Past exam of the year 1 of the computer science course of the University of Oxford / 2013 / Discrete mathematics, probability and continuous mathematics / 9 / a / ii / Solution

The factor $2x-x^3=x(2-x^2)$ is positive on $(0,\sqrt{2})$ and negative on $(\sqrt{2},2)$. For every nonzero $\alpha$, the proposed probability density function is therefore negative on an interval. The choice $\alpha =0$ has total integral zero. In fact,so no scaling could normalise it even without the sign obstruction. Consequently