The flow of water from a tap has a constant and moderate flow rate.

We are particularly interested in a part of the flow profile represented in appendix 1 by the curve $C$ in an orthonormal coordinate system.

\section*{Part A}
We consider that the curve $C$ given in appendix 1 is the graphical representation of a function $f$ differentiable on the interval $] 0 ; 1 ]$ which respects the following three conditions:

$$( H ) : f ( 1 ) = 0 \quad f ^ { \prime } ( 1 ) = 0.25 \quad \text { and } \lim _ { \substack { x \rightarrow 0 \\ x > 0 } } f ( x ) = - \infty .$$

\begin{enumerate}
  \item Can the function $f$ be a polynomial function of degree two? Why?
  \item Let $g$ be the function defined on the interval $]0 ; 1]$ by $g ( x ) = k \ln x$.\\
a. Determine the real number $k$ so that the function $g$ respects the three conditions $( H )$.\\
b. Does the representative curve of the function $g$ coincide with the curve $C$ ? Why?
  \item Let $h$ be the function defined on the interval $]0; 1]$ by $h ( x ) = \frac { a } { x ^ { 4 } } + b x$ where $a$ and $b$ are real numbers. Determine $a$ and $b$ so that the function $h$ respects the three conditions ( $H$ ).
\end{enumerate}

\section*{Part B}
We admit in this part that the curve $C$ is the graphical representation of a function $f$ continuous, strictly increasing, defined and differentiable on the interval $] 0 ; 1 ]$ with expression:

$$f ( x ) = \frac { 1 } { 20 } \left( x - \frac { 1 } { x ^ { 4 } } \right)$$

\begin{enumerate}
  \item Justify that the equation $f ( x ) = - 5$ admits on the interval $] 0 ; 1 ]$ a unique solution which will be denoted $\alpha$. Determine an approximate value of $\alpha$ to $10 ^ { - 2 }$ near.
  \item It is admitted that the volume of water in $\mathrm { cm } ^ { 3 }$, contained in the first 5 centimetres of the flow, is given by the formula: $V = \int _ { \alpha } ^ { 1 } \pi x ^ { 2 } f ^ { \prime } ( x ) \mathrm { d } x$.\\
a. Let $u$ be the function differentiable on $] 0; 1]$ defined by $u ( x ) = \frac { 1 } { 2 x ^ { 2 } }$. Determine its derivative function.\\
b. Determine the exact value of $V$. Using the approximate value of $\alpha$ obtained in question 1, give an approximate value of $V$.
\end{enumerate}