6.
$$f ( x ) = x ^ { 2 } - 3 x + 2 \cos \left( \frac { 1 } { 2 } x \right) , \quad 0 \leqslant x \leqslant \pi$$
- Show that the equation \(\mathrm { f } ( x ) = 0\) has a solution in the interval \(0.8 < x < 0.9\)
The curve with equation \(y = \mathrm { f } ( x )\) has a minimum point \(P\).
- Show that the \(x\)-coordinate of \(P\) is the solution of the equation
$$x = \frac { 3 + \sin \left( \frac { 1 } { 2 } x \right) } { 2 }$$
- Using the iteration formula
$$x _ { n + 1 } = \frac { 3 + \sin \left( \frac { 1 } { 2 } x _ { n } \right) } { 2 } , \quad x _ { 0 } = 2$$
find the values of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\), giving your answers to 3 decimal places.
- By choosing a suitable interval, show that the \(x\)-coordinate of \(P\) is 1.9078 correct to 4 decimal places.