\begin{exercice}[Prendre des initiatives]\par 
\tikzset{math3d/.style=
    {x= {(-0.353cm,-0.353cm)}, z={(0cm,1cm)},y={(1cm,0cm)}}}
$ABCDEFGH$ est un cube d'arête 6 cm.\\
On construit un point $M$ appartenant au segment $[AB]$ et le point $P$ appartenant au segment $[AE]$ tel que $AM~=~EP$. On construit alors le pavé droit $AMNTPQRS$ de telle façon que $AMNT$ soit un carré.\\

\begin{center}
%parallélépipède ABCDEFGH AB=3; BC=3 et AE=3
\begin{tikzpicture}[math3d,scale=1]
\coordinate (A) at (3,0,0);
\coordinate (B) at (3,3,0);
\coordinate (E) at (3,0,3);
\coordinate (F) at (3,3,3);
\coordinate (D) at (0,0,0);
\coordinate (H) at (0,0,3);
\coordinate (C) at (0,3,0);
\coordinate (G) at (0,3,3);


\coordinate (M) at (3,2,0);
\coordinate (P) at (3,0,1);
\coordinate (Q) at (3,2,1);
\coordinate (T) at (1,0,0);
\coordinate (S) at (1,0,1);
\coordinate (N) at (1,2,0);
\coordinate (R) at (1,2,1);



\draw[dashed,fill=green!10] (P) -- (S) -- (R)--(Q)--cycle;
\draw[fill=green!20]  (A) rectangle (Q);
\draw[dashed,fill=green!10]  (M) -- (N) -- (R) --(Q) -- cycle;
\draw  [dashed] (T) -- (N);
\draw [dashed] (A) -- (T) -- (S) ;

\draw  (A) rectangle (F);
\draw  (B) -- (C) -- (G) --(F) -- cycle;
\draw  [dashed] (D) -- (C);
\draw [dashed] (A) -- (D) -- (H) ;
\draw (E) -- (H) -- (G);

\begin{scriptsize}

\draw (A) node[below left] {$A$};
\draw (B) node[below right] {$B$};
\draw (C) node[right] {$C$};
\draw (D) node[left] {$D$};
\draw (E) node[above left] {$E$};
\draw (F) node[above ] {$F$};
\draw (G) node[above] {$G$};
\draw (H) node[above] {$H$};


\draw (M) node[below] {$M$};
\draw (N) node[below ] {$N$};
\draw (T) node[left] {$T$};
\draw (P) node[above left] {$P$};
\draw (Q) node[above left] {$Q$};
\draw (R) node[above] {$R$};
\draw (S) node[above] {$S$};
\end{scriptsize}


\end{tikzpicture}
\end{center}

Pour quelle position du point $M$ le volume du pavé $AMNTPQRS$ sera-t-il maximal?
\end{exercice}