Techniques of Differentiation - Derivatives of Natural Logarithmic Functions

The function \(\mathrm{f}\) is given by \(\mathrm{f}(x) = \frac{\ln 2x}{x}\), \(x > 0\).

1. 1. Show that \(\mathrm{f}'(x) = \frac{1 - \ln 2x}{x^2}\).
   2. Show that the graph of \(\mathrm{f}\) can have only one local maximum or minimum point.
   3. Find the coordinates of the maximum point on the graph of \(\mathrm{f}\).
2. By showing that the second derivative \(\mathrm{f}''(x) = \frac{2 \ln 2x - 3}{x^3}\) or otherwise, find the \(x\)-coordinates of the point of inflexion on the graph of \(\mathrm{f}\).