Let $f$ be the function defined on the interval $]0; +\infty[$ by $f(x) = \ln x$. For every strictly positive real number $a$, we define on $]0; +\infty[$ the function $g_a$ by $g_a(x) = ax^2$. We denote by $\mathscr{C}$ the curve representing the function $f$ and $\Gamma_a$ that of the function $g_a$ in a coordinate system of the plane. The purpose of the exercise is to study the intersection of the curves $\mathscr{C}$ and $\Gamma_a$ according to the values of the strictly positive real number $a$.
Part A
We have constructed in appendix 1 the curves $\mathscr{C}, \Gamma_{0,05}, \Gamma_{0,1}, \Gamma_{0,19}$ and $\Gamma_{0,4}$.

1. Name the different curves on the graph. No justification is required.
2. Use the graph to make a conjecture about the number of intersection points of $\mathscr{C}$ and $\Gamma_a$ according to the values (to be specified) of the real number $a$.

Part B
For a strictly positive real number $a$, we consider the function $h_a$ defined on the interval $]0; +\infty[$ by $$h_a(x) = \ln x - ax^2.$$

1. Justify that $x$ is the abscissa of a point $M$ belonging to the intersection of $\mathscr{C}$ and $\Gamma_a$ if and only if $h_a(x) = 0$.
2. a. We admit that the function $h_a$ is differentiable on $]0; +\infty[$ and we denote by $h_a'$ the derivative of the function $h_a$ on this interval. The variation table of the function $h_a$ is given below. Justify, by calculation, the sign of $h_a'(x)$ for $x$ belonging to $]0; +\infty[$.
   

   $x$	0		$\frac{1}{\sqrt{2a}}$
   $h_a'(x)$		+	0	-
   			$\frac{-1 - \ln(2a)}{2}$
   $h_a(x)$

   
   b. Recall the limit of $\frac{\ln x}{x}$ as $x \to +\infty$. Deduce the limit of the function $h_a$ as $x \to +\infty$. We do not ask you to justify the limit of $h_a$ at 0.
   
3. In this question and only in this question, we assume that $a = 0,1$. a. Justify that, in the interval $\left.]0; \frac{1}{\sqrt{0,2}}\right]$, the equation $h_{0,1}(x) = 0$ admits a unique solution. We admit that this equation also has only one solution in the interval $]\frac{1}{\sqrt{0,2}}; +\infty[$. b. What is the number of intersection points of $\mathscr{C}$ and $\Gamma_{0,1}$?
   
4. In this question and only in this question, we assume that $a = \frac{1}{2\mathrm{e}}$. a. Determine the value of the maximum of $h_{\frac{1}{2\mathrm{e}}}$. b. Deduce the number of intersection points of the curves $\mathscr{C}$ and $\Gamma_{\frac{1}{2\mathrm{e}}}$. Justify.
   
5. What are the values of $a$ for which $\mathscr{C}$ and $\Gamma_a$ have no intersection points? Justify.

$f ( x ) = x ^ { 3 } + x ^ { 2 } + c x + d$, where $c$ and $d$ are real numbers. Prove that if $c > \frac { 1 } { 3 }$, then $f$ has exactly one real root.

For $a > 1$, consider the function $f ( x ) = 2 x ^ { 3 } - 3 ( a + 1 ) x ^ { 2 } + 6 a x - 4 a + 2$. Let $b$ be one real root of the equation $f ( x ) = 0$. The following is a process for comparing the magnitudes of two numbers $a$ and $b$. $f ^ { \prime } ( x ) =$ (가) and since $a > 1$, $f ( x )$ has a (나) at $x = 1$. Since $f ( 1 ) < 0$ and $f ( b ) = 0$, $a$ (다) $b$.
What are the correct expressions for (가), (나), and (다) in the above process? [3 points]

(가)	(나)	(다)
(1) $6 ( x - 1 ) ( x - a )$	local minimum	$>$
(2) $6 ( x - 1 ) ( x - a )$	local minimum	$<$
(3) $6 ( x - 1 ) ( x - a )$	local maximum	$>$
(4) $6 ( x - a ) ( x - 1 )$	local maximum	$<$
(5) $6 ( x - a ) ( x - 1 )$	local maximum	$>$

The cubic equation in $x$, $\frac { 1 } { 3 } x ^ { 3 } - x = k$, has three distinct real roots $\alpha$, $\beta$, $\gamma$. For a real number $k$, let $m$ be the minimum value of $| \alpha | + | \beta | + | \gamma |$. Find the value of $m ^ { 2 }$. [4 points]

For the function $f ( x ) = x ( x + 1 ) ( x - 4 )$, answer the following. When the line $y = 5 x + k$ and the graph of the function $y = f ( x )$ intersect at two distinct points, what is the value of the positive number $k$? [4 points]
(1) 5
(2) $\frac { 11 } { 2 }$
(3) 6
(4) $\frac { 13 } { 2 }$
(5) 7

Find the positive value of $k$ such that the curve $y = 4 x ^ { 3 } - 12 x + 7$ and the line $y = k$ intersect at exactly 2 points. [3 points]

How many integers $k$ are there such that the equation $2 x ^ { 3 } - 3 x ^ { 2 } - 12 x + k = 0$ has three distinct real roots? [3 points]
(1) 20
(2) 23
(3) 26
(4) 29
(5) 32

19. Given the function $f ( x ) = x ^ { 3 } + a x ^ { 2 } + b ( a , b \in R )$.
(1) Discuss the monotonicity of $f ( x )$;
(2) If $b = c - a$ (where the real number c is a constant independent of a), when the function $f ( x )$ has three distinct zeros, the range of a is exactly $( - \infty , - 3 ) \cup \left( 1 , \frac { 3 } { 2 } \right) \cup \left( \frac { 3 } { 2 } , + \infty \right)$, find the value of c.

The polynomial $x ^ { 4 } + 4 x + c = 0$ has at least one real root if and only if
(A) $c < 2$.
(B) $c \leq 2$.
(C) $c < 3$.
(D) $c \leq 3$.

Let $a \in \mathbb{R}$ and let $f : \mathbb{R} \rightarrow \mathbb{R}$ be given by $$f(x) = x^5 - 5x + a$$ Then
(A) $f(x)$ has three real roots if $a > 4$
(B) $f(x)$ has only one real root if $a > 4$
(C) $f(x)$ has three real roots if $a < -4$
(D) $f(x)$ has three real roots if $-4 < a < 4$

Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a thrice differentiable function such that $f ( 0 ) = 0 , f ( 1 ) = 1 , f ( 2 ) = - 1 , f ( 3 ) = 2$ and $f ( 4 ) = - 2$. Then, the minimum number of zeros of $\left( 3 f ^ { \prime } f ^ { \prime \prime } + f f ^ { \prime \prime \prime } \right) ( x )$ is $\_\_\_\_$