Computer Science applicants should turn to page 14. For a positive whole number $n$, the function $f _ { n } ( x )$ is defined by $$f _ { n } ( x ) = \left( x ^ { 2 n - 1 } - 1 \right) ^ { 2 } .$$ (i) On the axes provided opposite, sketch the graph of $y = f _ { 2 } ( x )$ labelling where the graph meets the axes. (ii) On the same axes sketch the graph of $y = f _ { n } ( x )$ where $n$ is a large positive integer. (iii) Determine $$\int _ { 0 } ^ { 1 } f _ { n } ( x ) \mathrm { d } x$$ (iv) The positive constants $A$ and $B$ are such that $$\int _ { 0 } ^ { 1 } f _ { n } ( x ) \mathrm { d } x \leqslant 1 - \frac { A } { n + B } \text { for all } n \geqslant 1 .$$ Show that $$( 3 n - 1 ) ( n + B ) \geqslant A ( 4 n - 1 ) n ,$$ and explain why $A \leqslant 3 / 4$. (v) When $A = 3 / 4$, what is the smallest possible value of $B$ ? [Figure]
(i) [4 marks] We have $f _ { 2 } ( x ) = \left( x ^ { 3 } - 1 \right) ^ { 2 }$. Note that $f _ { 2 } ( x ) = 0$ only when $x = 1$ and $f _ { 2 } ( 0 ) = 1$; so the graph crosses the axes only at $( 1,0 )$ and $( 0,1 )$.
\section*{3. For APPLICANTS IN $\left\{ \begin{array} { l } \text { MATHEMATICS } \\ \text { MATHEMATICS \& STATISTICS } \\ \text { MATHEMATICS \& PHILOSOPHY } \\ \text { MATHEMATICS \& COMPUTER SCIENCE } \end{array} \right\}$ ONLY.}
Computer Science applicants should turn to page 14.\\
For a positive whole number $n$, the function $f _ { n } ( x )$ is defined by
$$f _ { n } ( x ) = \left( x ^ { 2 n - 1 } - 1 \right) ^ { 2 } .$$
(i) On the axes provided opposite, sketch the graph of $y = f _ { 2 } ( x )$ labelling where the graph meets the axes.\\
(ii) On the same axes sketch the graph of $y = f _ { n } ( x )$ where $n$ is a large positive integer.\\
(iii) Determine
$$\int _ { 0 } ^ { 1 } f _ { n } ( x ) \mathrm { d } x$$
(iv) The positive constants $A$ and $B$ are such that
$$\int _ { 0 } ^ { 1 } f _ { n } ( x ) \mathrm { d } x \leqslant 1 - \frac { A } { n + B } \text { for all } n \geqslant 1 .$$
Show that
$$( 3 n - 1 ) ( n + B ) \geqslant A ( 4 n - 1 ) n ,$$
and explain why $A \leqslant 3 / 4$.\\
(v) When $A = 3 / 4$, what is the smallest possible value of $B$ ?\\
\includegraphics[max width=\textwidth, alt={}, center]{373b6bde-f147-4e16-aaa9-78f86989fc96-11_1143_1109_153_470}\\