4. $$\left[ \begin{array} { l } \text { The Taylor series expansion of } \mathrm { f } ( x ) \text { about } x = a \text { is given by }
\mathrm { f } ( x ) = \mathrm { f } ( a ) + ( x - a ) \mathrm { f } ^ { \prime } ( a ) + \frac { ( x - a ) ^ { 2 } } { 2 ! } \mathrm { f } ^ { \prime \prime } ( a ) + \ldots + \frac { ( x - a ) ^ { r } } { r ! } \mathrm { f } ^ { ( r ) } ( a ) + \ldots \end{array} \right]$$ The curve with equation \(y = \mathrm { f } ( x )\) satisfies the differential equation $$\cos x \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + y ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + \sin x = 0$$ Given that \(\left( \frac { \pi } { 4 } , 1 \right)\) is a stationary point of the curve,

1. determine the nature of this stationary point, giving a reason for your answer.
2. Show that \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } = \sqrt { 2 } - 2\) at this stationary point.
3. Hence determine a series solution for \(y\), in ascending powers of \(\left( x - \frac { \pi } { 4 } \right)\) up to and including the term in \(\left( x - \frac { \pi } { 4 } \right) ^ { 3 }\), giving each coefficient in simplest form.