Exercise 2 -- Common to all candidates
Consider the functions $f$ and $g$ defined for all real $x$ by: $$f(x) = \mathrm{e}^{x} \quad \text{and} \quad g(x) = 1 - \mathrm{e}^{-x}.$$ The representative curves of these functions in an orthogonal coordinate system of the plane, denoted respectively $\mathscr{C}_{f}$ and $\mathscr{C}_{g}$, are provided in the appendix.
Part A
These curves appear to admit two common tangent lines. Draw these tangent lines as accurately as possible on the figure in the appendix.
Part B
In this part, the existence of these common tangent lines is admitted. Let $\mathscr{D}$ denote one of them. This line is tangent to the curve $\mathscr{C}_{f}$ at point A with abscissa $a$ and tangent to the curve $\mathscr{C}_{g}$ at point B with abscissa $b$.

1. a. Express in terms of $a$ the slope of the tangent line to the curve $\mathscr{C}_{f}$ at point A. b. Express in terms of $b$ the slope of the tangent line to the curve $\mathscr{C}_{g}$ at point B. c. Deduce that $b = -a$.
2. Prove that the real number $a$ is a solution of the equation $$2(x - 1)\mathrm{e}^{x} + 1 = 0.$$

Part C
Consider the function $\varphi$ defined on $\mathbb{R}$ by $$\varphi(x) = 2(x - 1)\mathrm{e}^{x} + 1$$

1. a. Calculate the limits of the function $\varphi$ at $-\infty$ and $+\infty$. b. Calculate the derivative of the function $\varphi$, then study its sign. c. Draw the variation table of the function $\varphi$ on $\mathbb{R}$. Specify the value of $\varphi(0)$.
2. a. Prove that the equation $\varphi(x) = 0$ has exactly two solutions in $\mathbb{R}$. b. Let $\alpha$ denote the negative solution of the equation $\varphi(x) = 0$ and $\beta$ the positive solution of this equation. Using a calculator, give the values of $\alpha$ and $\beta$ rounded to the nearest hundredth.

Part D
In this part, we prove the existence of these common tangent lines, which was admitted in Part B. Let E be the point on the curve $\mathscr{C}_{f}$ with abscissa $\alpha$ and F the point on the curve $\mathscr{C}_{g}$ with abscissa $-\alpha$ ($\alpha$ is the real number defined in Part C).

1. Prove that the line $(EF)$ is tangent to the curve $\mathscr{C}_{f}$ at point E.
2. Prove that $(EF)$ is tangent to $\mathscr{C}_{g}$ at point F.

On the graph below, we have drawn, in an orthonormal coordinate system $(\mathrm{O}, \vec{\imath}, \vec{\jmath})$, a curve $\mathscr{C}$ and the line $(\mathrm{AB})$ where A and B are the points with coordinates $(0;1)$ and $(-1;3)$ respectively.
We denote by $f$ the function differentiable on $\mathbb{R}$ whose representative curve is $\mathscr{C}$. We further assume that there exists a real number $a$ such that for all real $x$, $$f(x) = x + 1 + ax\mathrm{e}^{-x^{2}}$$

1. a. Justify that the curve $\mathscr{C}$ passes through point A. b. Determine the slope of the line (AB). c. Prove that for all real $x$, $$f^{\prime}(x) = 1 - a\left(2x^{2} - 1\right)\mathrm{e}^{-x^{2}}$$ d. We assume that the line $(\mathrm{AB})$ is tangent to the curve $\mathscr{C}$ at point A. Determine the value of the real number $a$.
2. According to the previous question, for all real $x$, $$f(x) = x + 1 - 3x\mathrm{e}^{-x^{2}} \text{ and } f^{\prime}(x) = 1 + 3\left(2x^{2} - 1\right)\mathrm{e}^{-x^{2}}.$$ a. Prove that for all real $x$ in the interval $]-1;0]$, $f(x) > 0$. b. Prove that for all real $x$ less than or equal to $-1$, $f^{\prime}(x) > 0$. c. Prove that there exists a unique real number $c$ in the interval $\left[-\frac{3}{2};-1\right]$ such that $f(c) = 0$. Justify that $c < -\frac{3}{2} + 2 \cdot 10^{-2}$.
3. We denote by $\mathscr{A}$ the area, expressed in square units, of the region defined by: $$c \leqslant x \leqslant 0 \quad \text{and} \quad 0 \leqslant y \leqslant f(x)$$ a. Write $\mathscr{A}$ in the form of an integral. b. We admit that the integral $I = \int_{-\frac{3}{2}}^{0} f(x)\,\mathrm{d}x$ is an approximate value of $\mathscr{A}$ to within $10^{-3}$. Calculate the exact value of the integral $I$.

The profile of a slide is modelled by the curve $\mathcal { C }$ representing the function $f$ defined on the interval [1;8] by
$$f ( x ) = ( a x + b ) \mathrm { e } ^ { - x } \text { where } a \text { and } b \text { are two natural integers. }$$
The curve $\mathcal { C }$ is drawn in an orthonormal coordinate system with unit of one metre.

Part A Modelling

1. We want the tangent to the curve $\mathcal { C }$ at its point with abscissa 1 to be horizontal. Determine the value of the integer $b$.
2. We want the top of the slide to be located between 3.5 and 4 metres high. Determine the value of the integer $a$.

Part B An amenity for visitors

We assume in the following that the function $f$ introduced in Part A is defined for all real $x \in [ 1 ; 8 ]$ by
$$f ( x ) = 10 x \mathrm { e } ^ { - x }$$
The retaining wall of the slide will be painted by an artist on a single face. In the quote he proposes, he asks for a flat fee of 300 euros plus 50 euros per square metre painted.

1. Let $g$ be the function defined on [ $1 ; 8$ ] by
   $$g ( x ) = 10 ( - x - 1 ) \mathrm { e } ^ { - x }$$
   Determine the derivative of the function $g$.
2. What is the amount of the artist's quote?

Part C A constraint to verify

Safety considerations require limiting the maximum slope of the slide. Consider a point $M$ on the curve $\mathcal { C }$, with abscissa different from 1. We call $\alpha$ the acute angle formed by the tangent to $\mathcal { C }$ at $M$ and the horizontal axis. The constraints require that the angle $\alpha$ be less than 55 degrees.

1. We denote $f ^ { \prime }$ the derivative of the function $f$ on the interval $[ 1 ; 8 ]$. We admit that, for all $x$ in the interval $[ 1 ; 8 ] , f ^ { \prime } ( x ) = 10 ( 1 - x ) \mathrm { e } ^ { - x }$. Study the variations of the function $f ^ { \prime }$ on the interval [ $1 ; 8$ ].
2. Let $x$ be a real number in the interval ] 1; 8] and let $M$ be the point with abscissa $x$ on the curve $\mathcal { C }$. Justify that $\tan \alpha = \left| f ^ { \prime } ( x ) \right|$.
3. Is the slide compliant with the imposed constraints?

The curves $\mathscr { C } _ { f }$ and $\mathscr { C } _ { g }$ given in appendix 1 are the graphical representations, in an orthonormal coordinate system ( $\mathrm { O } ; \vec { \imath } , \vec { \jmath }$ ), of two functions $f$ and $g$ defined on $[ 0 ; + \infty [$. We consider the points $\mathrm { A } ( 0,5 ; 1 )$ and $\mathrm { B } ( 0 ; - 1 )$ in the coordinate system ( $\mathrm { O } ; \vec { \imath } , \vec { \jmath }$ ). We know that O belongs to $\mathscr { C } _ { f }$ and that the line (OA) is tangent to $\mathscr { C } _ { f }$ at point O.

1. We assume that the function $f$ is written in the form $f ( x ) = ( a x + b ) \mathrm { e } ^ { - x ^ { 2 } }$ where $a$ and $b$ are real numbers. Determine the exact values of the real numbers $a$ and $b$, detailing the approach. From now on, we consider that $\boldsymbol { f } ( \boldsymbol { x } ) = \mathbf { 2 } \boldsymbol { x } \mathrm { e } ^ { - \boldsymbol { x } ^ { \mathbf { 2 } } }$ for all $\boldsymbol { x }$ belonging to $[ \mathbf { 0 } ; + \infty [$
2. a. We will admit that, for all real $x$ strictly positive, $f ( x ) = \frac { 2 } { x } \times \frac { x ^ { 2 } } { \mathrm { e } ^ { x ^ { 2 } } }$.
   Calculate $\lim _ { x \rightarrow + \infty } f ( x )$. b. Draw up, justifying it, the table of variations of the function $f$ on $[ 0 ; + \infty [$.
3. The function $g$ whose representative curve $\mathscr { C } _ { g }$ passes through the point $\mathrm { B } ( 0 ; - 1 )$ is a primitive of the function $f$ on $[ 0 ; + \infty [$. a. Determine the expression of $g ( x )$. b. Let $m$ be a strictly positive real number.
   Calculate $I _ { m } = \int _ { 0 } ^ { m } f ( t ) \mathrm { d } t$ as a function of $m$. c. Determine $\lim _ { m \rightarrow + \infty } I _ { m }$.
4. a. Justify that $f$ is a probability density function on $[ 0 ; + \infty [$. b. Let $X$ be a continuous random variable that admits the function $f$ as its probability density function. Justify that, for all real $x$ in $[ 0 ; + \infty [$, $P ( X \leqslant x ) = g ( x ) + 1$. c. Deduce the exact value of the real number $\alpha$ such that $P ( X \leqslant \alpha ) = 0,5$. d. Without using an approximate value of $\alpha$, construct in the coordinate system of appendix 1 the point with coordinates ( $\alpha ; 0$ ) leaving the construction lines visible. Then shade the region of the plane corresponding to $P ( X \leqslant \alpha )$.

A manufacturer must create a solid wooden gate made to measure for a homeowner. The opening of the enclosure wall (not yet built) cannot exceed 4 meters wide. The gate consists of two panels of width $a$ such that $0 < a \leqslant 2$.
In the chosen model, the closed gate has the shape illustrated in the figure. The sides $[\mathrm{AD}]$ and $[\mathrm{BC}]$ are perpendicular to the threshold [CD] of the gate. Between points A and B, the top of the panels has the shape of a portion of curve. This portion of curve is part of the graph of the function $f$ defined on $[-2 ; 2]$ by:
$$f ( x ) = - \frac { b } { 8 } \left( \mathrm { e } ^ { \frac { x } { b } } + \mathrm { e } ^ { - \frac { x } { b } } \right) + \frac { 9 } { 4 } \quad \text { where } b > 0 .$$
The coordinate system is chosen so that the points $\mathrm{A}, \mathrm{B}, \mathrm{C}$ and D have coordinates respectively $(-a ; f(-a))$, $(a ; f(a))$, $(a ; 0)$ and $(-a ; 0)$ and we denote S the vertex of the curve of $f$.
Part A

1. Show that, for all real $x$ belonging to the interval $[-2 ; 2], f(-x) = f(x)$. What can we deduce about the graph of the function $f$?
2. Let $f^{\prime}$ denote the derivative function of $f$. Show that, for all real $x$ in the interval $[-2 ; 2]$: $$f^{\prime}(x) = -\frac{1}{8}\left(\mathrm{e}^{\frac{x}{b}} - \mathrm{e}^{-\frac{x}{b}}\right)$$
3. Draw up the table of variations of the function $f$ on the interval $[-2 ; 2]$ and deduce the coordinates of point S as a function of $b$.

Part B
The height of the wall is $1.5\mathrm{~m}$. We want point S to be 2 m from the ground. We then seek the values of $a$ and $b$.

1. Justify that $b = 1$.
2. Show that the equation $f(x) = 1.5$ has a unique solution on the interval $[0 ; 2]$ and deduce an approximate value of $a$ to the nearest hundredth.
3. In this question, we choose $a = 1.8$ and $b = 1$. The customer decides to automate his gate if the mass of a panel exceeds 60 kg. The density of the wooden planks used to manufacture the panels is equal to $20\mathrm{~kg\cdot m^{-2}}$. What does the customer decide?

Part C
We keep the values $a = 1.8$ and $b = 1$. To cut the panels, the manufacturer pre-cuts planks. He has a choice between two forms of pre-cut planks: either a rectangle OCES, or a trapezoid OCHG. In the second method, the line (GH) is tangent to the graph of the function $f$ at point F with abscissa 1.
Form 1 is the simplest, but visually form 2 seems more economical. Evaluate the savings achieved in terms of wood surface area by choosing form 2 rather than form 1. We recall the formula giving the area of a trapezoid. By denoting $b$ and $B$ respectively the lengths of the small base and the large base of the trapezoid (parallel sides) and $h$ the height of the trapezoid: $$\text{Area} = \frac{b + B}{2} \times h$$

Exercise 1 (6 points) -- Part A
Let $a$ and $b$ be real numbers. We consider a function $f$ defined on $[ 0 ; + \infty [$ by $$f ( x ) = \frac { a } { 1 + \mathrm { e } ^ { - b x } }$$ The curve $\mathscr { C } _ { f }$ representing the function $f$ in an orthogonal coordinate system is given. The curve $\mathscr { C } _ { f }$ passes through the point $\mathrm { A } ( 0 ; 0.5 )$. The tangent line to the curve $\mathscr { C } _ { f }$ at point A passes through the point B(10; 1).

1. Justify that $a = 1$.

We then obtain, for all real $x \geqslant 0$, $$f ( x ) = \frac { 1 } { 1 + \mathrm { e } ^ { - b x } }$$

1. It is admitted that the function $f$ is differentiable on $\left[ 0 ; + \infty \left[ \right. \right.$ and we denote $f ^ { \prime }$ its derivative function. Verify that, for all real $x \geqslant 0$ $$f ^ { \prime } ( x ) = \frac { b \mathrm { e } ^ { - b x } } { \left( 1 + \mathrm { e } ^ { - b x } \right) ^ { 2 } }$$
2. Using the data from the problem statement, determine $b$.

In the plane with a coordinate system, we consider the curve $\mathscr{C}_{f}$ representative of a function $f$, twice differentiable on the interval $]0 ; +\infty[$. The curve $\mathscr{C}_{f}$ admits a horizontal tangent line $T$ at point A(1;4).

1. Specify the values $f(1)$ and $f^{\prime}(1)$.

We admit that the function $f$ is defined for every real number $x$ in the interval $]0 ; +\infty[$ by:
$$f(x) = \frac{a + b \ln x}{x} \text{ where } a \text{ and } b \text{ are two real numbers.}$$

1. Prove that, for every strictly positive real number $x$, we have: $$f^{\prime}(x) = \frac{b - a - b \ln x}{x^{2}}$$
2. Deduce the values of the real numbers $a$ and $b$.

In the rest of the exercise, we admit that the function $f$ is defined for every real number $x$ in the interval $]0; +\infty[$ by:
$$f(x) = \frac{4 + 4 \ln x}{x}$$

1. Determine the limits of $f$ at 0 and at $+\infty$.
2. Determine the variation table of $f$ on the interval $]0 ; +\infty[$.
3. Prove that, for every strictly positive real number $x$, we have: $$f^{\prime\prime}(x) = \frac{-4 + 8 \ln x}{x^{3}}$$
4. Show that the curve $\mathscr{C}_{f}$ has a unique inflection point B whose coordinates you will specify.

The graph opposite gives the graphical representation $\mathscr { C } _ { f }$ in an orthogonal coordinate system of a function $f$ defined and differentiable on $\mathbb { R }$. We denote $f ^ { \prime }$ the derivative function of $f$. We are given points A with coordinates (0; 5) and B with coordinates (1; 20). Point C is the point on the curve $\mathscr { C } _ { f }$ with abscissa $-2.5$. The line (AB) is tangent to the curve $\mathscr { C } _ { f }$ at point A. Questions 1 to 3 relate to this same function $f$.
We admit that the function $f$ represented above is defined on $\mathbb { R }$ by $f ( x ) = ( a x + b ) \mathrm { e } ^ { x }$, where $a$ and $b$ are two real numbers and that its curve intersects the x-axis at the point with coordinates ($-0.5$; 0). We can assert that: a. $a = 10$ and $b = 5$ b. $a = 2.5$ and $b = -0.5$ c. $a = -1.5$ and $b = 5$ d. $a = 0$ and $b = 5$

Let $f$ be a function defined and differentiable on $\mathbb { R }$. We consider the points $\mathrm { A } ( 1 ; 3 )$ and $\mathrm { B } ( 3 ; 5 )$. We give below $\mathscr { C } _ { f }$ the representative curve of $f$ in an orthogonal coordinate system of the plane, as well as the tangent line (AB) to the curve $\mathscr { C } _ { f }$ at point A.
The three parts of the exercise can be worked on independently.

Part A

1. Determine graphically the values of $f ( 1 )$ and $f ^ { \prime } ( 1 )$.
2. The function $f$ is defined by the expression $f ( x ) = \ln \left( a x ^ { 2 } + 1 \right) + b$, where $a$ and $b$ are positive real numbers. a. Determine the expression of $f ^ { \prime } ( x )$. b. Determine the values of $a$ and $b$ using the previous results.

Part B

It is admitted that the function $f$ is defined on $\mathbb { R }$ by $$f ( x ) = \ln \left( x ^ { 2 } + 1 \right) + 3 - \ln ( 2 )$$

1. Show that $f$ is an even function.
2. Determine the limits of $f$ at $+ \infty$ and at $- \infty$.
3. Determine the expression of $f ^ { \prime } ( x )$. Study the direction of variation of the function $f$ on $\mathbb { R }$. Draw up the table of variations of $f$ showing the exact value of the minimum as well as the limits of $f$ at $- \infty$ and $+ \infty$.
4. Using the table of variations of $f$, give the values of the real number $k$ for which the equation $f ( x ) = k$ admits two solutions.
5. Solve the equation $f ( x ) = 3 + \ln 2$.

Part C

We recall that the function $f$ is defined on $\mathbb{R}$ by $f ( x ) = \ln \left( x ^ { 2 } + 1 \right) + 3 - \ln ( 2 )$.

1. Conjecture, by graphical reading, the abscissas of any inflection points of the curve $\mathscr { C } _ { f }$.
2. Show that, for any real number $x$, we have: $f ^ { \prime \prime } ( x ) = \frac { 2 \left( 1 - x ^ { 2 } \right) } { \left( x ^ { 2 } + 1 \right) ^ { 2 } }$.
3. Deduce the largest interval on which the function $f$ is convex.

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 $b$.

16. The tangent line to the curve $y = x + \ln x$ at the point $( 1,1 )$ is tangent to the curve $y = a x ^ { 2 } + ( a + 2 ) x + 1$. Then $a = $ $\_\_\_\_$ .

III. Solution Questions

17 (This question is worth 12 points). In $\triangle A B C$, $D$ is a point on $BC$, $AD$ bisects $\angle B A C$, and $B D = 2 D C$.
(1) Find $\frac { \sin \angle B } { \sin \angle C }$; (II) If $\angle B A C = 60 ^ { \circ }$, find $\angle B$.

The slope of the tangent line to the curve $y = ( ax + 1 ) e ^ { x }$ at the point $( 0,1 )$ is $- 2$. Then $a = $ $\_\_\_\_$.

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

If the function $$f ( x ) = \begin{cases} \frac { x ^ { 2 } - 2 x + A } { \sin x } & \text { if } x \neq 0 \\ B & \text { if } x = 0 \end{cases}$$ is continuous at $x = 0$, then
(A) $A = 0 , B = 0$
(B) $A = 0 , B = - 2$
(C) $A = 1 , B = 1$
(D) $A = 1 , B = 0$

If the function $$f ( x ) = \begin{cases} \frac { x ^ { 2 } - 2 x + A } { \sin x } & \text { if } x \neq 0 \\ B & \text { if } x = 0 \end{cases}$$ is continuous at $x = 0$, then
(A) $A = 0 , B = 0$
(B) $A = 0 , B = - 2$
(C) $A = 1 , B = 1$
(D) $A = 1 , B = 0$

If the function $$f ( x ) = \begin{cases} \frac { x ^ { 2 } - 2 x + A } { \sin x } & \text { if } x \neq 0 \\ B & \text { if } x = 0 \end{cases}$$ is continuous at $x = 0$, then
(A) $A = 0 , B = 0$
(B) $A = 0 , B = - 2$
(C) $A = 1 , B = 1$
(D) $A = 1 , B = 0$

Let $g ( x ) = \frac { ( x - 1 ) ^ { n } } { \log \cos ^ { m } ( x - 1 ) } ; 0 < x < 2 , m$ and $n$ are integers, $m \neq 0 , n > 0$, and let $p$ be the left hand derivative of $| x - 1 |$ at $x = 1$.
If $\lim _ { x \rightarrow 1 + } g ( x ) = p$, then
(A) $n = 1 , m = 1$
(B) $n = 1 , m = - 1$
(C) $n = 2 , m = 2$
(D) $n > 2 , m = n$

If a function $f(x)$ defined by $$f(x) = \begin{cases} ae^{x} + be^{-x}, & -1 \leq x < 1 \\ cx^{2}, & 1 \leq x \leq 3 \\ ax^{2} + 2cx, & 3 < x \leq 4 \end{cases}$$ be continuous for some $a, b, c \in R$ and $f'(0) + f'(2) = e$, then the value of $a$ is
(1) $\frac{1}{e^{2} - 3e + 13}$
(2) $\frac{e}{e^{2} - 3e - 13}$
(3) $\frac{e}{e^{2} + 3e + 13}$
(4) $\frac{e}{e^{2} - 3e + 13}$

If the function $f ( x ) = \left\{ \begin{array} { c c } k _ { 1 } ( x - \pi ) ^ { 2 } - 1 , & x \leq \pi \\ k _ { 2 } \cos x , & x > \pi \end{array} \right.$ is twice differentiable, then the ordered pair $\left( k _ { 1 } , k _ { 2 } \right)$ is equal to:
(1) $\left( \frac { 1 } { 2 } , 1 \right)$
(2) $( 1,0 )$
(3) $\left( \frac { 1 } { 2 } , - 1 \right)$
(4) $( 1,1 )$

If the function $f ( x ) = \left\{ \begin{array} { l l } \frac { \log _ { e } \left( 1 - x + x ^ { 2 } \right) + \log _ { e } \left( 1 + x + x ^ { 2 } \right) } { \sec x - \cos x } , & x \in \left( \frac { - \pi } { 2 } , \frac { \pi } { 2 } \right) - \{ 0 \} \\ k & , x = 0 \end{array} \right.$ is continuous at $x = 0$, then $k$ is equal to:
(1) 1
(2) $- 1$
(3) $e$
(4) 0

Let $f$ and $g$ be twice differentiable functions on $R$ such that $f ^ { \prime \prime } ( x ) = g ^ { \prime \prime } ( x ) + 6 x$ $f ^ { \prime } ( 1 ) = 4 g ^ { \prime } ( 1 ) - 3 = 9$ $f ( 2 ) = 3 g ( 2 ) = 12$ Then which of the following is NOT true ? (1) $g ( - 2 ) - f ( - 2 ) = 20$ (2) If $- 1 < x < 2$, then $| f ( x ) - g ( x ) | < 8$ (3) $\left| f ^ { \prime } ( x ) - g ^ { \prime } ( x ) \right| < 6 \Rightarrow - 1 < x < 1$ (4) There exists $x _ { 0 } \in \left( 1 , \frac { 3 } { 2 } \right)$ such that $f \left( x _ { 0 } \right) = g \left( x _ { 0 } \right)$

Let $a$ be a constant. Assume that the function
$$f(x) = 2\sin^3 x + a\sin 2x + \frac{9}{2}\cos 2x - 9\cos x - 2ax + 6$$
takes a local extremum at $x = \frac{\pi}{3}$. We consider about the maximum and minimum values of $f(x)$ over the interval $0 \leqq x \leqq \frac{\pi}{2}$.
(1) Since $f(x)$ takes a local extremum at $x = \frac{\pi}{3}$, it follows that $a = \frac{\mathbf{A}}{\mathbf{B}}$.
Hence the derivative $f'(x)$ of $f(x)$ can be expressed as
$$f'(x) = \mathbf{C}\cos x(\mathbf{D}\cos x - 1)(\sin x - \mathbf{E}).$$
(2) It can be seen from the result of (1) that $f(x)$ over $0 \leqq x \leqq \frac{\pi}{2}$ takes the maximum value at $x = \mathbf{F}$ and the minimum value at $x = \mathbf{G}$, where $\mathbf{F}$ and $\mathbf{G}$ are the appropriate expressions from among (0) $\sim$ (4) below. (0) $0$
(1) $\frac{\pi}{6}$
(2) $\frac{\pi}{4}$
(3) $\frac{\pi}{3}$
(4) $\frac{\pi}{2}$