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 $a$.

\textbf{Part A}

We have constructed in appendix 1 the curves $\mathscr{C}, \Gamma_{0,05}, \Gamma_{0,1}, \Gamma_{0,19}$ and $\Gamma_{0,4}$.

\begin{enumerate}
  \item Name the different curves on the graph. No justification is required.
  \item 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 $a$.
\end{enumerate}

\textbf{Part B}

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

\begin{enumerate}
  \item Justify that $x$ is the abscissa of a point $M$ belonging to the intersection of $\mathscr{C}$ and $\Gamma_a$ if and only if $h_a(x) = 0$.
  \item 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 $]0; +\infty[$.

\begin{center}
\begin{tabular}{ | c | | c c c c | }
\hline
$x$ & 0 &  & $\frac{1}{\sqrt{2a}}$ &  \\
\hline
$h_a'(x)$ &  & + & 0 & - \\
\hline
 &  &  & $\frac{-1 - \ln(2a)}{2}$ &  \\
 &  &  &  &  \\
$h_a(x)$ &  &  &  &  \\
 &  &  &  &  \\
\end{tabular}
\end{center}

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.

  \item 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 $\Gamma_{0,1}$?

  \item 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.

  \item What are the values of $a$ for which $\mathscr{C}$ and $\Gamma_a$ have no intersection points? Justify.
\end{enumerate}