1.1. The profile of a vertical drop along a rollercoaster track is modelled using part of the function  $f(x)=10-({x}^{3}-3{x}^{2}-x)$. The graph of  $h=f(x),x\geq0,h\geq0,$  is shown below.

h is the height, in metres, above the ground of the front carriage of the rollercoaster.

x is the horizontal distance from the origin, in metres, of the front carriage of the rollercoaster.

(i) Show that the equation$f(x)=15$ can be rewritten as  ${x}^{3}-3{x}^{2}-x+5=0$.

(ii) Find the derivative of ${x}^{3}-3{x}^{2}-x+5=0.$

1.2. The rollercoaster briefly stops at the point labelled S on the graph, where $f(x)=15$.

Use the Newton-Raphson method with initial value ${x}_{0}=2.6$ to find the x-coordinate of point S. Give your answer to three significant figures.

1.3. State a possible problem if  ${x}_{0}=1.7$were to be used as the initial value in part (b).