2009 HCI P1 Q3

The diagram below shows the graph of \(y=\mathrm{{f}'}\left( x \right)\). The curve passes through the origin and has turning points at \(\left( -3,0 \right)\) and \(\left( -1.5,-1 \right)\). The \(x\)-axis and \(x=1\) are the two asymptotes of the curve.

1. Find the range of values of \(x\) for which the graph of \(y=\mathrm{f}\left( x \right)\) is strictly increasing and concave downwards.[1]
2. State the \(x\)-coordinates of all the stationary points of the graph of \(y=\mathrm{f}\left( x \right)\) and determine the nature of each point.[2]
3. Given that \(\mathrm{f}\left( 0 \right)=1\), sketch the graph of \(y=\mathrm{f}\left( x \right)\) for \(x<1\). Your sketch should indicate clearly all stationary points, asymptotes and intersections with the axes.[2]