We are interested in the evolution of the height of a corn plant as a function of time.

We decide to model this growth by a logistic function of the type:
$$h ( t ) = \frac { a } { 1 + b \mathrm { e } ^ { - 0,04 t } }$$
where $a$ and $b$ are positive real constants, $t$ is the time variable expressed in days and $h ( t )$ denotes the height of the plant, expressed in metres.

We know that initially, for $t = 0$, the plant measures $0,1 \mathrm{~m}$ and that its height tends towards a limiting height of 2 m.

\textbf{Part 1.} Determine the constants $a$ and $b$ so that the function $h$ corresponds to the growth of the corn plant studied.

\textbf{Part 2.} We now consider that the growth of the corn plant is given by the function $f$ defined on $[0;250]$ by
$$f ( t ) = \frac { 2 } { 1 + 19 \mathrm { e } ^ { - 0,04 t } }$$

\begin{enumerate}
  \item Determine $f ^ { \prime } ( t )$ as a function of $t$ ($f ^ { \prime }$ denoting the derivative function of the function $f$). Deduce the variations of the function $f$ on the interval $[ 0 ; 250 ]$.
  \item Calculate the time required for the corn plant to reach a height greater than $1,5 \mathrm{~m}$.
  \item a. Verify that for all real $t$ belonging to the interval $[ 0 ; 250 ]$ we have $f ( t ) = \frac { 2 \mathrm { e } ^ { 0,04 t } } { \mathrm { e } ^ { 0,04 t } + 19 }$.

Show that the function $F$ defined on the interval $[ 0 ; 250]$ by $F ( t ) = 50 \ln \left( \mathrm { e } ^ { 0,04 t } + 19 \right)$ is an antiderivative of the function $f$.

b. Determine the average value of $f$ on the interval $[ 50 ; 100 ]$. Give an approximate value to $10 ^ { - 2 }$ and interpret this result.
  \item We are interested in the growth rate of the corn plant; it is given by the derivative function of the function $f$. The growth rate is maximum for a value of $t$. Using the graph given in the appendix, determine an approximate value of this. Then estimate the height of the plant.
\end{enumerate}