2022 TGM P1 Q10

The diagram shows the graph of \(y = \mathrm{f}'(x)\) with equations of asymptotes of \(y = -2x\) and \(x = 0\). The curve cuts the \(x\)-axis at \(A\), and has a maximum point at \(B\). The coordinates of \(A\) and \(B\) are \((a, 0)\) and \((b, -2)\) respectively.

1. Based on the graph of \(y = \mathrm{f}'(x)\), find the range of values of \(x\), with a reason, for which the graph of \(y = \mathrm{f}(x)\) is

   1. strictly increasing,[2]
   2. concave upwards.[2]

Showing clearly, in terms of \(a\) and \(b\) where possible, the equations of any asymptote(s), the coordinates of any turning point(s) and any point(s) where the curve cuts the axes. Sketch on separate diagrams, the graphs of

2. \(y = \mathrm{f}''(x)\),

   [3]

3. \(y = \frac{1}{\mathrm{f}'(x)}\),

   [3]