Past exam of the mathematical and Computational Biology course of the University of Cambridge / 2025 / Paper 1 / 5 / b / 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 $x\downarrow 0$, 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.

The left panel shows the required sketch; the right panel is used for question 6(e).