8. The curve \(C\) has equation \(y = \mathrm { f } ( x )\), where
$$\mathrm { f } ( x ) = 3 \ln x + \frac { 1 } { x } , \quad x > 0$$
The point \(P\) is a stationary point on \(C\).
- Calculate the \(x\)-coordinate of \(P\).
- Show that the \(y\)-coordinate of \(P\) may be expressed in the form \(k - k \ln k\), where \(k\) is a constant to be found.
The point \(Q\) on \(C\) has \(x\)-coordinate 1 .
- Find an equation for the normal to \(C\) at \(Q\).
The normal to \(C\) at \(Q\) meets \(C\) again at the point \(R\).
- Show that the \(x\)-coordinate of \(R\)
- satisfies the equation \(6 \ln x + x + \frac { 2 } { x } - 3 = 0\),
- lies between 0.13 and 0.14 .