5.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2a7c2530-a93c-4a26-bc37-c20c0f40c8f2-4_484_581_287_843}
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\caption{Figure 2}
\end{figure}
Figure 2 shows part of the curve \(T\) with equation \(y = \cos 2 x\) and the circle \(C _ { 1 }\) that touches \(T\) at \(\left( \frac { \pi } { 4 } , 0 \right)\) and \(\left( \frac { 3 \pi } { 4 } , 0 \right)\) ．
（a）Find the radius of \(C _ { 1 }\)
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2a7c2530-a93c-4a26-bc37-c20c0f40c8f2-4_486_586_1199_841}
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\caption{Figure 3}
\end{figure}
Figure 3 shows a sketch of part of \(T\) and part of a circle \(C _ { 2 }\) that touches \(T\) at the point \(P\) with coordinates \(\left( \frac { \pi } { 2 } , - 1 \right)\) ．For values of \(x\) close to \(\frac { \pi } { 2 }\) the curve \(T\) lies inside \(C _ { 2 }\) as shown in Figure 3.
（b）Without doing any calculation，explain why the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) for \(C _ { 2 }\) at \(P\) is less than the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) for \(T\) at \(P\) ．
The radius of \(C _ { 2 }\) is \(r\) ．
（c）Use the result from part（b）to find a value of \(k\) such that \(r > k\) ．
Given that \(C _ { 2 }\) cuts \(T\) at the point \(( 0,1 )\) ，
（d）find the value of \(r\) ．