Exercise 1 (5 points)
The plane is equipped with an orthogonal coordinate system $(\mathrm{O}, \mathrm{I}, \mathrm{J})$.
Consider the function $f$ defined on the interval $]0; 1]$ by $$f(x) = x(1 - \ln x)^2.$$
a. Determine an expression for the derivative of $f$ and verify that for all $x \in ]0; 1]$, $f'(x) = (\ln x + 1)(\ln x - 1)$.
b. Study the variations of the function $f$ and draw its variation table on the interval $]0; 1]$ (it will be admitted that the limit of the function $f$ at 0 is zero).

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.

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 $f ( x ) = \left( x ^ { 2 } - 2 x - 7 \right) e ^ { x }$, 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

Let $f : (0, \infty) \rightarrow \mathbb{R}$ be given by $$f(x) = \int_{\frac{1}{x}}^{x} e^{-\left(t + \frac{1}{t}\right)} \frac{dt}{t}$$ Then
(A) $f(x)$ is monotonically increasing on $[1, \infty)$
(B) $f(x)$ is monotonically decreasing on $(0,1)$
(C) $f(x) + f\left(\frac{1}{x}\right) = 0$, for all $x \in (0, \infty)$
(D) $f\left(2^x\right)$ is an odd function of $x$ on $\mathbb{R}$

Let $f(x) = 2x + \tan^{-1} x$ and $g(x) = \log_e\left(\sqrt{1 + x^2} + x\right)$, $x \in [0, 3]$. Then
(1) There exists $x \in (0, 3)$ such that $f'(x) < g'(x)$
(2) $\max f(x) > \max g(x)$
(3) There exist $0 < x_1 < x_2 < 3$ such that $f(x) < g(x)$, $\forall x \in (x_1, x_2)$
(4) $\min f'(x) = 1 + \max g'(x)$

Q1 Let $f(x)=\log(4x-\log x)$, where $\log$ is the natural logarithm. We are to find a local extremum of $f(x)$ by using $f''(x)$.
For $\mathbf{K}$ and $\mathbf{L}$, choose the most appropriate answer from among the choices (0)$\sim$(6) below.
First of all, we have
$$\begin{aligned} f'(x) &= \frac{\mathbf{A}-\frac{\mathbf{B}}{x}}{4x-\log x} \\ f''(x) &= \frac{\frac{1}{x^{\mathbf{C}}}(4x-\log x)-\left(\mathbf{A}-\frac{\square}{x}\right)^{\mathbf{D}}}{(4x-\log x)^2} \end{aligned}$$
which give
$$\begin{aligned} f'\left(\frac{\mathbf{E}}{\mathbf{F}}\right) &= 0 \\ f''\left(\frac{\mathbf{E}}{\mathbf{F}}\right) &= \frac{\mathbf{GH}}{\mathbf{I}+\log\mathbf{J}}. \end{aligned}$$
Since
$$f''\left(\frac{\mathbf{E}}{\mathbf{F}}\right) \mathbf{K} \, 0,$$
$f(x)$ has a $\mathbf{L}$ at $x=\frac{\mathbf{E}}{\mathbf{F}}$, and this value is $\log(\mathbf{M}+\log\mathbf{N})$.
(0) $=$ (1) $>$ (2) $\geqq$ (3) $<$ (4) $\leqq$ (5) local maximum (6) local minimum

For each of $\mathbf{A} \sim \mathbf{I}$ in the following sentences, choose the appropriate answer from among (0) $\sim$ (9) at the bottom of this page.
We are to compare the magnitudes of $a ^ { a + 1 }$ and $( a + 1 ) ^ { a }$ by using the properties of the function $f ( x ) = \dfrac { \log x } { x }$, where $a > 0$.
(1) Since the derivative of $f ( x )$ is
$$f ^ { \prime } ( x ) = \frac { \mathbf { A } - \log x } { x^{\mathbf{B}} } ,$$
the interval on $x$ in which $f ( x )$ monotonically increases is
$$\mathbf { C } < x \leqq \mathbf { D }$$
and the interval on $x$ in which $f ( x )$ monotonically decreases is
$$\mathbf { E } \leq x .$$
(2) When we set $p = a ^ { a + 1 }$, $q = ( a + 1 ) ^ { a }$, we have
$$\log p - \log q = \left( a ^ { \mathbf { F } } + a \right) \{ f ( a ) - f ( a + \mathbf { G } ) \} .$$
Hence we see that
$$\text { if } \quad 0 < a < \tfrac{3}{2} \quad \text { then } \quad p \quad \mathbf { H } \quad q ,$$
and
$$\text { if } \quad 3 < a \quad \text { then } \quad p \quad \mathbf { I } \quad q .$$
(0) 0 (1) 1 (2) 2 (3) 3 (4) $e$ (5) $e + 1$ (6) $\dfrac{1}{e}$ (7) $>$ (8) $=$ (9) $<$

We are to find the range of the values of $k$ such that the inequality
$$\frac { \log 3 x } { 4 x + 1 } \leqq \log \left( \frac { 2 k x } { 4 x + 1 } \right) \tag{1}$$
holds for all positive real numbers $x$, where $\log$ is the natural logarithm.
(1) For $\mathbf{A}$ and $\mathbf{B}$ in the following sentences, choose the correct answer from among (0) $\sim$ (8) below.
By transforming inequality (1) we obtain
$$\log k \geqq \mathbf { A } . \tag{2}$$
Here, when the right side of (2) is denoted by $g ( x )$ and this $g ( x )$ is differentiated with respect to $x$, we have
$$g ^ { \prime } ( x ) = \mathbf { B } .$$
(0) $\frac { \log 3 x } { 4 x + 1 } - \log ( 4 x + 1 ) - \log 2 x$
(1) $\frac { \log 3 x } { 4 x + 1 } - \log ( 4 x + 1 ) + \log 2 x$
(2) $\frac { \log 3 x } { 4 x + 1 } + \log ( 4 x + 1 ) + \log 2 x$
(3) $\frac { \log 3 x } { 4 x + 1 } + \log ( 4 x + 1 ) - \log 2 x$
(4) $\frac { 4 \log 3 x } { ( 4 x + 1 ) ^ { 2 } }$
(5) $\frac { 3 x + 2 + \log 3 x } { ( 4 x + 1 ) ^ { 2 } }$ (6) $- \frac { 4 \log 3 x } { ( 4 x + 1 ) ^ { 2 } }$ (7) $\frac { 3 x - 2 - \log 2 x } { ( 4 x + 1 ) ^ { 2 } }$ (8) $- \frac { 3 \log 2 x } { ( 4 x + 1 ) ^ { 2 } }$
(2) In the following sentences, for $\mathbf { E } , \mathbf { F }$ and $\mathbf { G }$, choose the correct answer from among (0) $\sim$ (3) below. For the other blanks, enter the correct number.
Over the interval $0 < x < \frac { \mathbf { C } } { \mathbf{D} }$, $g ( x )$ is $\mathbf { E }$ and over the interval $\frac { \mathbf{C} } { \mathbf{D} } < x$, $g ( x )$ is $\mathbf { F }$. Hence at $x = \frac { \mathbf { C } } { \mathbf{D} }$, $g ( x )$ is $\mathbf { G }$.
From the above, the range of the value of $k$ such that inequality (1) holds for all positive real numbers $x$ is
$$k \geqq \frac { \mathbf { H } } { \mathbf { I } }$$
(0) increasing
(1) decreasing
(2) maximized
(3) minimized