= Solution
The function is strictly <concave> on $x>0$. Its <derivative> is positive for $0<x<1$, zero at $x=1$, and negative for $x>1$. Thus it rises from the limiting value $-1$ as $x\downarrow0$, touches the horizontal axis at its unique <maximum> $(1,0)$, and then decreases to $-\infty$. It never crosses above the axis. The left endpoint is not included, and the slope tends to $+\infty$ there.
\Image[/past-exam-of-the-mathematical-and-computational-biology-course-of-the-university-of-cambridge/2025/paper-1-sketches.png]
{title=Concave logarithmic function and the beta posterior for the censored observations}
The left panel shows the required sketch; the right panel is used for question 6(e).
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