\textbf{EXERCISE B}\\
Main topics covered: Natural logarithm function, differentiation

This exercise consists of two parts.\\
Some results from the first part will be used in the second.

\section*{Part 1: Study of an auxiliary function}
Let the function $f$ defined on the interval $[1 ; 4]$ by:
$$f ( x ) = - 30 x + 50 + 35 \ln x$$

\begin{enumerate}
  \item Recall that $f ^ { \prime }$ denotes the derivative function of the function $f$.\\
a. For every real number $x$ in the interval $[ 1 ; 4 ]$, show that:
$$f ^ { \prime } ( x ) = \frac { 35 - 30 x } { x }$$
b. Draw up the sign table of $f ^ { \prime } ( x )$ on the interval $[ 1 ; 4 ]$.\\
c. Deduce the variations of $f$ on this same interval.
  \item Justify that the equation $f ( x ) = 0$ has a unique solution, denoted $\alpha$, on the interval $[1;4]$ then give an approximate value of $\alpha$ to $10 ^ { - 3 }$ near.
  \item Draw up the sign table of $f ( x )$ for $x \in [ 1 ; 4 ]$.
\end{enumerate}

\section*{Part 2: Optimisation}
A company sells fruit juice. For $x$ thousand litres sold, with $x$ a real number in the interval $[ 1 ; 4 ]$, the analysis of sales leads to modelling the profit $B ( x )$ by the expression given in thousands of euros by:
$$B ( x ) = - 15 x ^ { 2 } + 15 x + 35 x \ln x$$

\begin{enumerate}
  \item According to the model, calculate the profit made by the company when it sells 2500 litres of fruit juice.\\
Give an approximate value to the nearest euro of this profit.
  \item For all $x$ in the interval $[ 1 ; 4 ]$, show that $B ^ { \prime } ( x ) = f ( x )$ where $B ^ { \prime }$ denotes the derivative function of $B$.
  \item a. Using the results from part 1, give the variations of the function $B$ on the interval $[1;4]$.\\
b. Deduce the quantity of fruit juice, to the nearest litre, that the company must sell in order to achieve maximum profit.
\end{enumerate}