4. 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 $0 \leq x \leq 1$. Of course, this value, which we shall call $m$, will depend on $p$.\\
(ii) Show that if $p \geq 1$ then $m = 4 - 2 p$.\\
(iii) What is the value of $m$ if $p \leq 0$ ?\\
(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 $- 2 \leq p \leq 2$.\\
\includegraphics[max width=\textwidth, alt={}, center]{2943001e-0059-4191-9307-3bd89df4b942-13_944_940_1391_532}\\
5.\\
\includegraphics[max width=\textwidth, alt={}, center]{2943001e-0059-4191-9307-3bd89df4b942-14_670_593_228_641}

The diagram represents an array of $N$ electric lights arranged in a circle. Initially, each light may be set to be ON or OFF in an arbitrary way. After one second the settings are updated according to the following rule which determine the new state of a bulb in terms of the initial states of that bulb and the one just next to it in the clockwise direction.\\
if initially bulb $n$ and bulb $n + 1$ are in the same state (i.e. either both OFF or both ON) then after 1 second bulb $n$ will be OFF;\\
if initially bulb $n$ and bulb $n + 1$ are in different states (i.e. one OFF the other ON) then after 1 second bulb $n$ will be ON ;\\
Of course, if $n = N$, we replace $n + 1$ with 1 in the above.\\
Subsequently, the settings are updated each second by reapplying the same rule.\\
(i) Explain why after one second there cannot be exactly one bulb ON .\\
(ii) More generally, explain why after one second there cannot be an odd number of bulbs ON.\\
(iii) Show that the state of bulb $n$ after 2 seconds is completely determined by the initial states of bulbs $n$ and $n + 2$ (with appropriate modifications when $n = N$ or $n = N - 1 )$.\\
(iv) The initial states of which bulbs determine the state of bulb $n$ after 4 seconds?\\
(v) Show that if $N = 8$ then, irrespective of the initial settings, all bulbs will eventually be OFF. How long will this take?