The purpose of this exercise is to study the function $f$ defined on the interval $]0; +\infty[$ by:
$$f(x) = x \ln(x^2) - \frac{1}{x}$$
Part A: graphical readings
Below is the representative curve $(\mathscr{C}_f)$ of the function $f$, as well as the line $(T)$, tangent to the curve $(\mathscr{C}_f)$ at point A with coordinates $(1; -1)$. This tangent also passes through the point $B(0; -4)$.

1. Read graphically $f'(1)$ and give the reduced equation of the tangent $(T)$.
2. Give the intervals on which the function $f$ appears to be convex or concave. What does point A appear to represent for the curve $(\mathscr{C}_f)$?

Part B: analytical study

1. Determine, by justifying, the limit of $f$ at $+\infty$, then its limit at 0.
2. It is admitted that the function $f$ is twice differentiable on the interval $]0; +\infty[$. a. Determine $f'(x)$ for $x$ belonging to the interval $]0; +\infty[$. b. Show that for all $x$ belonging to the interval $]0; +\infty[$, $$f''(x) = \frac{2(x+1)(x-1)}{x^3}.$$
3. a. Study the convexity of the function $f$ on the interval $]0; +\infty[$. b. Study the variations of the function $f'$, then the sign of $f'(x)$ for $x$ belonging to the interval $]0; +\infty[$. Deduce the direction of variation of the function $f$ on the interval $]0; +\infty[$.
4. a. Show that the equation $f(x) = 0$ has a unique solution $\alpha$ on the interval $]0; +\infty[$. b. Give the value of $\alpha$ rounded to the nearest hundredth and show that $\alpha$ satisfies: $$\alpha^2 = \exp\left(\frac{1}{\alpha^2}\right)$$

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.

3. We consider the function $f$ defined on $] 0 ; + \infty \left[ \right.$ whose representative curve $C _ { f }$ is given in an orthonormal coordinate system in the figure (Fig. 1) on page 5. We specify that:

• $T$ is the tangent to $C _ { f }$ at point $A$ with abscissa 8;
• The $x$-axis is the horizontal tangent to $C _ { f }$ at the point with abscissa 1.

[Figure]
Fig. 1
Statement 3: According to the graph, the function $f$ is convex on its domain of definition.

Find the slope of a line L that satisfies both of the following properties: (i) L is tangent to the graph of $y = x ^ { 3 }$. (ii) L passes through the point $( 0, 200 )$.

12. Let $f ( x ) = \sqrt { x }$. We draw a tangent to the curve $y = f ( x )$ at the point on the curve whose $x$ coordinate is equal to 4 . Where does this tangent intersect the $X$-axis?
(a) $x = 4$
(b) $x = - 2$
(c) $x = - 4$
(d) $x = 2$

The following shows the graph of a continuous function $y = f ( x )$ and two distinct points $\mathrm { P } ( a , f ( a ) ) , \mathrm { Q } ( b , f ( b ) )$ on this graph.
When the function $F ( x )$ satisfies $F ^ { \prime } ( x ) = f ( x )$, which of the following statements in $\langle$Remarks$\rangle$ are always correct? [4 points]
$\langle$Remarks$\rangle$ ㄱ. The function $F ( x )$ is increasing on the interval $[ a , b ]$. ㄴ. $\frac { F ( b ) - F ( a ) } { b - a }$ is equal to the slope of line PQ. ㄷ. $\int _ { a } ^ { b } \{ f ( x ) - f ( b ) \} d x \leqq \frac { ( b - a ) \{ f ( a ) - f ( b ) \} } { 2 }$
(1) ㄱ
(2) ㄴ
(3) ㄱ, ㄷ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

For the quartic function $f ( x ) = x ^ { 4 } - 4 x ^ { 3 } + 6 x ^ { 2 } + 4$, the slope of the tangent line at the point $( a , b )$ on the graph is 4. Find the value of $a ^ { 2 } + b ^ { 2 }$. [3 points]

As shown in the figure, let $\mathrm { Q } _ { 1 }$ be the point where the tangent line to the circle $x ^ { 2 } + y ^ { 2 } = 1$ at point $\mathrm { P } _ { 1 }$ meets the $x$-axis. The area of triangle $\mathrm { P } _ { 1 } \mathrm { OQ } _ { 1 }$ is $\frac { 1 } { 4 }$. Let $\mathrm { P } _ { 2 }$ be the point obtained by rotating $\mathrm { P } _ { 1 }$ about the origin O by $\frac { \pi } { 4 }$, and let $\mathrm { Q } _ { 2 }$ be the point where the tangent line at $\mathrm { P } _ { 2 }$ meets the $x$-axis. What is the area of triangle $\mathrm { P } _ { 2 } \mathrm { OQ } _ { 2 }$? (Here, point $\mathrm { P } _ { 1 }$ is in the first quadrant.) [3 points]
(1) 1
(2) $\frac { 5 } { 4 }$
(3) $\frac { 3 } { 2 }$
(4) $\frac { 7 } { 4 }$
(5) 2

For a natural number $n$, when two points $\mathrm { P } _ { n - 1 } , \mathrm { P } _ { n }$ are on the graph of the function $y = x ^ { 2 }$, the point $\mathrm { P } _ { n + 1 }$ is determined according to the following rule.
(a) The coordinates of the two points $\mathrm { P } _ { 0 } , \mathrm { P } _ { 1 }$ are $(0,0)$ and $(1,1)$, respectively.
(b) The point $\mathrm { P } _ { n + 1 }$ is the intersection of the line passing through point $\mathrm { P } _ { n }$ and perpendicular to the line $\mathrm { P } _ { n - 1 } \mathrm { P } _ { n }$ and the graph of the function $y = x ^ { 2 }$. (Here, $\mathrm { P } _ { n }$ and $\mathrm { P } _ { n + 1 }$ are distinct points.) Let $l _ { n } = \overline { \mathrm { P } _ { n - 1 } \mathrm { P } _ { n } }$. What is the value of $\lim _ { n \rightarrow \infty } \frac { l _ { n } } { n }$? [3 points]
(1) $2 \sqrt { 3 }$
(2) $2 \sqrt { 2 }$
(3) 2
(4) $\sqrt { 3 }$
(5) $\sqrt { 2 }$

[Calculus] As shown in the figure, let Q be the point where the tangent line to the circle $x ^ { 2 } + y ^ { 2 } = 1$ at point P on the circle meets the $x$-axis. For point $\mathrm { A } ( - 1,0 )$ and the origin O, let $\angle \mathrm { PAO } = \theta$. Find the value of $\lim _ { \theta \rightarrow \frac { \pi } { 4 } - 0 } \frac { \overline { \mathrm { PQ } } - \overline { \mathrm { OQ } } } { \theta - \frac { \pi } { 4 } }$. (where point P is in the first quadrant) [3 points]
(1) 2
(2) $\sqrt { 3 }$
(3) $\frac { 3 } { 2 }$
(4) 1
(5) $\frac { \sqrt { 2 } } { 2 }$

On the coordinate plane, what is the slope of the tangent line to the curve $y ^ { 3 } = \ln \left( 5 - x ^ { 2 } \right) + x y + 4$ at the point $( 2,2 )$? [3 points]
(1) $- \frac { 3 } { 5 }$
(2) $- \frac { 1 } { 2 }$
(3) $- \frac { 2 } { 5 }$
(4) $- \frac { 3 } { 10 }$
(5) $- \frac { 1 } { 5 }$

The equation of the tangent line to the curve $y = - x ^ { 3 } + 4 x$ at the point $( 1,3 )$ is $y = a x + b$. Find the value of $10 a + b$. (where $a , b$ are constants) [4 points]

The equation of the tangent line to the graph of the cubic function $f(x) = x^3 + ax^2 + 9x + 3$ at the point $(1, f(1))$ is $y = 2x + b$. What is the value of $a + b$? (Here, $a, b$ are constants.) [4 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

The function $$f(x) = \begin{cases} x^3 + ax & (x < 1) \\ bx^2 + x + 1 & (x \geq 1) \end{cases}$$ is differentiable at $x = 1$. What is the value of $a + b$? (Here, $a, b$ are constants.) [4 points]
(1) 5
(2) 6
(3) 7
(4) 8
(5) 9

Two polynomial functions $f ( x ) , g ( x )$ satisfy the following conditions. (가) $g ( x ) = x ^ { 3 } f ( x ) - 7$ (나) $\lim _ { x \rightarrow 2 } \frac { f ( x ) - g ( x ) } { x - 2 } = 2$
When the equation of the tangent line to the curve $y = g ( x )$ at the point $( 2 , g ( 2 ) )$ is $y = a x + b$, find the value of $a ^ { 2 } + b ^ { 2 }$. (where $a , b$ are constants.) [4 points]

For the curve $y = x ^ { 3 } - a x + b$, the slope of the line perpendicular to the tangent line at the point $( 1,1 )$ is $- \frac { 1 } { 2 }$. For the two constants $a$ and $b$, find the value of $a + b$. [4 points]

The tangent line to the curve $y = x ^ { 3 } - 3 x ^ { 2 } + 2 x + 2$ at point $\mathrm { A } ( 0,2 )$ is perpendicular to a line passing through point A. What is the $x$-intercept of this line? [3 points]
(1) 4
(2) 6
(3) 8
(4) 10
(5) 12

For a cubic function $f ( x )$, the tangent line to the curve $y = f ( x )$ at the point $( 0,0 )$ and the tangent line to the curve $y = x f ( x )$ at the point $( 1,2 )$ coincide. What is the value of $f ^ { \prime } ( 2 )$? [4 points]
(1) $-18$
(2) $-17$
(3) $-16$
(4) $-15$
(5) $-14$

What is the $x$-intercept of the tangent line drawn from the point $(0, 4)$ to the curve $y = x ^ { 3 } - x + 2$? [3 points]
(1) $- \frac { 1 } { 2 }$
(2) $- 1$
(3) $- \frac { 3 } { 2 }$
(4) $- 2$
(5) $- \frac { 5 } { 2 }$

For a real number $a > \sqrt{2}$, define the function $f(x)$ as $$f(x) = -x^3 + ax^2 + 2x$$ The tangent line to the curve $y = f(x)$ at the point $\mathrm{O}(0,0)$ intersects the curve $y = f(x)$ at another point A. The tangent line to the curve $y = f(x)$ at point A intersects the $x$-axis at point B. If point A lies on the circle with diameter OB, find the value of $\overline{\mathrm{OA}} \times \overline{\mathrm{AB}}$. [4 points]

For a positive number $a$, let the function $f ( x )$ be $$f ( x ) = x ^ { 3 } + 3 a x ^ { 2 } - 9 a ^ { 2 } x + 4$$ When the line $y = 5$ is tangent to the curve $y = f ( x )$, what is the value of $f ( 2 )$? [4 points]
(1) 11
(2) 12
(3) 13
(4) 14
(5) 15

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 $\angle B$.

18. (This question is worth 13 points) Given the function $f ( x ) = \ln \frac { 1 + x } { 1 - x }$. (I) Find the equation of the tangent line to the curve $y = f ( x )$ at the point $( 0 , f ( 0 ) )$; (II) Prove: When $x \in ( 0,1 )$, $f ( x ) > 2 \left( x + \frac { x ^ { 3 } } { 3 } \right)$; (III) Let the real number $k$ be such that $f ( x ) > k \left( x + \frac { x ^ { 3 } } { 3 } \right)$ holds for all $x \in ( 0,1 )$. Find the maximum value of $k$.

(12 points)
Let $A$ and $B$ be two points on the curve $C: y = \frac{x^2}{4}$, and the sum of the $x$-coordinates of $A$ and $B$ is 4.
(1) Find the slope of line $AB$;
(2) Find the equation of line $AB$.

Let $f ( x ) = x ^ { 3 } + ( a - 1 ) x ^ { 2 } + a x$. If $f ( x )$ is an odd function, then the equation of the tangent line to $y = f ( x )$ at the point $( 0,0 )$ is
A. $y = - 2 x$
B. $y = - x$
C. $y = 2 x$
D. $y = x$