\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.\\
In this question we shall consider the function $f ( x )$ defined by

$$f ( x ) = x ^ { 2 } - 2 p x + 3$$

where $p$ is a constant.\\
(i) Show that the function $f ( x )$ has one stationary value in the range $0 < x < 1$ if $0 < p < 1$, and no stationary values in that range otherwise.

In the remainder of the question we shall be interested in the smallest value attained by $f ( x )$ in the range $0 \leqslant x \leqslant 1$. Of course, this value, which we shall call $m$, will depend on $p$.\\
(ii) Show that if $p \geqslant 1$ then $m = 4 - 2 p$.\\
(iii) What is the value of $m$ if $p \leqslant 0$ ?\\
(iv) Obtain a formula for $m$ in terms of $p$, valid for $0 < p < 1$.\\
(v)Using the axes opposite, sketch the graph of $m$ as a function of $p$ in the range $- 2 \leqslant p \leqslant 2$.\\
\includegraphics[max width=\textwidth, alt={}, center]{d1f785ad-1c78-46e3-9768-075d08bf73c7-11_1104_1109_1361_470}