Maximum and point of inflexion of a logarithmic quotient and an unknown constant from the area between them

Let \(f(x)=\dfrac{1+\ln{x}}{kx}\), where \(x>0\) and \(k\in\mathbb{R}^+\).

1. Show that \(f'(x)=-\dfrac{\ln{x}}{kx^2}\).[3]

The graph of \(f\) has a single maximum point, \(\mathrm{P}\).

2. Find the \(x\)-coordinate of \(\mathrm{P}\).[3]

It is given that \(f''(x)=\dfrac{2\ln{x}-1}{kx^3}\). The graph of \(f\) has a single point of inflexion, \(\mathrm{Q}\).

3. Show that the \(x\)-coordinate of \(\mathrm{Q}\) is \(\sqrt{\mathrm{e}}\).[3]

The shaded region \(R\) lies between the graph of \(f\) and the \(x\)-axis, and is bounded on the left and on the right by the vertical lines through \(\mathrm{P}\) and through \(\mathrm{Q}\).

4. The area of \(R\) is \(\frac{5}{2}\). Find the value of \(k\).[7]