- The curve \(C\) has equation
$$y = \frac { \ln \left( x ^ { 2 } + k \right) } { x ^ { 2 } + k } \quad x \in \mathbb { R }$$
where \(k\) is a positive constant.
- Show that
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { A x \left( B - \ln \left( x ^ { 2 } + k \right) \right) } { \left( x ^ { 2 } + k \right) ^ { 2 } }$$
where \(A\) and \(B\) are constants to be found.
Given that \(C\) has exactly three turning points,
- find the \(x\) coordinate of each of these points. Give your answer in terms of \(k\) where appropriate.
- find the upper limit to the value for \(k\).