2.
$$f ( x ) = \frac { 3 } { 2 } x ^ { 2 } + \frac { 4 } { 3 x } + 2 x - 5 , \quad x < 0$$
The equation \(\mathrm { f } ( x ) = 0\) has a single root \(\alpha\).
- Show that \(\alpha\) lies in the interval \([ - 3 , - 2.5 ]\)
- Taking - 3 as a first approximation to \(\alpha\), apply the Newton-Raphson procedure once to \(\mathrm { f } ( x )\) to obtain a second approximation to \(\alpha\). Give your answer to 3 decimal places.
- Use linear interpolation once on the interval \([ - 3 , - 2.5 ]\) to find another approximation to \(\alpha\), giving your answer to 3 decimal places.