h(a)$ C) $g(a) < h(a) < f(a)$ D) $h(a) < f(a) < g(a)$ E) $h(a) < g(a) < f(a)$