6.
$$\mathrm { f } ( x ) = x \cos \left( \frac { x } { 3 } \right) \quad x > 0$$
- Find \(\mathrm { f } ^ { \prime } ( x )\)
- Show that the equation \(\mathrm { f } ^ { \prime } ( x ) = 0\) can be written as
$$x = k \arctan \left( \frac { k } { x } \right)$$
where \(k\) is an integer to be found.
- Starting with \(x _ { 1 } = 2.5\) use the iteration formula
$$x _ { n + 1 } = k \arctan \left( \frac { k } { x _ { n } } \right)$$
with the value of \(k\) found in part (b), to calculate the values of \(x _ { 2 }\) and \(x _ { 6 }\) giving your answers to 3 decimal places.
- Using a suitable interval and a suitable function that should be stated, show that a root of \(\mathrm { f } ^ { \prime } ( x ) = 0\) is 2.581 correct to 3 decimal places.
In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.