Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2009 / Paper 1 / 8

The graph is nonnegative, tends to $+\infty$ as $x\downarrow 0$ and as $x\to \infty$, and vanishes only at $x=1$. Its derivative is $2\log x/x$, negative before one and positive after one, so $(1,0)$ is the unique global minimum. The second derivative is $2(1-\log x)/x^2$: the graph is convex for $0<x<e$ and concave for $x>e$, with an inflection point at $(e,1)$. These features determine the sketch in the left panel.

Write $y=\frac{1}{2}\sin (2x)$. It is an odd function with amplitude $1/2$ and period $\pi$. Its zeros on the specified interval are $-\pi ,-\pi /2,0,\pi /2,\pi$. The maxima are at $x=-3\pi /4,\pi /4$ and the minima at $x=-\pi /4,3\pi /4$, with values $1/2$ and $-1/2$. The middle panel of the preceding figure shows the two full oscillations.