7
\includegraphics[max width=\textwidth, alt={}, center]{a257f49d-5c8f-4c23-be78-46619b746fde-10_353_689_262_726}
The diagram shows the curve \(y = \frac { \tan ^ { - 1 } x } { \sqrt { x } }\) and its maximum point \(M\) where \(x = a\).
- Show that \(a\) satisfies the equation
$$a = \tan \left( \frac { 2 a } { 1 + a ^ { 2 } } \right)$$
- Verify by calculation that \(a\) lies between 1.3 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.