1.
$$f ( x ) = 2 x ^ { 3 } + x - 10$$
- Show that the equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval \([ 1.5,2 ]\)
The only real root of \(\mathrm { f } ( x ) = 0\) is \(\alpha\)
The iterative formula
$$x _ { n + 1 } = \left( 5 - \frac { 1 } { 2 } x _ { n } \right) ^ { \frac { 1 } { 3 } } , \quad x _ { 0 } = 1.5$$
can be used to find an approximate value for \(\alpha\)
- Calculate \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\), giving your answers to 4 decimal places.
- By choosing a suitable interval, show that \(\alpha = 1.6126\) correct to 4 decimal places.