We consider the function $f$ defined on $]0; +\infty[$ by
$$f(x) = x^2 - 8\ln(x)$$
where ln denotes the natural logarithm function. We admit that $f$ is differentiable on $]0; +\infty[$, we denote by $f'$ its derivative function.

\begin{enumerate}
  \item Determine $\lim_{x \rightarrow 0} f(x)$.
  \item We admit that, for all $x > 0$, $f(x) = x^2\left(1 - 8\frac{\ln(x)}{x^2}\right)$.

Deduce the limit: $\lim_{x \rightarrow +\infty} f(x)$.
  \item Show that, for all real $x$ in $]0; +\infty[$, $f'(x) = \frac{2(x^2 - 4)}{x}$.
  \item Study the variations of $f$ on $]0; +\infty[$ and draw up its complete variation table. We will specify the exact value of the minimum of $f$ on $]0; +\infty[$.
  \item Prove that, on the interval $]0; 2]$, the equation $f(x) = 0$ admits a unique solution $\alpha$ (we will not seek to determine the value of $\alpha$).
  \item We admit that, on the interval $[2; +\infty[$, the equation $f(x) = 0$ admits a unique solution $\beta$ (we will not seek to determine the value of $\beta$). Deduce the sign of $f$ on the interval $]0; +\infty[$.
  \item For any real number $k$, we consider the function $g_k$ defined on $]0; +\infty[$ by:
$$g_k(x) = x^2 - 8\ln(x) + k$$
Using the variation table of $f$, determine the smallest value of $k$ for which the function $g_k$ is positive on the interval $]0; +\infty[$.
\end{enumerate}