3. Let

$$f ( x ) = \left( c - \frac { 1 } { c } - x \right) \left( 4 - 3 x ^ { 2 } \right)$$

where $c$ is a positive constant and $x$ varies over the real numbers.\\
(i) Show that $f ( x )$ has one maximum and one minimum.\\
(ii) Show that the difference between the values of $f ( x )$ at its turning points is

$$\frac { 4 } { 9 } \left( c + \frac { 1 } { c } \right) ^ { 3 }$$

(iii) What is the least value that the difference in (ii) can have for $c > 0$ ?



\begin{enumerate}
  \setcounter{enumi}{3}
  \item The absolute value $| x |$ of a real number $x$ is defined by
\end{enumerate}

$$| x | = \begin{cases} x & \text { if } x \geqslant 0 \\ - x & \text { if } x < 0 \end{cases}$$

(So, for example, $| 3 | = 3$ and $| - 5 | = 5$.)\\
(i) Sketch the locus of points ( $x , y$ ) in the first quadrant ( $x \geqslant 0 , y \geqslant 0$ ) of the $x - y$ plane that satisfy the equation $x + y = 1$.\\
(ii) Hence, sketch the set of points $( x , y )$ in the whole $x - y$ plane that satisfy the equation $| x | + | y | = 1$.\\
(iii) Give the equations of the straight lines that contain the sides of the figure that you have drawn in (ii).\\
(iv) If $| x | + | y | = 1$, what are the maximum and minimum values of $\sqrt { x ^ { 2 } + y ^ { 2 } }$ ? At which points $( x , y )$ are they attained?

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{455b2f35-67b1-4743-8350-741ca5c51568-12_530_521_1274_372}
\captionsetup{labelformat=empty}
\caption{(i)}
\end{center}
\end{figure}

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{455b2f35-67b1-4743-8350-741ca5c51568-12_530_527_1271_1131}
\captionsetup{labelformat=empty}
\caption{(ii)}
\end{center}
\end{figure}

\begin{enumerate}
  \setcounter{enumi}{4}
  \item The game of Hexaglide is played on a board with $3 \times 3$ squares; White and Black each have 3 pieces, and they begin as shown in the first diagram. White moves first, and the players take turns to move one of their pieces forwards or backwards to an adjacent empty square. (Pieces never move sideways or diagonally, and they are never captured or removed from the board.)\\
\includegraphics[max width=\textwidth, alt={}, center]{455b2f35-67b1-4743-8350-741ca5c51568-14_271_271_641_441}\\
\includegraphics[max width=\textwidth, alt={}, center]{455b2f35-67b1-4743-8350-741ca5c51568-14_267_267_645_845}\\
\includegraphics[max width=\textwidth, alt={}, center]{455b2f35-67b1-4743-8350-741ca5c51568-14_269_272_645_1245}\\
(i) In a practice game, White plays without any Black pieces on the board, and, from his usual starting position, reaches the position shown in the second diagram. How many different sequences of moves end with this position if White makes (a) 6 moves, (b) 7 moves, (c) 8 moves?
\end{enumerate}

For parts (ii) and (iii) of this question, give 'yes' or 'no' answers in the grids opposite; for both parts, you need not show your working or explain your answers.\\
In the real game, White and Black both play, and a player wins if he can trap his opponent's pieces so that they cannot move: the third diagram shows a win for White.\\
(ii) (a) Is it possible to reach the position shown in the third diagram?\\
(b) Is it possible to reach a position where Black has won?\\
(c) Can White play so as to ensure that either he wins or the game goes on forever?\\
(d) Can Black play so as to ensure that either he wins or the game goes on forever?\\
(iii) In an advanced version of the game, the board has $4 \times 4$ squares, and each player has 4 pieces. What would be the answers to the four questions in part (ii) in this case?

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{455b2f35-67b1-4743-8350-741ca5c51568-15_657_461_445_705}
\captionsetup{labelformat=empty}
\caption{(ii)}
\end{center}
\end{figure}

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{455b2f35-67b1-4743-8350-741ca5c51568-15_657_455_1468_699}
\captionsetup{labelformat=empty}
\caption{(iii)}
\end{center}
\end{figure}