Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 40 / 5 / a

A buyer accepts a posted price precisely when their valuation exceeds it; equality has probability zero. The independent uniform valuations therefore give probability $p^2$ that neither accepts, for $0\le p\le 1$. The seller's expected revenue isThe interior critical point is $p=1/\sqrt{3}$, with negative second derivative; the endpoint revenues are zero. Prices above one make no sale and nonpositive prices cannot improve on this positive optimum. HenceWhich willing buyer is served if both accept does not affect this revenue.