10 A curve has equation \(y = \mathrm { f } ( x )\) and it is given that
$$\mathrm { f } ^ { \prime } ( x ) = \left( \frac { 1 } { 2 } x + k \right) ^ { - 2 } - ( 1 + k ) ^ { - 2 }$$
where \(k\) is a constant. The curve has a minimum point at \(x = 2\).
- Find \(\mathrm { f } ^ { \prime \prime } ( x )\) in terms of \(k\) and \(x\), and hence find the set of possible values of \(k\).
It is now given that \(k = - 3\) and the minimum point is at \(\left( 2,3 \frac { 1 } { 2 } \right)\). - Find \(\mathrm { f } ( x )\).
- Find the coordinates of the other stationary point and determine its nature.
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