2. A curve \(C\) has equation \(y = \mathrm { e } ^ { 4 x } + x ^ { 4 } + 8 x + 5\)
- Show that the \(x\) coordinate of any turning point of \(C\) satisfies the equation
$$x ^ { 3 } = - 2 - \mathrm { e } ^ { 4 x }$$
- On the axes given on page 5, sketch, on a single diagram, the curves with equations
- \(y = x ^ { 3 }\),
- \(y = - 2 - e ^ { 4 x }\)
On your diagram give the coordinates of the points where each curve crosses the \(y\)-axis and state the equation of any asymptotes.
- Explain how your diagram illustrates that the equation \(x ^ { 3 } = - 2 - e ^ { 4 x }\) has only one root.
The iteration formula
$$x _ { n + 1 } = \left( - 2 - \mathrm { e } ^ { 4 x _ { n } } \right) ^ { \frac { 1 } { 3 } } , \quad x _ { 0 } = - 1$$
can be used to find an approximate value for this root.
- Calculate the values of \(x _ { 1 }\) and \(x _ { 2 }\), giving your answers to 5 decimal places.
- Hence deduce the coordinates, to 2 decimal places, of the turning point of the curve \(C\).
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