Specimen Paper SPEC/5/TZ0 P1 Q9

Let \(f(x) = \frac{\ln{5x}}{kx}\) where \(x > 0, k \in \mathbb{R}^+\).

1. Show that \(f'(x) = \frac{1 - \ln{5x}}{kx^2}\).

   [3]

The graph of \(f\) has exactly one maximum point \(\mathrm{P}\)

2. Find the \(x\)-coordinate of \(\mathrm{P}\).

   [3]

The second derivative of \(f\) is given by \(f''(x) = \frac{2\ln{5x} - 3}{kx^3}\). The graph of \(f\) has exactly one point of inflexion \(\mathrm{Q}\).

3. Show that the \(x\)-coordinate of \(\mathrm{Q}\) is \(\frac{1}{5}\mathrm{e}^{\frac{3}{2}}\).

   [3]

The region \(R\) is enclosed by the graph of \(f\), the \(x\)-axis, and the vertical lines through the maximum point \(\mathrm{P}\) and the point of inflexion \(\mathrm{Q}\).

4. Given that the area of \(R\) is \(3\), find the value of \(k\).

   [7]