For a real number $t$, let $f(t)$ denote the slope of the line passing through the origin and tangent to the curve $y = \frac{1}{e^x} + e^t$. For the constant $a$ satisfying $f(a) = -e\sqrt{e}$, find the value of $f'(a)$. [3 points]
(1) $-\frac{1}{3}e\sqrt{e}$
(2) $-\frac{1}{2}e\sqrt{e}$
(3) $-\frac{2}{3}e\sqrt{e}$
(4) $-\frac{5}{6}e\sqrt{e}$
(5) $-e\sqrt{e}$

11. Given the function $f ( x ) = a x \ln x , x \in ( 0 , + \infty )$, where $a$ is a real number, and $f ^ { \prime } ( x )$ is the derivative of $f ( x )$. If $f ^ { \prime } ( 1 ) = 3$, then the value of $a$ is $\_\_\_\_$.

21. (This question is worth 13 points) Given $\mathrm { a } > 0$, the function $\mathrm { f } ( \mathrm { x } ) = \mathrm { a } e ^ { x } \cos x$ for $\mathrm { x } \in [ 0 , + \infty )$. Let $x _ { n }$ denote the $n$-th (where $n \in \mathbb { N } ^ { * }$) extremum point of $f ( x )$ in increasing order. (I) Prove that: the sequence $\left\{ f \left( \mathrm { x } _ { \mathrm { n } } \right) \right\}$ is a geometric sequence; (II) If for all $n \in \mathbb { N } ^ { * }$, the inequality $x _ { n } \leq \left| f \left( x _ { n } \right) \right|$ always holds, find the range of $a$.

21. Given the function $f ( x ) = - 2 ( x + a ) \ln x + x ^ { 2 } - 2 a x - 2 a ^ { 2 } + a$, where $a > 0$.
(1) Let $g ( x )$ be the derivative of $f ( x )$. Discuss the monotonicity of $g ( x )$;
(2) Prove: there exists $a \in ( 0, 1 )$ such that $f ( x ) \geq 0$ holds on the interval $(1, + \infty)$, and $f ( x ) = 0$ has a unique solution in $(1, + \infty)$.

The monotone increasing interval of the function $f(x) = \ln(x^2 - 2x - 9)$ is
A. $(-\infty, -2)$
B. $(-\infty, 1)$
C. $(1, +\infty)$
D. $(4, +\infty)$

The tangent line to the curve $y = a \mathrm { e } ^ { x } + x \ln x$ at the point $( 1 , a \mathrm { e } )$ has equation $y = 2 x + b$ . Then
A. $a = \mathrm { e } , b = - 1$
B. $a = \mathrm { e } , b = 1$
C. $a = \mathrm { e } ^ { - 1 } , b = 1$
D. $a = \mathrm { e } ^ { - 1 } , b = - 1$

6. The tangent line to the curve $y = a \mathrm { e } ^ { x } + x \ln x$ at the point $( 1 , a \mathrm { e } )$ has equation $y = 2 x + b$ . Then
A. $a = \mathrm { e } , \quad b = - 1$
B. $a = \mathrm { e } , b = 1$
C. $a = \mathrm { e } ^ { - 1 } , b = 1$
D. $a = \mathrm { e } ^ { - 1 } , b = - 1$

7. The tangent line to the curve $y = a \mathrm { e } ^ { x } + x \ln x$ at the point $( 1 , a e )$ has equation $y = 2 x + b$ . Then
A. $a = \mathrm { e } , b = - 1$
B. $a = \mathrm { e } , b = 1$
C. $a = \mathrm { e } ^ { - 1 } , b = 1$
D. $a = \mathrm { e } ^ { - 1 } , b = - 1$

9. If the line $y = k x - 2$ is tangent to the curve $y = 1 + 3 \ln x$, then $k =$
A. $2$ B. $\frac { 1 } { 3 }$
C. $3$ D. $\frac { 1 } { 2 }$

Given the function $f ( x ) = \frac { \mathrm { e } ^ { x } } { x + a }$. If $f ^ { \prime } ( 1 ) = \frac { \mathrm { e } } { 4 }$, then $a =$ $\_\_\_\_$ .

Given $f(x) = ax - \frac{\sin x}{\cos^{2} x} , \quad x \in \left(0 , \frac{\pi}{2}\right)$ ,
(1) When $a = 8$ , discuss the monotonicity of $f(x)$ ;
(2) If $f(x) < \sin 2x$ , find the range of values for $a$ .

(17 points) Given function $f ( x ) = \ln \frac { x } { 2 - x } + a x + b ( x - 1 ) ^ { 3 }$ .
(1) If $b = 0$ and $f ^ { \prime } ( x ) \geqslant 0$ , find the minimum value of $a$ ;
(2) Prove that the curve $y = f ( x )$ is centrally symmetric;
(3) If $f ( x ) > - 2$ if and only if $1 < x < 2$ , find the range of $b$ .

Given $f ( x ) = x + k \ln ( 1 + x )$, the tangent line to the curve at point $( t , f ( t ) ) ( t > 0 )$ is $l$.
(1) If the slope of tangent line $l$ is $k = - 1$, find the monotonic intervals of $f ( x )$;
(2) Prove that tangent line $l$ does not pass through $( 0,0 )$;
(3) Given $A ( t , f ( t ) ) , C ( 0 , f ( t ) ) , O ( 0,0 )$, where $t > 0$, and tangent line $l$ intersects the $y$-axis at point $B$. When $2 S _ { \triangle ACO } = 15 S _ { \triangle ABC }$, how many points $A$ satisfy the condition? (Reference data: $1.09 < \ln 3 < 1.10, 1.60 < \ln 5 < 1.61, 1.94 < \ln 7 < 1.95$.)

If the line $y = 2x + 5$ is tangent to the curve $y = \mathrm{e}^x + x + a$, then $a = $ $\_\_\_\_$ .

Given is the function $f : x \mapsto \ln ( x - 3 )$ with maximal domain $D$ and derivative function $f ^ { \prime }$. (1a) [2 marks] State $D$ and the zero of $f$. (1b) [3 marks] Determine the point $x \in D$ for which $f ^ { \prime } ( x ) = 2$ holds.
Given is the function $g : x \mapsto \frac { 1 } { x ^ { 2 } } - 1$ defined in $\mathbb { R } \backslash \{ 0 \}$. (2a) [2 marks] State an equation of the horizontal asymptote of the graph of $g$ and the range of $g$.
(2b) [3 marks] Calculate the value of the integral $\int _ { \frac { 1 } { 2 } } ^ { 2 } g ( x ) \mathrm { dx }$.
A polynomial function $f$ defined in $\mathbb { R }$, which is not linear, with first derivative function $f ^ { \prime }$ and second derivative function $f ^ { \prime \prime }$ has the following properties:

• $f$ has a zero at $x _ { 1 }$.
• It holds that $f ^ { \prime } \left( x _ { 2 } \right) = 0$ and $f ^ { \prime \prime } \left( x _ { 2 } \right) \neq 0$.
• $f ^ { \prime }$ has a local minimum at the point $x _ { 3 }$.

Figure 1 shows the positions of $x _ { 1 } , x _ { 2 }$ and $x _ { 3 }$. [Figure]
(3a) [2 marks] Justify that the degree of $f$ is at least 3.
(3b) [3 marks] Sketch a possible graph of $f$ in Figure 1.
Figure 2 shows the graph of the function $g$ defined in $\mathbb { R }$, whose only extreme points are $( - 1 \mid 1 )$ and $( 0 \mid 0 )$, as well as the point $P$.
[Figure]
Fig. 2
(4a) [2 marks] State the coordinates of the minimum point of the graph of the function $h$ defined in $\mathbb { R }$ with $h ( x ) = - g ( x - 3 )$.
Subtask Part A 4b $( 3 \mathrm { marks } )$ The graph of an antiderivative of $g$ passes through $P$. Sketch this graph in Figure 2.
Given is the function $f : x \mapsto 2 e ^ { - \frac { 1 } { 8 } x ^ { 2 } }$ defined in $\mathbb { R }$. Figure 3 shows the graph $G _ { f }$ of $f$, which has the x-axis as a horizontal asymptote.
[Figure]
Fig. 3

Let $n \in \mathbb{N}$.
a) Show that the function $F_n$ is of class $C^\infty$ on $\mathbb{R}$.
b) For $x \in ]-1,1[$, give a simple expression for $F_n'(x)$. Justify the calculation carefully.

Let $n \in \mathbb{N}^*$.
a) Show that $\arccos(x) \sim \sqrt{2(1-x)}$ as $x \rightarrow 1$.
b) Deduce the calculation of $F_n'(1)$ and $F_n'(-1)$.

Throughout the rest of this problem, we set $T_0(x) = 1$. For $n \in \mathbb{N}^*$, we denote by $T_n$ the polynomial function satisfying $T_n(x) = 2^{1-n} F_n(x)$ for all $x \in \mathbb{R}$.
Show that, for all $n \in \mathbb{N}^*$ and all real $x$, the following relation holds: $$\left(1 - x^2\right) T_n''(x) - x T_n'(x) + n^2 T_n(x) = 0.$$

We consider the function $f : \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x) = \mathrm{e}^{-\frac{1}{2}x^{2}}$ for all $x \in \mathbb{R}$.
Show that there exists a real number $M > 0$ such that $f$ is an $M$-Lipschitz function.

For $x \in \mathbb{R}^{+}$, we define $$f(x) = \int_{0}^{\infty} \frac{1 - \cos t}{t^{2}} \mathrm{e}^{-xt} \mathrm{~d}t$$ Show that $f$ is defined and continuous on $[0, +\infty[$ and of class $C^{2}$ on $]0, +\infty[$.

For $x \in \mathbb{R}^{+}$, we define $$f(x) = \int_{0}^{\infty} \frac{1 - \cos t}{t^{2}} \mathrm{e}^{-xt} \mathrm{~d}t$$ Determine the limits of $f$ and $f^{\prime}$ at $+\infty$.

For $x \in \mathbb{R}^{+}$, we define $$f(x) = \int_{0}^{\infty} \frac{1 - \cos t}{t^{2}} \mathrm{e}^{-xt} \mathrm{~d}t$$ Express $f^{\prime\prime}$ on $]0, +\infty[$ using standard functions and deduce that $$\forall x > 0, \quad f^{\prime}(x) = \ln(x) - \frac{1}{2} \ln\left(x^{2} + 1\right)$$

Verify that $\varphi$ is of class $\mathscr{C}^{0}$ on $[0, +\infty[$ and $\mathscr{C}^{\infty}$ on $]0, +\infty[$. Give the limit of the derivative $\varphi'(t)$ of $\varphi$ as $t$ tends to 0 in $]0, +\infty[$.
Where $\varphi$ is defined by $$\varphi(t) = \begin{cases} 0 & \text{if } t = 0 \\ -t \ln(t) & \text{otherwise.} \end{cases}$$

We define the function $\theta : \mathbb { R } \rightarrow \mathbb { C }$ by $$\begin{cases} \theta ( x ) = 0 & \text { if } x \leqslant 0 \\ \theta ( x ) = \exp \left( - \frac { \ln ^ { 2 } x } { 4 \pi ^ { 2 } } + \mathrm { i } \frac { \ln x } { 2 \pi } \right) & \text { if } x > 0 \end{cases}$$
Deduce that $\lim _ { \substack { x \rightarrow 0 \\ x > 0 } } \left| \theta ^ { ( n ) } ( x ) \right| = 0$. One may perform the change of variable $y = - \ln x$.

We define the function $\theta : \mathbb { R } \rightarrow \mathbb { C }$ by $$\begin{cases} \theta ( x ) = 0 & \text { if } x \leqslant 0 \\ \theta ( x ) = \exp \left( - \frac { \ln ^ { 2 } x } { 4 \pi ^ { 2 } } + \mathrm { i } \frac { \ln x } { 2 \pi } \right) & \text { if } x > 0 \end{cases}$$
Demonstrate that $\theta$ is of class $C ^ { \infty }$ on $\mathbb { R }$.