3.
$$f ( x ) = x ^ { 2 } + 5 x - 2 \sec x , \quad x \in \mathbb { R } , \quad - \frac { \pi } { 2 } < x < \frac { \pi } { 2 } .$$
- Show that the equation \(\mathrm { f } ( x ) = 0\) has a root in the interval [1,1.5].
A more accurate estimate of this root is to be found using iterations of the form
$$x _ { n + 1 } = \arccos \mathrm { g } \left( x _ { n } \right) .$$
- Find a suitable form for \(\mathrm { g } ( x )\) and use this formula with \(x _ { 0 } = 1.25\) to find \(x _ { 1 } , x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\). Give the value of \(x _ { 4 }\) to 3 decimal places.
The curve \(y = \mathrm { f } ( x )\) has a stationary point at \(P\).
- Show that the \(x\)-coordinate of \(P\) is 1.0535 correct to 5 significant figures.