Find the coordinates of the points A and B.

Given that the graph of the function has exactly one point of inflexion, find its coordinates.

\(f'(x) = {(\ln x)^2} + \frac{{2x\ln x}}{x}\left( { = {{(\ln x)}^2} + 2\ln x = \ln x(\ln x + 2)} \right)\)     M1A1

\(f'(x) = 0{\text{ }}( \Rightarrow x = 1,{\text{ }}x = {e^{ - 2}})\)     M1

Note: Award M1 for an attempt to solve \(f'(x) = 0\).

\(A({e^{ - 2}},\,4{e^{ - 2}})\) and B(1, 0)     A1A1

Note: The final A1 is independent of prior working.

[5 marks]

\(f''(x) = \frac{2}{x}(\ln x + 1)\)     A1

\(f''(x) = 0{\text{ }}\left( { \Rightarrow x = {e^{ - 1}}} \right)\)     (M1)

inflexion point \(({e^{ - 1}},{\text{ }}{e^{ - 1}})\)     A1 

Note: M1 for attempt to solve \(f''(x) = 0\).

[3 marks]

This was answered very well. Candidates are very familiar with this type of question. Some lost a couple of marks by failing to find their final y coordinates, though only the weakest struggled with differentiation and so made little progress.

This was answered very well. Candidates are very familiar with this type of question. Some lost a couple of marks by failing to find their final y coordinates, though only the weakest struggled with differentiation and so made little progress.