6.
$$f ( x ) = \frac { 2 \left( x ^ { 3 } + 3 \right) } { \sqrt { x } } - 9 , \quad x > 0$$
The equation \(\mathrm { f } ( x ) = 0\) has two real roots \(\alpha\) and \(\beta\), where \(0.4 < \alpha < 0.5\) and \(1.2 < \beta < 1.3\)
- Taking 0.45 as a first approximation to \(\alpha\), apply the Newton-Raphson procedure once to \(\mathrm { f } ( x )\) to find a second approximation to \(\alpha\), giving your answer to 3 decimal places.
[0pt] - Use linear interpolation once on the interval [1.2, 1.3] to find an approximation to \(\beta\), giving your answer to 3 decimal places.