If $f ^ { \prime } ( x ) > 0$ for all real numbers $x$ and $\int _ { 4 } ^ { 7 } f ( t ) d t = 0$, which of the following could be a table of values for the function $f$ ?
(A)

$x$	$f ( x )$
4	$-4$
5	$-3$
7	0

(B)

$x$	$f ( x )$
4	$-4$
5	$-2$
7	5

(C)

$x$	$f ( x )$
4	$-4$
5	6
7	3

(D)

$x$	$f ( x )$
4	0
5	0
7	0

(E)

$x$	$f ( x )$
4	0
5	4
7	6

For each of the following statements, indicate whether it is true or false. You will justify each answer.
Statement 1: For all real numbers $a$ and $b$, $\left( \mathrm{e}^{a+b} \right)^{2} = \mathrm{e}^{2a} + \mathrm{e}^{2b}$.
Statement 2: In the plane with a coordinate system, the tangent line at point A with abscissa 0 to the representative curve of the function $f$ defined on $\mathbb{R}$ by $f(x) = -2 + (3-x)\mathrm{e}^{x}$ has the reduced equation $y = 2x + 1$.
Statement 3: $\lim_{x \rightarrow +\infty} \left( \mathrm{e}^{2x} - \mathrm{e}^{x} + \frac{3}{x} \right) = 0$.
Statement 4: The equation $1 - x + \mathrm{e}^{-x} = 0$ has a unique solution belonging to the interval $[0 ; 2]$.
Statement 5: The function $g$ defined on $\mathbb{R}$ by $g(x) = x^{2} - 5x + \mathrm{e}^{x}$ is convex.

For each of the following statements, indicate whether it is true or false. Justify each answer.

1. Statement 1: For all real $x : 1 - \frac { 1 - \mathrm { e } ^ { x } } { 1 + \mathrm { e } ^ { x } } = \frac { 2 } { 1 + \mathrm { e } ^ { - x } }$.
   
2. We consider the function $g$ defined on $\mathbb { R }$ by $g ( x ) = \frac { \mathrm { e } ^ { x } } { \mathrm { e } ^ { x } + 1 }$. Statement 2: The equation $g ( x ) = \frac { 1 } { 2 }$ admits a unique solution in $\mathbb { R }$.
   
3. We consider the function $f$ defined on $\mathbb { R }$ by $f ( x ) = x ^ { 2 } \mathrm { e } ^ { - x }$ and we denote $\mathscr { C }$ its curve in an orthonormal coordinate system. Statement 3: The $x$-axis is tangent to the curve $\mathscr { C }$ at only one point.
   
4. We consider the function $h$ defined on $\mathbb { R }$ by $h ( x ) = \mathrm { e } ^ { x } \left( 1 - x ^ { 2 } \right)$. Statement 4: In the plane equipped with an orthonormal coordinate system, the curve representing the function $h$ does not admit an inflection point.
   
5. Statement 5: $\lim _ { x \rightarrow + \infty } \frac { \mathrm { e } ^ { x } } { \mathrm { e } ^ { x } + x } = 0$.
   
6. Statement 6: For all real $x , 1 + \mathrm { e } ^ { 2 x } \geqslant 2 \mathrm { e } ^ { x }$.

Exercise 3 — 5 points Theme: exponential function, algorithms For each of the following statements, indicate whether it is true or false. Each answer must be justified. An unjustified answer receives no points.

1. Statement: The function $f$ defined on $\mathbb{R}$ by $f(x) = \mathrm{e}^x - x$ is convex.
2. Statement: The equation $(2\mathrm{e}^x - 6)(\mathrm{e}^x + 2) = 0$ has $\ln(3)$ as its unique solution in $\mathbb{R}$.
3. Statement: $$\lim_{x \to +\infty} \frac{\mathrm{e}^{2x} - 1}{\mathrm{e}^x - x} = 0.$$
4. Let $f$ be the function defined on $\mathbb{R}$ by $f(x) = (6x + 5)\mathrm{e}^{3x}$ and $F$ the function defined on $\mathbb{R}$ by: $F(x) = (2x + 1)\mathrm{e}^{3x} + 4$. Statement: $F$ is the antiderivative of $f$ on $\mathbb{R}$ that takes the value 5 when $x = 0$.
5. We consider the function \texttt{mystere} defined below which takes a list $L$ of numbers as a parameter. We recall that \texttt{len(L)} represents the length of list $L$. \begin{verbatim} def mystere(L) : S = 0 for i in range(len(L)) : S = S + L[i] return S / len(L) \end{verbatim} Statement: The execution of \texttt{mystere([1, 9, 9, 5, 0, 3, 6, 12, 0, 5])} returns 50.

For the function $f ( x ) = \frac { 4 ^ { x } } { 4 ^ { x } + 2 }$, select all correct statements from . [4 points]
ㄱ. $f \left( \frac { 1 } { 2 } \right) = \frac { 1 } { 2 }$ ㄴ. $f ( x ) + f ( 1 - x ) = 1$ ㄷ. $\sum _ { k = 1 } ^ { 100 } f \left( \frac { k } { 101 } \right) = 50$
(1) ㄱ
(2) ㄴ
(3) ㄷ
(4) ㄱ, ㄴ
(5) ㄱ, ㄴ, ㄷ

On the coordinate plane, the two points where the two curves $y = \left| \log _ { 2 } x \right|$ and $y = \left( \frac { 1 } { 2 } \right) ^ { x }$ meet are $\mathrm { P } \left( x _ { 1 } , y _ { 1 } \right) , \mathrm { Q } \left( x _ { 2 } , y _ { 2 } \right) \left( x _ { 1 } < x _ { 2 } \right)$, and the point where the two curves $y = \left| \log _ { 2 } x \right|$ and $y = 2 ^ { x }$ meet is $\mathrm { R } \left( x _ { 3 } , y _ { 3 } \right)$. Which of the following statements in are correct? [4 points]
ㄱ. $\frac { 1 } { 2 } < x _ { 1 } < 1$ ㄴ. $x _ { 2 } y _ { 2 } - x _ { 3 } y _ { 3 } = 0$ ㄷ. $x _ { 2 } \left( x _ { 1 } - 1 \right) > y _ { 1 } \left( y _ { 2 } - 1 \right)$
(1) ㄱ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

In the coordinate plane, let the two points where the curves $y = \left| \log _ { 2 } x \right|$ and $y = \left( \frac { 1 } { 2 } \right) ^ { x }$ meet be $\mathrm { P } \left( x _ { 1 } , y _ { 1 } \right) , \mathrm { Q } \left( x _ { 2 } , y _ { 2 } \right) \left( x _ { 1 } < x _ { 2 } \right)$, and let the point where the curves $y = \left| \log _ { 2 } x \right|$ and $y = 2 ^ { x }$ meet be $\mathrm { R } \left( x _ { 3 } , y _ { 3 } \right)$. Which of the following are correct? Choose all that apply from $\langle$Remarks$\rangle$. [4 points]
$\langle$Remarks$\rangle$ ㄱ. $\frac { 1 } { 2 } < x _ { 1 } < 1$ ㄴ. $x _ { 2 } y _ { 2 } - x _ { 3 } y _ { 3 } = 0$ ㄷ. $x _ { 2 } \left( x _ { 1 } - 1 \right) > y _ { 1 } \left( y _ { 2 } - 1 \right)$
(1) ㄱ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

6. If $a > b$, then
A. $\ln ( a - b ) > 0$
B. $3 ^ { a } < 3 ^ { b }$
C. $a ^ { 3 } - b ^ { 3 } > 0$
D. $| a | > | b |$

Given that $\left( x _ { 1 } , y _ { 1 } \right) , \left( x _ { 2 } , y _ { 2 } \right)$ are points on $y = 2 ^ { x }$, which of the following is correct?
A. $\log _ { 2 } \frac { y _ { 1 } + y _ { 2 } } { 2 } > \frac { x _ { 1 } + x _ { 2 } } { 2 }$
B. $\log _ { 2 } \frac { y _ { 1 } + y _ { 2 } } { 2 } < \frac { x _ { 1 } + x _ { 2 } } { 2 }$
C. $\log _ { 2 } \frac { y _ { 1 } + y _ { 2 } } { 2 } > x _ { 1 } + x _ { 2 }$
D. $\log _ { 2 } \frac { y _ { 1 } + y _ { 2 } } { 2 } < x _ { 1 }$

Let $a$ and $b$ be two reals satisfying $a < b$. Show that $\forall \lambda \in [0,1], \mathrm{e}^{\lambda a+(1-\lambda) b} \leqslant \lambda \mathrm{e}^{a}+(1-\lambda) \mathrm{e}^{b}$.

If $f : \mathbb { R } \rightarrow \mathbb { R }$ is a differentiable function such that $f ^ { \prime } ( x ) > 2 f ( x )$ for all $x \in \mathbb { R }$, and $f ( 0 ) = 1$, then
[A] $f ( x )$ is increasing in $( 0 , \infty )$
[B] $f ( x )$ is decreasing in $( 0 , \infty )$
[C] $f ( x ) > e ^ { 2 x }$ in $( 0 , \infty )$
[D] $f ^ { \prime } ( x ) < e ^ { 2 x }$ in $( 0 , \infty )$

Let $f : \mathbb { R } \rightarrow \mathbb { R }$ and $g : \mathbb { R } \rightarrow \mathbb { R }$ be two non-constant differentiable functions. If $$f ^ { \prime } ( x ) = \left( e ^ { ( f ( x ) - g ( x ) ) } \right) g ^ { \prime } ( x ) \text { for all } x \in \mathbb { R }$$ and $f ( 1 ) = g ( 2 ) = 1$, then which of the following statement(s) is (are) TRUE?
(A) $f ( 2 ) < 1 - \log _ { \mathrm { e } } 2$
(B) $f ( 2 ) > 1 - \log _ { \mathrm { e } } 2$
(C) $g ( 1 ) > 1 - \log _ { \mathrm { e } } 2$
(D) $g ( 1 ) < 1 - \log _ { e } 2$

Let $b$ be a nonzero real number. Suppose $f: \mathbb{R} \rightarrow \mathbb{R}$ is a differentiable function such that $f(0) = 1$. If the derivative $f'$ of $f$ satisfies the equation $$f'(x) = \frac{f(x)}{b^{2} + x^{2}}$$ for all $x \in \mathbb{R}$, then which of the following statements is/are TRUE?
(A) If $b > 0$, then $f$ is an increasing function
(B) If $b < 0$, then $f$ is a decreasing function
(C) $f(x)f(-x) = 1$ for all $x \in \mathbb{R}$
(D) $f(x) - f(-x) = 0$ for all $x \in \mathbb{R}$

Let the exponential function $f(x) = 1.2^{x}$. Select the correct options.
(1) $f(0) > 0$
(2) $f(10) > 10$
(3) On the coordinate plane, the graph of $y = 1.2^{x}$ intersects the line $y = x$
(4) On the coordinate plane, the graphs of $y = 1.2^{x}$ and $y = \log(1.2^{x})$ are symmetric about the line $y = x$
(5) For any positive real number $b$, $\log_{1.2} b \neq 1.2^{b}$

A function f is defined on the set of real numbers as
$$f ( x ) = 1 + e ^ { - x }$$
Accordingly, I. The range of function f is $( 1 , \infty )$. II. Function f is decreasing on its domain. III. The line $y = 0$ is a horizontal asymptote of function f. Which of the following statements are true?
A) Only II
B) Only III
C) I and II