\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.\\
Let

$$f ( x ) = \left\{ \begin{array} { c c } 
x + 1 & \text { for } 0 \leqslant x \leqslant 1 ; \\
2 x ^ { 2 } - 6 x + 6 & \text { for } 1 \leqslant x \leqslant 2 .
\end{array} \right.$$

(i) On the axes provided below, sketch a graph of $y = f ( x )$ for $0 \leqslant x \leqslant 2$, labelling any turning points and the values attained at $x = 0,1,2$.\\
(ii) For $1 \leqslant t \leqslant 2$, define

$$g ( t ) = \int _ { t - 1 } ^ { t } f ( x ) \mathrm { d } x$$

Express $g ( t )$ as a cubic in $t$.\\
(iii) Calculate and factorize $g ^ { \prime } ( t )$.\\
(iv) What are the minimum and maximum values of $g ( t )$ for $t$ in the range $1 \leqslant t \leqslant 2$ ?\\
\includegraphics[max width=\textwidth, alt={}, center]{43f3ba53-7486-4997-ada9-58d9009d4293-10_1008_624_1544_715}