Computer Science and Computer Science \& Philosophy applicants should turn to page 14. The graphs of $y = x ^ { 3 } - x$ and $y = m ( x - a )$ are drawn on the axes below. Here $m > 0$ and $a \leqslant - 1$. The line $y = m ( x - a )$ meets the $x$-axis at $A = ( a , 0 )$, touches the cubic $y = x ^ { 3 } - x$ at $B$ and intersects again with the cubic at $C$. The $x$-coordinates of $B$ and $C$ are respectively $b$ and $c$. [Figure] (i) Use the fact that the line and cubic touch when $x = b$, to show that $m = 3 b ^ { 2 } - 1$. (ii) Show further that $$a = \frac { 2 b ^ { 3 } } { 3 b ^ { 2 } - 1 }$$ (iii) If $a = - 10 ^ { 6 }$, what is the approximate value of $b$ ? (iv) Using the fact that $$x ^ { 3 } - x - m ( x - a ) = ( x - b ) ^ { 2 } ( x - c )$$ (which you need not prove), show that $c = - 2 b$. (v) $R$ is the finite region bounded above by the line $y = m ( x - a )$ and bounded below by the cubic $y = x ^ { 3 } - x$. For what value of $a$ is the area of $R$ largest? Show that the largest possible area of $R$ is $\frac { 27 } { 4 }$.
\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 and Computer Science \& Philosophy applicants should turn to page 14.
The graphs of $y = x ^ { 3 } - x$ and $y = m ( x - a )$ are drawn on the axes below. Here $m > 0$ and $a \leqslant - 1$.
The line $y = m ( x - a )$ meets the $x$-axis at $A = ( a , 0 )$, touches the cubic $y = x ^ { 3 } - x$ at $B$ and intersects again with the cubic at $C$. The $x$-coordinates of $B$ and $C$ are respectively $b$ and $c$.\\
\includegraphics[max width=\textwidth, alt={}, center]{fc6f49e6-218d-48e2-a799-b90a6c6181d7-10_630_762_1005_644}\\
(i) Use the fact that the line and cubic touch when $x = b$, to show that $m = 3 b ^ { 2 } - 1$.\\
(ii) Show further that
$$a = \frac { 2 b ^ { 3 } } { 3 b ^ { 2 } - 1 }$$
(iii) If $a = - 10 ^ { 6 }$, what is the approximate value of $b$ ?\\
(iv) Using the fact that
$$x ^ { 3 } - x - m ( x - a ) = ( x - b ) ^ { 2 } ( x - c )$$
(which you need not prove), show that $c = - 2 b$.\\
(v) $R$ is the finite region bounded above by the line $y = m ( x - a )$ and bounded below by the cubic $y = x ^ { 3 } - x$. For what value of $a$ is the area of $R$ largest?
Show that the largest possible area of $R$ is $\frac { 27 } { 4 }$.