\section*{Exercise 2}
4 points

\section*{Common to all candidates}
When the tail of a wall lizard breaks, it regrows on its own in about sixty days.\\
During regrowth, the length in centimeters of the lizard's tail is modeled as a function of the number of days.\\
This length is modeled by the function $f$ defined on $[ 0 ; + \infty [$ by:

$$f ( x ) = 10 \mathrm { e } ^ { u ( x ) }$$

where $u$ is the function defined on $[ 0 ; + \infty [$ by:

$$u ( x ) = - \mathrm { e } ^ { 2 - \frac { x } { 10 } }$$

It is admitted that the function $f$ is differentiable on $\left[ 0 ; + \infty \left[ \right. \right.$ and we denote $f ^ { \prime }$ its derivative function.

\begin{enumerate}
  \item Verify that for all positive $x$ we have $f ^ { \prime } ( x ) = - u ( x ) \mathrm { e } ^ { u ( x ) }$.
\end{enumerate}

Deduce the direction of variation of the function $f$ on $[ 0 ; + \infty [$.\\
2. a. Calculate $f ( 20 )$.

Deduce an estimate, rounded to the nearest millimeter, of the length of the lizard's tail after twenty days of regrowth.\\
b. According to this model, can the lizard's tail measure 11 cm?\\
3. We wish to determine after how many days the growth rate is maximum.

It is admitted that the growth rate after $x$ days is given by $f ^ { \prime } ( x )$.\\
It is admitted that the derivative function $f ^ { \prime }$ is differentiable on $\left[ 0 ; + \infty \left[ \right. \right.$, we denote $f ^ { \prime \prime }$ the derivative function of $f ^ { \prime }$ and it is admitted that:

$$f ^ { \prime \prime } ( x ) = \frac { 1 } { 10 } u ( x ) \mathrm { e } ^ { u ( x ) } ( 1 + u ( x ) )$$

a. Determine the variations of $f ^ { \prime }$ on $[ 0 ; + \infty [$.\\
b. Deduce after how many days the growth rate of the length of the lizard's tail is maximum.