8.
$$\mathrm { f } ( x ) = 2 x ^ { - \frac { 2 } { 3 } } + \frac { 1 } { 2 } x - \frac { 1 } { 3 x - 5 } - \frac { 5 } { 2 } \quad x \neq \frac { 5 } { 3 }$$
The table below shows values of \(\mathrm { f } ( x )\) for some values of \(x\), with values of \(\mathrm { f } ( x )\) given to 4 decimal places where appropriate.
| \(x\) | 1 | 2 | 3 | 4 | 5 |
| \(\mathrm { f } ( x )\) | 0.5 | | - 0.2885 | | 0.5834 |
- Complete the table giving the values to 4 decimal places.
The equation \(\mathrm { f } ( x ) = 0\) has exactly one positive root, \(\alpha\).
Using the values in the completed table and explaining your reasoning,
- determine an interval of width one that contains \(\alpha\).
- Hence use interval bisection twice to obtain an interval of width 0.25 that contains \(\alpha\).
Given also that the equation \(\mathrm { f } ( x ) = 0\) has a negative root, \(\beta\), in the interval \([ - 1 , - 0.5 ]\)
- use linear interpolation once on this interval to find an approximation for \(\beta\).
Give your answer to 3 significant figures.