A treatment protocol for a disease in children involves long-term infusion of an appropriate medication. The concentration of the medication in the blood over time is modeled by the function $C$ defined on the interval $[0; +\infty[$ by:

$$C ( t ) = \frac { d } { a } \left( 1 - \mathrm { e } ^ { - \frac { a } { 80 } t } \right)$$

\textbf{Part B: study of functions}

\begin{enumerate}
  \item Let $f$ be the function defined on $]0; +\infty[$ by:

$$f ( x ) = \frac { 105 } { x } \left( 1 - \mathrm { e } ^ { - \frac { 3 } { 40 } x } \right)$$

Prove that, for every real $x$ in $]0; +\infty[$, $f ^ { \prime } ( x ) = \frac { 105 g ( x ) } { x ^ { 2 } }$, where $g$ is the function defined on $[0; +\infty[$ by:

$$g ( x ) = \frac { 3 x } { 40 } \mathrm { e } ^ { - \frac { 3 } { 40 } x } + \mathrm { e } ^ { - \frac { 3 } { 40 } x } - 1$$

  \item The variation table of the function $g$ is given:

\begin{center}
\begin{tabular}{ | c | l l | }
\hline
$x$ & 0 & $+\infty$ \\
\hline
 & 0 &  \\
$g ( x )$ &  & - 1 \\
\hline
\end{tabular}
\end{center}

Deduce the monotonicity of the function $f$.\\
The limits of the function $f$ are not required.

  \item Show that the equation $f ( x ) = 5.9$ has a unique solution on the interval $[1; 80]$.\\
Deduce that this equation has a unique solution on the interval $]0; +\infty[$.\\
Give an approximate value of this solution to the nearest tenth.
\end{enumerate}