6. \(\quad f ( x ) = 2 x ^ { 2 } + 3 \ln ( 2 - x ) , \quad x \in \mathbb { R } , \quad x < 2\).
- Show that the equation \(\mathrm { f } ( x ) = 0\) can be written in the form
$$x = 2 - \mathrm { e } ^ { k x ^ { 2 } }$$
where \(k\) is a constant to be found.
The root, \(\alpha\), of the equation \(\mathrm { f } ( x ) = 0\) is 1.9 correct to 1 decimal place.
- Use the iterative formula
$$x _ { n + 1 } = 2 - \mathrm { e } ^ { k x _ { n } ^ { 2 } }$$
with \(x _ { 0 } = 1.9\) and your value of \(k\), to find \(\alpha\) correct to 3 decimal places.
You should show the result of each iteration. - Solve the equation \(\mathrm { f } ^ { \prime } ( x ) = 0\).