We consider the function $f$ defined for every real $x$ in the interval $]0; +\infty[$ by:

$$f(x) = 5x^2 + 2x - 2x^2\ln(x).$$

We denote by $\mathscr{C}_f$ the representative curve of $f$ in an orthogonal reference frame of the plane.\\
We admit that $f$ is twice differentiable on the interval $]0; +\infty[$.\\
We denote by $f'$ its derivative and $f''$ its second derivative.

\begin{enumerate}
  \item a. Prove that the limit of the function $f$ at 0 is equal to 0.\\
b. Determine the limit of the function $f$ at $+\infty$.
  \item Determine $f'(x)$ for every real $x$ in the interval $]0; +\infty[$.
  \item a. Prove that for every real $x$ in the interval $]0; +\infty[$
$$f''(x) = 4(1 - \ln(x)).$$
b. Deduce the largest interval on which the curve $\mathscr{C}_f$ is above its tangent lines.\\
c. Draw the variation table of the function $f'$ on the interval $]0; +\infty[$.\\
(We will admit that $\lim_{\substack{x \to 0 \\ x > 0}} f'(x) = 2$ and that $\lim_{x \to +\infty} f'(x) = -\infty$.)
  \item a. Show that the equation $f'(x) = 0$ admits in the interval $]0; +\infty[$ a unique solution $\alpha$ for which we will give an enclosure of amplitude $10^{-2}$.\\
b. Deduce the sign of $f'(x)$ on the interval $]0; +\infty[$ as well as the variation table of the function $f$ on the interval $]0; +\infty[$.
  \item a. Using the equality $f'(\alpha) = 0$, prove that:
$$\ln(\alpha) = \frac{4\alpha + 1}{2\alpha}.$$
Deduce that $f(\alpha) = \alpha^2 + \alpha$.\\
b. Deduce an enclosure of amplitude $10^{-1}$ of the maximum of the function $f$.
\end{enumerate}