8 The curve with equation \(y = \frac { \sinh x } { x ^ { 2 } }\), for \(x > 0\), has one turning point.
- Show that the \(x\)-coordinate of the turning point satisfies the equation \(x - 2 \tanh x = 0\).
- Use the Newton-Raphson method, with a first approximation \(x _ { 1 } = 2\), to find the next two approximations, \(x _ { 2 }\) and \(x _ { 3 }\), to the positive root of \(x - 2 \tanh x = 0\).
- By considering the approximate errors in \(x _ { 1 }\) and \(x _ { 2 }\), estimate the error in \(x _ { 3 }\). (You are not expected to evaluate \(x _ { 4 }\).)