5.
$$f ( x ) = 30 + \frac { 7 } { \sqrt { x } } - x ^ { 5 } , \quad x > 0$$
The only real root, \(\alpha\), of the equation \(\mathrm { f } ( x ) = 0\) lies in the interval [2,2.1].
[0pt]
- Starting with the interval [2,2.1], use interval bisection twice to find an interval of width 0.025 that contains \(\alpha\).
- Taking 2 as a first approximation to \(\alpha\), apply the Newton-Raphson process once to \(\mathrm { f } ( x )\) to find a second approximation to \(\alpha\), giving your answer to 2 decimal places.