9 This question is about the equation \(\mathrm { f } ( x ) = 0\), where \(\mathrm { f } ( x ) = x ^ { 4 } - x - \frac { 1 } { 3 x - 2 }\).
Fig. 9.1 shows the curve \(y = f ( x )\).
Fig. 9.1
\includegraphics[max width=\textwidth, alt={}, center]{60e1e785-c34b-48ef-a63f-13a25fee186e-06_940_929_518_239}
- Show, by calculation, that the equation \(\mathrm { f } ( x ) = 0\) has a root between \(x = 1\) and \(x = 2\).
- Fig. 9.2 shows part of a spreadsheet being used to find a root of the equation.
\begin{table}[h]
\captionsetup{labelformat=empty}
\caption{Fig. 9.2}
| A | B |
| 1 | \(x\) | \(f ( x )\) |
| 2 | 1.5 | 3.1625 |
| 3 | 1.25 | 0.619977679 |
| 4 | 1.125 | - 0.250466087 |
| 5 | | |
- Determine a root of the equation \(\mathrm { f } ( x ) = 0\). Give your answer correct to \(\mathbf { 1 }\) decimal place.
- Fig. 9.3 shows a similar spreadsheet being used to search for another root of \(\mathrm { f } ( x ) = 0\).
\begin{table}[h]
\captionsetup{labelformat=empty}
\caption{Fig. 9.3}
| A | B |
| 1 | x | f(x) |
| 2 | 0 | 0.5 |
| 3 | 1 | -1 |
| 4 | 0.5 | 1.5625 |
| 5 | 0.75 | -4.4336 |
| 6 | 0.6 | 4.5296 |
| 7 | 0.7 | -10.4599 |
| 8 | 0.65 | 19.5285 |
| 9 | 0.675 | -40.4674 |
| 10 | 0.6625 | 79.5301 |
| 11 | 0.66875 | -160.4687 |
| 10 | | |
- Explain why it looks from rows 2 and 3 of the spreadsheet as if there is a root between 0 and 1.
- Explain why this process will not find a root between 0 and 1 .