Let $f$ and $g$ be the functions defined on $\mathbb{R}$ by $$f(x) = \mathrm{e}^{x} \quad \text{and} \quad g(x) = 2\mathrm{e}^{\frac{x}{2}} - 1.$$ We denote $\mathcal{C}_f$ and $\mathcal{C}_g$ the representative curves of functions $f$ and $g$ in an orthogonal coordinate system.

1. Prove that the curves $\mathcal{C}_f$ and $\mathcal{C}_g$ have a common point with abscissa 0 and that at this point, they have the same tangent line $\Delta$ whose equation we will determine.
2. Study of the relative position of curve $\mathcal{C}_g$ and line $\Delta$
   Let $h$ be the function defined on $\mathbb{R}$ by $h(x) = 2\mathrm{e}^{\frac{x}{2}} - x - 2$. a. Determine the limit of function $h$ at $-\infty$. b. Justify that, for every real $x \neq 0, h(x) = x\left(\frac{\mathrm{e}^{\frac{x}{2}}}{\frac{x}{2}} - 1 - \frac{2}{x}\right)$. Deduce the limit of function $h$ at $+\infty$. c. We denote $h'$ the derivative function of function $h$ on $\mathbb{R}$. For every real $x$, calculate $h'(x)$ and study the sign of $h'(x)$ according to the values of $x$. d. Draw the variation table of function $h$ on $\mathbb{R}$. e. Deduce that, for every real $x, 2\mathrm{e}^{\frac{x}{2}} - 1 \geqslant x + 1$. f. What can we deduce about the relative position of curve $\mathcal{C}_g$ and line $\Delta$?
3. Study of the relative position of curves $\mathcal{C}_f$ and $\mathcal{C}_g$. a. For every real $x$, expand the expression $\left(\mathrm{e}^{\frac{x}{2}} - 1\right)^2$. b. Determine the relative position of curves $\mathcal{C}_f$ and $\mathcal{C}_g$.

Part B

In this part, $k$ denotes a strictly positive real number. We consider the function $f$ defined on $\mathbb { R }$ by $$f ( x ) = ( x - 1 ) \mathrm { e } ^ { - k x } + 1 .$$ We admit that the function $f$ is differentiable on $\mathbb { R }$ and we denote $f ^ { \prime }$ its derivative function. In the plane with an orthonormal coordinate system ( $\mathrm { O } ; \mathrm { I } , \mathrm { J }$ ), we denote $\mathscr { C } _ { f }$ the representative curve of the function $f$. The tangent line $T$ to the curve $\mathscr { C } _ { f }$ at point A with abscissa 1 intersects the ordinate axis at a point denoted B.

1. a. Prove that for all real $x$, $$f ^ { \prime } ( x ) = \mathrm { e } ^ { - k x } ( - k x + k + 1 )$$ b. Prove that the ordinate of point B is equal to $g ( k )$ where $g$ is the function defined in Part A, with $g ( x ) = 1 - \mathrm{e}^{-x}$.
2. Using Part A, prove that point B belongs to the segment [OJ].

We consider a function $f$ defined on the interval $] 0 ; + \infty [$. We admit that it is twice differentiable on the interval $] 0 ; + \infty[$. We denote by $f ^ { \prime }$ its derivative function and $f ^ { \prime \prime }$ its second derivative function.
In an orthogonal coordinate system, we have drawn:

• the representative curve of $f$, denoted $\mathscr { C } _ { f }$ on the interval $] 0$; 3 ];
• the line $T _ { \mathrm { A } }$, tangent to $\mathscr { C } _ { f }$ at point $\mathrm { A } ( 1 ; 2 )$;
• the line $T _ { \mathrm { B } }$ tangent to $\mathscr { C } _ { f }$ at point $\mathrm { B } ( \mathrm { e } ; \mathrm { e } )$.

We further specify that the tangent $T _ { \mathrm { A } }$ passes through point $\mathrm { C } ( 3 ; 0 )$.
Part A: Graphical readings
Answer the following questions by justifying them using the graph.

1. Determine the derivative number $f ^ { \prime } ( 1 )$.
2. How many solutions does the equation $f ^ { \prime } ( x ) = 0$ have in the interval $] 0$; 3 ]?
3. What is the sign of $f ^ { \prime \prime } ( 0{,}2 )$ ?

Part B: Study of the function $f$
We admit in this part that the function $f$ is defined on the interval $] 0 ; + \infty [$ by $$f ( x ) = x \left[ 2 ( \ln x ) ^ { 2 } - 3 \ln x + 2 \right]$$ where ln denotes the natural logarithm function.

1. Solve in $\mathbb { R }$ the equation $2 X ^ { 2 } - 3 X + 2 = 0$. Deduce that $\mathscr { C } _ { f }$ does not intersect the x-axis.
2. Determine, by justifying, the limit of $f$ as $+ \infty$. We will admit that the limit of $f$ at 0 is equal to 0.
3. We admit that for all $x$ belonging to $] 0 ; + \infty [$, $f ^ { \prime } ( x ) = 2 ( \ln x ) ^ { 2 } + \ln x - 1$. a. Show that for all $x$ belonging to $] 0 ; + \infty [$, $f ^ { \prime \prime } ( x ) = \frac { 1 } { x } ( 4 \ln x + 1 )$. b. Study the convexity of the function $f$ on the interval $] 0 ; + \infty [$ and specify the exact value of the abscissa of the inflection point. c. Show that the curve $\mathscr { C } _ { f }$ is above the tangent $T _ { \mathrm { B } }$ on the interval $] 1 ; + \infty [$.

Part C: Area calculation

1. Justify that the tangent $T _ { \mathrm { B } }$ has the reduced equation $y = 2 x - \mathrm { e }$.
2. Using integration by parts, show that $$\int _ { 1 } ^ { \mathrm { e } } x \ln x \mathrm {~d} x = \frac { \mathrm { e } ^ { 2 } + 1 } { 4 }$$
3. We denote by $\mathscr { A }$ the area of the shaded region in the figure, bounded by the curve $\mathscr { C } _ { f }$, the tangent $T _ { \mathrm { B } }$, and the lines with equations $x = 1$ and $x = \mathrm { e }$. We admit that $\int _ { 1 } ^ { \mathrm { e } } x ( \ln x ) ^ { 2 } \mathrm {~d} x = \frac { \mathrm { e } ^ { 2 } - 1 } { 4 }$. Deduce the exact value of $\mathscr { A }$ in square units.

Let P be a point on the curve $y = 2 e ^ { - x }$ at $\mathrm { P } \left( t , 2 e ^ { - t } \right)$ $(t > 0)$. Let A be the foot of the perpendicular from P to the $y$-axis, and let B be the point where the tangent line at P intersects the $y$-axis. What is the value of $t$ that maximizes the area of triangle APB? [4 points]
(1) 1
(2) $\frac { e } { 2 }$
(3) $\sqrt { 2 }$
(4) 2
(5) $e$

For a real number $t$, let the equation of the tangent line to the curve $y = e ^ { x }$ at the point $\left( t , e ^ { t } \right)$ be $y = f ( x )$. Let $g ( t )$ be the minimum value of the real number $k$ such that the function $y = | f ( x ) + k - \ln x |$ is differentiable on the entire set of positive real numbers. For two real numbers $a , b ( a < b )$, let $\int _ { a } ^ { b } g ( t ) d t = m$. Which of the following statements in the given options are correct? [4 points]
$\langle$Options$\rangle$
ㄱ. There exist two real numbers $a , b ( a < b )$ such that $m < 0$. ㄴ. If $g ( c ) = 0$ for a real number $c$, then $g ( - c ) = 0$. ㄷ. If $m$ is minimized when $a = \alpha , b = \beta ( \alpha < \beta )$, then
$$\frac { 1 + g ^ { \prime } ( \beta ) } { 1 + g ^ { \prime } ( \alpha ) } < - e ^ { 2 }$$
(1) ㄱ
(2) ㄴ
(3) ㄱ, ㄴ
(4) ㄱ, ㄷ
(5) ㄱ, ㄴ, ㄷ

Given functions $f ( x ) = x ^ { 3 } - x , g ( x ) = x ^ { 2 } + a$. The tangent line to the curve $y = f ( x )$ at the point $\left( x _ { 1 } , f \left( x _ { 1 } \right) \right)$ is also tangent to the curve $y = g ( x )$ at some point.
(1) If $x _ { 1 } = - 1$ , find $a$ ;
(2) If $x_1 \neq 0$, prove that $a > \frac{1}{4}$ .

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: $1.09 < \ln 3 < 1.10, 1.60 < \ln 5 < 1.61, 1.94 < \ln 7 < 1.95$.)

Given function $f ( x ) = \mathrm { e } ^ { x } - a x - a ^ { 3 }$.
(1) When $a = 1$, find the equation of the tangent line to the curve $y = f ( x )$ at the point $( 1 , f ( 1 ) )$;
(2) If $f ( x )$ has a local minimum value that is negative, find the range of values for $a$.

Let $S$ be the set of all values of $x$ for which the tangent to the curve $y = f ( x ) = x ^ { 3 } - x ^ { 2 } - 2 x$ at ( $x , y$ ) is parallel to the line segment joining the points $( 1 , f ( 1 ) )$ and $( - 1 , f ( - 1 ) )$, then $S$ is equal to
(1) $\left\{ - \frac { 1 } { 3 } , - 1 \right\}$
(2) $\left\{ - \frac { 1 } { 3 } , 1 \right\}$
(3) $\left\{ \frac { 1 } { 3 } , 1 \right\}$
(4) $\left\{ \frac { 1 } { 3 } , - 1 \right\}$

If the normal to the curve $y ( x ) = \int _ { 0 } ^ { x } \left( 2 t ^ { 2 } - 15 t + 10 \right) d t$ at a point $( a , b )$ is parallel to the line $x + 3 y = - 5 , a > 1$, then the value of $| a + 6 b |$ is equal to $\_\_\_\_$.

Q1 Let $a > 0$. Consider two curves
$$C_1: y = e^{6x}$$ $$C_2: y = ax^2.$$
We are to find the condition on $a$ such that there exist two straight lines, each of which is tangent to both $C_1$ and $C_2$.
The equation of the tangent to $C_1$ at a point $(t, e^{6t})$ is
$$y = \mathbf{A}e^{6t}x - e^{6t}(\mathbf{B}t - \mathbf{C})$$
This is tangent also to $C_2$ under the condition that the quadratic equation
$$ax^2 = \mathbf{A}e^{6t}x - e^{6t}(\mathbf{B}t - \mathbf{C})$$
has just one solution. Hence, the equation
$$\mathbf{D}e^{12t} - ae^{6t}(\mathbf{E}t - \mathbf{F}) = 0$$
must hold for $a$ and $t$. From this equation we obtain
$$a = \frac{\mathbf{D}}{\mathbf{E}t - \mathbf{F}}e^{6t}$$
Let $f(t)$ denote the right side of this equation. The condition under which there exist two straight lines each of which is tangent to both $C_1$ and $C_2$, is that the straight line $s = a$ intersects the graph of $s = f(t)$ at two points.
Now, the derivative of $f(t)$ is
$$f'(t) = \frac{108e^{6t}(\mathbf{G}t - \mathbf{H})}{(\mathbf{E}t - \mathbf{F})^2}.$$
Hence the condition on $a$ that we are seeking is
$$a > \square e^{\square}.$$
Note that $\lim_{t \to \infty} \frac{e^t}{t} = \infty$.

In the coordinate plane, a ``Bézier curve'' determined by four points $A$, $B$, $C$, $D$ refers to a polynomial function of degree at most 3 whose graph passes through points $A$ and $D$, and whose tangent line at point $A$ passes through point $B$, and whose tangent line at point $D$ passes through point $C$. Let $y = f(x)$ be the ``Bézier curve'' determined by the four points $A(0, 0)$, $B(1, 4)$, $C(3, 2)$, $D(4, 0)$. Answer the following questions.
(1) Let the equation of the tangent line to the graph of $y = f(x)$ at point $D$ be $y = ax + b$, where $a$ and $b$ are real numbers. Find the values of $a$ and $b$. (2 points)
(2) Prove that the polynomial $f(x)$ is divisible by $x^{2} - 4x$. (2 points)
(3) Find $f(x)$. (4 points)
(4) Find the value of the definite integral $\int_{2}^{6} |8f(x)|\, dx$. (4 points)

Consider vectors $\vec{a}$ and $\vec{b}$ on the coordinate plane satisfying $|\vec{a}| + |\vec{b}| = 9$ and $|\vec{a} - \vec{b}| = 7$. Let $|\vec{a}| = x$, where $1 < x < 8$, and let the angle between $\vec{a}$ and $\vec{b}$ be $\theta$. Using the triangle formed by vectors $\vec{a}$, $\vec{b}$, and $\vec{a} - \vec{b}$, we can express $\cos\theta$ in terms of $x$ as $f(x) = \frac{c}{9x - x^2} + d$, where $c$ and $d$ are constants with $c > 0$, with domain $\{x \mid 1 < x < 8\}$. Using the linear approximation (first-order approximation) of $f(x)$, find the approximate value of $\cos\theta$ when $x = 4.96$. (Non-multiple choice question, 4 points)