Let $k$ be a strictly positive real number.\\
The purpose of this exercise is to determine the number of solutions of the equation

$$\ln ( x ) = k x$$

with parameter $k$.

\textbf{1. Graphical conjectures:}\\
Based on the graph (showing the curve $y = \ln(x)$, the line $y = x$ and the line $y = 0{,}2x$), conjecture the number of solutions of the equation $\ln ( x ) = k x$ for $k = 1$ then for $k = 0{,}2$.

\textbf{2. Study of the case $k = 1$:}

We consider the function $f$, defined and differentiable on $] 0 ; + \infty [$, by:

$$f ( x ) = \ln ( x ) - x .$$

We denote $f ^ { \prime }$ the derivative function of the function $f$.\\
a. Calculate $f ^ { \prime } ( x )$.\\
b. Study the direction of variation of the function $f$ on $] 0 ; + \infty [$.

Draw the variation table of the function $f$ showing the exact value of the extrema if there are any.\\
The limits at the boundaries of the domain of definition are not expected.\\
c. Deduce the number of solutions of the equation $\ln ( x ) = x$.

\textbf{3. Study of the general case:}\\
$k$ is a strictly positive real number.\\
We consider the function $g$ defined on $] 0 ; + \infty [$ by:

$$g ( x ) = \ln ( x ) - k x .$$

We admit that the variation table of the function $g$ is as follows:

\begin{center}
\begin{tabular}{ | c | c c c | }
\hline
$x$ & 0 & $\frac { 1 } { k }$ & $+ \infty$ \\
\hline
$g ( x )$ & $\longrightarrow$ & $g \left( \frac { 1 } { k } \right)$ &  \\
 & $- \infty$ &  & $- \infty$ \\
\hline
\end{tabular}
\end{center}

a. Give, as a function of the sign of $g \left( \frac { 1 } { k } \right)$, the number of solutions of the equation $g ( x ) = 0$.\\
b. Calculate $g \left( \frac { 1 } { k } \right)$ as a function of the real number $k$.\\
c. Show that $g \left( \frac { 1 } { k } \right) > 0$ is equivalent to $\ln ( k ) < - 1$.\\
d. Determine the set of values of $k$ for which the equation $\ln ( x ) = k x$ has exactly two solutions.\\
e. Give, according to the values of $k$, the number of solutions of the equation $\ln ( x ) = k x$.