\section*{Exercise 2 (5 points)}
\section*{Part A}
A craftsman creates chocolate candies whose shape recalls the profile of the local mountain. The base of such a candy is modeled by the shaded surface defined in an orthonormal coordinate system with unit 1 cm.

This surface is bounded by the x-axis and the graph denoted $\mathscr { C } _ { f }$ of the function $f$ defined on $[ - 1 ; 1 ]$ by:
$$f ( x ) = \left( 1 - x ^ { 2 } \right) \mathrm { e } ^ { x } .$$

The objective of this part is to calculate the volume of chocolate needed to manufacture a chocolate candy.

\begin{enumerate}
  \item a. Justify that for all $x$ belonging to the interval $[ - 1 ; 1 ]$ we have $f ( x ) \geqslant 0$.\\
b. Show using integration by parts that:
$$\int _ { - 1 } ^ { 1 } f ( x ) \mathrm { d } x = 2 \int _ { - 1 } ^ { 1 } x \mathrm { e } ^ { x } \mathrm {~d} x .$$
  \item The volume $V$ of chocolate, in $\mathrm { cm } ^ { 3 }$, needed to manufacture a candy is given by:
$$V = 3 \times S$$
where $S$ is the area, in $\mathrm { cm } ^ { 2 }$, of the colored surface.\\
Deduce that this volume, rounded to $0.1 \mathrm {~cm} ^ { 3 }$, equals $4.4 \mathrm {~cm} ^ { 3 }$.
\end{enumerate}

\section*{Part B}
We now consider the profit realized by the craftsman on the sale of these chocolate candies as a function of the weekly sales volume.

This profit can be modeled by the function $B$ defined on the interval $[ 0.01 ; + \infty [$ by:
$$B ( q ) = 8 q ^ { 2 } [ 2 - 3 \ln ( q ) ] - 3 .$$

The profit is expressed in tens of euros and the quantity $q$ in hundreds of candies.\\
We admit that the function $B$ is differentiable on $\left[ 0.01 ; + \infty \left[ \right. \right.$. We denote $B ^ { \prime }$ its derivative function.

\begin{enumerate}
  \item a. Determine $\lim _ { q \rightarrow + \infty } B ( q )$.\\
b. Show that, for all $q \geqslant 0.01 , B ^ { \prime } ( q ) = 8 q ( 1 - 6 \ln ( q ) )$.\\
c. Study the sign of $B ^ { \prime } ( q )$, and deduce the direction of variation of $B$ on $[ 0.01 ; + \infty [$. Draw the complete variation table of function $B$.\\
d. What is the maximum profit, to the nearest euro, that the craftsman can expect?
  \item a. Show that the equation $B ( q ) = 10$ has a unique solution $\beta$ on the interval $[1.2; + \infty [$.\\
Give an approximate value of $\beta$ to $10 ^ { - 3 }$ near.\\
b. We admit that the equation $B ( q ) = 10$ has a unique solution $\alpha$ on $[ 0.01 ; 1.2 [$. We are given $\alpha \approx 0.757$.\\
Deduce the minimum and maximum number of chocolate candies to sell to achieve a profit greater than 100 euros.
\end{enumerate}