Computer Science and Computer Science \& Philosophy applicants should turn to page 14. Let $g ( x )$ be the function defined by $$g ( x ) = \begin{cases} ( x - 1 ) ^ { 2 } + 1 & \text { if } x \geqslant 0 \\ 3 - ( x + 1 ) ^ { 2 } & \text { if } x \leqslant 0 \end{cases}$$ and for $x \neq 0$ write $m ( x )$ for the gradient of the chord between ( $0 , g ( 0 )$ ) and ( $x , g ( x )$ ). (i) Sketch the graph $y = g ( x )$ for $- 3 \leqslant x \leqslant 3$. (ii) Write down expressions for $m ( x )$ in the two cases $x \geqslant 0$ and $x < 0$. (iii) Show that $m ( x ) + 2 = x$ for $x > 0$. What is the value of $m ( x ) + 2$ when $x < 0$ ? (iv) Explain why $g$ has derivative - 2 at 0 . (v) Suppose that $p < q$ and that $h ( x )$ is a cubic with a maximum at $x = p$ and a minimum at $x = q$. Show that $h ^ { \prime } ( x ) < 0$ whenever $p < x < q$. Suppose that $c$ and $d$ are real numbers and that there is a cubic $h ( x )$ with a maximum at $x = - 1$ and a minimum at $x = 1$ such that $h ^ { \prime } ( 0 ) = - 3 c$ and $h ( 0 ) = d$. (vi) Show that $c > 0$ and find a formula for $h ( x )$ in terms of $c$ and $d$ (and $x$ ). (vii) Show that there are no values of $c$ and $d$ such that the graphs of $y = g ( x )$ and $y = h ( x )$ are the same for $- 3 \leqslant x \leqslant 3$.
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\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.
Let $g ( x )$ be the function defined by
$$g ( x ) = \begin{cases} ( x - 1 ) ^ { 2 } + 1 & \text { if } x \geqslant 0 \\ 3 - ( x + 1 ) ^ { 2 } & \text { if } x \leqslant 0 \end{cases}$$
and for $x \neq 0$ write $m ( x )$ for the gradient of the chord between ( $0 , g ( 0 )$ ) and ( $x , g ( x )$ ).\\
(i) Sketch the graph $y = g ( x )$ for $- 3 \leqslant x \leqslant 3$.\\
(ii) Write down expressions for $m ( x )$ in the two cases $x \geqslant 0$ and $x < 0$.\\
(iii) Show that $m ( x ) + 2 = x$ for $x > 0$. What is the value of $m ( x ) + 2$ when $x < 0$ ?\\
(iv) Explain why $g$ has derivative - 2 at 0 .\\
(v) Suppose that $p < q$ and that $h ( x )$ is a cubic with a maximum at $x = p$ and a minimum at $x = q$. Show that $h ^ { \prime } ( x ) < 0$ whenever $p < x < q$.
Suppose that $c$ and $d$ are real numbers and that there is a cubic $h ( x )$ with a maximum at $x = - 1$ and a minimum at $x = 1$ such that $h ^ { \prime } ( 0 ) = - 3 c$ and $h ( 0 ) = d$.\\
(vi) Show that $c > 0$ and find a formula for $h ( x )$ in terms of $c$ and $d$ (and $x$ ).\\
(vii) Show that there are no values of $c$ and $d$ such that the graphs of $y = g ( x )$ and $y = h ( x )$ are the same for $- 3 \leqslant x \leqslant 3$.