10. A curve \(C\) has equation
$$y = 4 x ^ { 3 } - 9 x + \frac { k } { x } \quad x > 0$$
where \(k\) is a constant.
The point \(P\) with \(x\) coordinate \(\frac { 1 } { 2 }\) lies on \(C\).
Given that \(P\) is a stationary point of \(C\),
- show that \(k = - \frac { 3 } { 2 }\)
- Determine the nature of the stationary point at \(P\), justifying your answer.
The curve \(C\) has a second stationary point.
- Using algebra, find the \(x\) coordinate of this second stationary point.
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