4. (i)
$$f ( x ) = \frac { ( 2 x + 5 ) ^ { 2 } } { x - 3 } \quad x \neq 3$$
- Find \(\mathrm { f } ^ { \prime } ( x )\) in the form \(\frac { P ( x ) } { Q ( x ) }\) where \(P ( x )\) and \(Q ( x )\) are fully factorised quadratic expressions.
- Hence find the range of values of \(x\) for which \(\mathrm { f } ( x )\) is increasing.
(ii)
$$g ( x ) = x \sqrt { \sin 4 x } \quad 0 \leqslant x < \frac { \pi } { 4 }$$
The curve with equation \(y = g ( x )\) has a maximum at the point \(M\).
Show that the \(x\) coordinate of \(M\) satisfies the equation
$$\tan 4 x + k x = 0$$
where \(k\) is a constant to be found.