2.
$$f ( x ) = x ^ { 3 } + 3 x ^ { 2 } + 4 x - 12$$
- Show that the equation \(\mathrm { f } ( x ) = 0\) can be written as
$$x = \sqrt { } \left( \frac { 4 ( 3 - x ) } { ( 3 + x ) } \right) , \quad x \neq - 3$$
The equation \(x ^ { 3 } + 3 x ^ { 2 } + 4 x - 12 = 0\) has a single root which is between 1 and 2
- Use the iteration formula
$$x _ { n + 1 } = \sqrt { } \left( \frac { 4 \left( 3 - x _ { n } \right) } { \left( 3 + x _ { n } \right) } \right) , n \geqslant 0$$
with \(x _ { 0 } = 1\) to find, to 2 decimal places, the value of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\).
The root of \(\mathrm { f } ( x ) = 0\) is \(\alpha\).
- By choosing a suitable interval, prove that \(\alpha = 1.272\) to 3 decimal places.