9 Let \(\mathrm { f } ( x ) = \frac { \mathrm { e } ^ { 2 x } + 1 } { \mathrm { e } ^ { 2 x } - 1 }\), for \(x > 0\).
- The equation \(x = \mathrm { f } ( x )\) has one root, denoted by \(a\).
Verify by calculation that \(a\) lies between 1 and 1.5.
- Use an iterative formula based on the equation in part (a) to determine \(a\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
- Find \(\mathrm { f } ^ { \prime } ( x )\). Hence find the exact value of \(x\) for which \(\mathrm { f } ^ { \prime } ( x ) = - 8\).