This exercise is a multiple choice questionnaire. For each of the following questions, only one of the four proposed answers is correct. No justification is required. A wrong answer, no answer, or multiple answers, neither gives nor takes away points.

1. We consider the function $f$ defined on $\mathbb{R}$ by $f(x) = 2x\mathrm{e}^x$.
   The number of solutions on $\mathbb{R}$ of the equation $f(x) = -\dfrac{73}{100}$ is equal to:
   a. 0
   b. 1
   c. 2
   d. infinitely many.
   
2. We consider the function $g$ defined on $\mathbb{R}$ by: $$g(x) = \frac{x+1}{\mathrm{e}^x}.$$ The limit of the function $g$ at $-\infty$ is equal to:
   a. $-\infty$
   b. $+\infty$
   c. $0$
   d. it does not exist.
   
3. We consider the function $h$ defined on $\mathbb{R}$ by: $$h(x) = (4x - 16)\mathrm{e}^{2x}.$$ We denote $\mathscr{C}_h$ the representative curve of $h$ in an orthogonal coordinate system. We can affirm that:
   a. $h$ is convex on $\mathbb{R}$.
   b. $\mathscr{C}_h$ has an inflection point at $x = 3$.
   c. $h$ is concave on $\mathbb{R}$.
   d. $\mathscr{C}_h$ has an inflection point at $x = 3.5$.
   
4. We consider the function $k$ defined on the interval $]0; +\infty[$ by: $$k(x) = 3\ln(x) - x.$$ We denote $\mathscr{C}$ the representative curve of the function $k$ in an orthonormal coordinate system. We denote $T$ the tangent line to the curve $\mathscr{C}$ at the point with abscissa $x = \mathrm{e}$. An equation of $T$ is:
   a. $y = (3 - \mathrm{e})x$
   b. $y = \left(\dfrac{3 - \mathrm{e}}{\mathrm{e}}\right)x$
   c. $y = \left(\dfrac{3}{\mathrm{e}} - 1\right)x + 1$
   d. $y = (\mathrm{e} - 1)x + 1$
   
5. We consider the equation $[\ln(x)]^2 + 10\ln(x) + 21 = 0$, with $x \in ]0; +\infty[$.
   The number of solutions of this equation is equal to:
   a. 0
   b. 1
   c. 2
   d. infinitely many.

Consider the function $f$ defined on $] 0 ; + \infty [$ by
$$f ( x ) = x ^ { 2 } - x \ln ( x ) .$$
We admit that $f$ is twice differentiable on $] 0 ; + \infty [$. We denote $f ^ { \prime }$ the derivative function of $f$ and $f ^ { \prime \prime }$ the derivative function of $f ^ { \prime }$.
Part A: Study of the function $f$

1. Determine the limits of the function $f$ at 0 and at $+ \infty$.
2. For all strictly positive real $x$, calculate $f ^ { \prime } ( x )$.
3. Show that for all strictly positive real $x$: $$f ^ { \prime \prime } ( x ) = \frac { 2 x - 1 } { x }$$
4. Study the variations of the function $f ^ { \prime }$ on $] 0 ; + \infty [$, then draw up the table of variations of the function $f ^ { \prime }$ on $] 0 ; + \infty [$. Care should be taken to show the exact value of the extremum of the function $f ^ { \prime }$ on $] 0 ; + \infty [$. The limits of the function $f ^ { \prime }$ at the boundaries of the domain of definition are not expected.
5. Show that the function $f$ is strictly increasing on $] 0 ; + \infty [$.

Part B: Study of an auxiliary function for solving the equation $f ( x ) = x$
We consider in this part the function $g$ defined on $] 0 ; + \infty [$ by
$$g ( x ) = x - \ln ( x )$$
We admit that the function $g$ is differentiable on $] 0 ; + \infty [$, we denote $g ^ { \prime }$ its derivative.

1. For all strictly positive real, calculate $g ^ { \prime } ( x )$, then draw up the table of variations of the function $g$. The limits of the function $g$ at the boundaries of the domain of definition are not expected.
2. We admit that 1 is the unique solution of the equation $g ( x ) = 1$. Solve, on the interval $] 0 ; + \infty [$, the equation $f ( x ) = x$.

Part C: Study of a recursive sequence
We consider the sequence $( u _ { n } )$ defined by $u _ { 0 } = \frac { 1 } { 2 }$ and for all natural integer $n$,
$$u _ { n + 1 } = f \left( u _ { n } \right) = u _ { n } ^ { 2 } - u _ { n } \ln \left( u _ { n } \right) .$$

1. Show by induction that for all natural integer $n$: $$\frac { 1 } { 2 } \leqslant u _ { n } \leqslant u _ { n + 1 } \leqslant 1 .$$
2. Justify that the sequence $( u _ { n } )$ converges. We call $\ell$ the limit of the sequence $( u _ { n } )$ and we admit that $\ell$ satisfies the equality $f ( \ell ) = \ell$.
3. Determine the value of $\ell$.

We consider the function $f$ defined on the interval $]-\infty; 1[$ by $$f(x) = \frac{\mathrm{e}^x}{x-1}$$ We admit that the function $f$ is differentiable on the interval $]-\infty; 1[$. We call $\mathscr{C}$ its representative curve in a coordinate system.

1. 1. [a.] Determine the limit of the function $f$ at 1.
   2. [b.] Deduce from this a graphical interpretation.
2. Determine the limit of the function $f$ at $-\infty$.
3. 1. [a.] Show that for every real number $x$ in the interval $]-\infty; 1[$, we have $$f'(x) = \frac{(x-2)\mathrm{e}^x}{(x-1)^2}$$
   2. [b.] Draw up, by justifying, the table of variations of the function $f$ on the interval $]-\infty; 1[$.
4. We admit that for every real number $x$ in the interval $]-\infty; 1[$, we have $$f''(x) = \frac{\left(x^2 - 4x + 5\right)\mathrm{e}^x}{(x-1)^3}.$$
   1. [a.] Study the convexity of the function $f$ on the interval $]-\infty; 1[$.
   2. [b.] Determine the reduced equation of the tangent line $T$ to the curve $\mathscr{C}$ at the point with abscissa 0.
   3. [c.] Deduce from this that, for every real number $x$ in the interval $]-\infty; 1[$, we have: $$\mathrm{e}^x \geqslant (-2x-1)(x-1).$$
5. 1. [a.] Justify that the equation $f(x) = -2$ admits a unique solution $\alpha$ on the interval $]-\infty; 1[$.
   2. [b.] Using a calculator, determine an interval containing $\alpha$ with amplitude $10^{-2}$.

The function $f(x) = e^{2x}$ has derivative:
(A) $e^{2x}$
(B) $2e^{x}$
(C) $2e^{2x}$
(D) $4e^{2x}$
(E) $e^{x}$

A function $f$ is defined by $f ( x ) = e ^ { x }$ if $x < 1$ and $f ( x ) = \log _ { e } ( x ) + a x ^ { 2 } + b x$ if $x \geq 1$. Here $a$ and $b$ are unknown real numbers. Can $f$ be differentiable at $x = 1$ ?
(A) $f$ is not differentiable at $x = 1$ for any $a$ and $b$.
(B) There exist unique numbers $a$ and $b$ for which $f$ is differentiable at $x = 1$.
(C) $f$ is differentiable at $x = 1$ whenever $a + b = e$.
(D) $f$ is differentiable at $x = 1$ regardless of the values of $a$ and $b$.

Let $f : \mathbb{R} \longrightarrow \mathbb{R}$ be defined as $$f(x) = \begin{cases} x^2 \sin\left(\frac{1}{x^2}\right), & \text{if } x \neq 0 \\ 0, & \text{otherwise} \end{cases}$$ Choose the correct statement(s) from below:
(A) $f$ is continuous;
(B) $f$ is discontinuous at 0;
(C) $f$ is differentiable;
(D) $f$ is continuously differentiable.

Recall the function $\arctan(x)$, also denoted as $\tan^{-1}(x)$. Complete the sentence: $$\arctan(20202019) + \arctan(20202021) \quad\underline{\hspace{2cm}}\quad 2\arctan(20202020),$$ because in the relevant region, the graph of $y = \arctan(x)$ $\_\_\_\_$.
Fill in the first blank with one of the following: is less than / is equal to / is greater than. Fill in the second blank with a single correct reason consisting of one of the following phrases: is bounded / is continuous / has positive first derivative / has negative first derivative / has positive second derivative / has negative second derivative / has an inflection point.

(Calculus) For the function $f(x) = 4\ln x + \ln(10 - x)$, which of the following statements in are correct? [3 points]
ㄱ. The maximum value of function $f(x)$ is $13\ln 2$. ㄴ. The equation $f(x) = 0$ has two distinct real roots. ㄷ. The graph of function $y = e^{f(x)}$ is concave downward on the interval $(4, 8)$.
(1) ㄱ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

For a quartic function $f ( x )$ with leading coefficient 1, the function $g ( x )$ satisfies the following conditions. (가) When $- 1 \leqq x < 1$, $g ( x ) = f ( x )$. (나) For all real numbers $x$, $g ( x + 2 ) = g ( x )$.
Which of the following statements in are correct? [4 points]
Remarks ㄱ. If $f ( - 1 ) = f ( 1 )$ and $f ^ { \prime } ( - 1 ) = f ^ { \prime } ( 1 )$, then $g ( x )$ is differentiable on the entire set of real numbers. ㄴ. If $g ( x )$ is differentiable on the entire set of real numbers, then $f ^ { \prime } ( 0 ) f ^ { \prime } ( 1 ) < 0$. ㄷ. If $g ( x )$ is differentiable on the entire set of real numbers and $f ^ { \prime } ( 1 ) > 0$, then there exists $c$ in the interval $( - \infty , - 1 )$ such that $f ^ { \prime } ( c ) = 0$.
(1) ᄀ
(2) ᄂ
(3) ᄀ, ᄃ
(4) ㄴ,ㄷ
(5) ᄀ, ᄂ, ᄃ

What is the value of $\lim _ { x \rightarrow 0 } \frac { e ^ { x } - 1 } { 5 x }$? [2 points]
(1) 5
(2) $e$
(3) 1
(4) $\frac { 1 } { e }$
(5) $\frac { 1 } { 5 }$

For the function $f ( x ) = 2 x \cos x$ with domain $\{ x \mid 0 \leq x \leq \pi \}$, which of the following are correct? Choose all that apply from . [4 points]
Remarks ㄱ. If $f ^ { \prime } ( a ) = 0$, then $\tan a = \frac { 1 } { a }$. ㄴ. There exists $a$ in the interval $\left( \frac { \pi } { 4 } , \frac { \pi } { 3 } \right)$ where the function $f ( x )$ has a local maximum value at $x = a$. ㄷ. On the interval $\left[ 0 , \frac { \pi } { 2 } \right]$, the number of distinct real roots of the equation $f ( x ) = 1$ is 2.
(1) ㄱ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

For the function $f ( x ) = x \ln x + 13 x$, find the value of $f ^ { \prime } ( 1 )$. [3 points]

For the function $f ( x ) = 5 e ^ { 3 x - 3 }$, find the value of $f ^ { \prime } ( 1 )$. [3 points]

For the function $f ( x ) = \cos x + 4 e ^ { 2 x }$, find the value of $f ^ { \prime } ( 0 )$. [3 points]

For the function $f ( x ) = e ^ { x + 1 } - 1$ and a natural number $n$, let the function $g ( x )$ be defined as $$g ( x ) = 100 | f ( x ) | - \sum _ { k = 1 } ^ { n } \left| f \left( x ^ { k } \right) \right|$$ Find the sum of all natural numbers $n$ such that $g ( x )$ is differentiable on the entire set of real numbers. [4 points]

For the function $f ( x ) = 4 \sin 7 x$, find the value of $f ^ { \prime } ( 2 \pi )$. [3 points]

As shown in the figure, in the coordinate plane, the circle $x ^ { 2 } + y ^ { 2 } = 1$ and the curve $y = \ln ( x + 1 )$ meet at point A in the first quadrant. For point $\mathrm { B } ( 1,0 )$, let H be the foot of the perpendicular from point P on arc AB to the $y$-axis, and let Q be the intersection of segment PH and the curve $y = \ln ( x + 1 )$. Let $\angle \mathrm { POB } = \theta$. If $S ( \theta )$ is the area of triangle OPQ and $L ( \theta )$ is the length of segment HQ, and $\lim _ { \theta \rightarrow + 0 } \frac { S ( \theta ) } { L ( \theta ) } = k$, find the value of $60 k$. (Here, $0 < \theta < \frac { \pi } { 6 }$ and O is the origin.) [4 points]

What is the value of $\lim _ { x \rightarrow 0 } \frac { e ^ { 6 x } - 1 } { \ln ( 1 + 3 x ) }$? [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

For the function $f ( x ) = e ^ { - x } \int _ { 0 } ^ { x } \sin \left( t ^ { 2 } \right) d t$, which of the following statements are correct? [4 points]
ㄱ. $f ( \sqrt { \pi } ) > 0$ ㄴ. There exists at least one $a$ in the open interval $( 0 , \sqrt { \pi } )$ such that $f ^ { \prime } ( a ) > 0$. ㄷ. There exists at least one $b$ in the open interval $( 0 , \sqrt { \pi } )$ such that $f ^ { \prime } ( b ) = 0$.
(1) ㄱ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

For the function $f ( x ) = \ln \left( x ^ { 2 } + 1 \right)$, find the value of $f ^ { \prime } ( 1 )$. [3 points]

What is the value of $\lim _ { x \rightarrow 0 } \frac { x ^ { 2 } + 5 x } { \ln ( 1 + 3 x ) }$? [2 points]
(1) $\frac { 7 } { 3 }$
(2) 2
(3) $\frac { 5 } { 3 }$
(4) $\frac { 4 } { 3 }$
(5) 1

For a cubic function $f ( x )$ with leading coefficient $6 \pi$, the function $g ( x ) = \frac { 1 } { 2 + \sin ( f ( x ) ) }$ has a local maximum or minimum at $x = \alpha$, and when all $\alpha \geq 0$ are listed in increasing order as $\alpha _ { 1 }$, $\alpha _ { 2 } , \alpha _ { 3 } , \alpha _ { 4 } , \alpha _ { 5 } , \cdots$, the function $g ( x )$ satisfies the following conditions. (가) $\alpha _ { 1 } = 0$ and $g \left( \alpha _ { 1 } \right) = \frac { 2 } { 5 }$. (나) $\frac { 1 } { g \left( \alpha _ { 5 } \right) } = \frac { 1 } { g \left( \alpha _ { 2 } \right) } + \frac { 1 } { 2 }$ When $g ^ { \prime } \left( - \frac { 1 } { 2 } \right) = a \pi$, find the value of $a ^ { 2 }$. (Here, $0 < f ( 0 ) < \frac { \pi } { 2 }$.) [4 points]

What is the value of $\lim _ { x \rightarrow 0 } \frac { 6 x } { e ^ { 4 x } - e ^ { 2 x } }$? [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

For the function $f ( x ) = \left( x ^ { 2 } - 2 x - 7 \right) e ^ { x }$, let the local maximum value and local minimum value be $a$ and $b$ respectively. What is the value of $a \times b$? [3 points]
(1) - 32
(2) - 30
(3) - 28
(4) - 26
(5) - 24

Find the value of $\lim_{x \rightarrow 0} \frac{\ln(1+3x)}{\ln(1+5x)}$. [2 points]
(1) $\frac{1}{5}$
(2) $\frac{2}{5}$
(3) $\frac{3}{5}$
(4) $\frac{4}{5}$
(5) 1