\textbf{Exercise 1} (5 points)

The Delmas chocolate factory decides to market new confectionery: chocolate drops in the shape of a water droplet. To do this, it must manufacture custom moulds that must meet the following constraint: for this range of sweets to be profitable, the chocolate factory must be able to produce at least 80 with 1 litre of liquid chocolate paste.

\section*{Part A: modelling by a function}
The half-perimeter of the upper face of the drop will be modelled by a portion of the curve of the function $f$ defined on $]0;+\infty[$ by:
$$f(x) = \frac{x^2 - 2x - 2 - 3\ln x}{x}.$$

\begin{enumerate}
  \item Let $\varphi$ be the function defined on $]0;+\infty[$ by:
$$\varphi(x) = x^2 - 1 + 3\ln x.$$
a. Calculate $\varphi(1)$ and the limit of $\varphi$ at 0.\\
b. Study the variations of $\varphi$ on $]0;+\infty[$. Deduce the sign of $\varphi(x)$ according to the values of $x$.

  \item a. Calculate the limits of $f$ at the boundaries of its domain of definition.\\
b. Show that on $]0;+\infty[$: $f'(x) = \dfrac{\varphi(x)}{x^2}$. Deduce the variation table of $f$.\\
c. Prove that the equation $f(x) = 0$ has a unique solution $\alpha$ on $]0;1]$. Determine using a calculator an approximate value of $\alpha$ to $10^{-2}$ near.\\
It is admitted that the equation $f(x) = 0$ also has a unique solution $\beta$ on $[1;+\infty[$ with $\beta \approx 3.61$ to $10^{-2}$ near.\\
d. Let $F$ be the function defined on $]0;+\infty[$ by:
$$F(x) = \frac{1}{2}x^2 - 2x - 2\ln x - \frac{3}{2}(\ln x)^2.$$
Show that $F$ is an antiderivative of $f$ on $]0;+\infty[$.
\end{enumerate}

\section*{Part B: solving the problem}
In this part, calculations will be performed with the approximate values to $10^{-2}$ near of $\alpha$ and $\beta$ from Part A.\\
To obtain the shape of the droplet, we consider the representative curve $C$ of the function $f$ restricted to the interval $[\alpha;\beta]$ as well as its reflection $C'$ with respect to the horizontal axis.\\
The two curves $C$ and $C'$ delimit the upper face of the drop. For aesthetic reasons, the chocolatier would like his drops to have a thickness of $0.5$ cm. Under these conditions, would the profitability constraint be respected?