In the solution of the problem posed here , the function $Q(x)=x(x-1)$ suddenly comes out to match the equality condition.
I would like to see the intuition behind the function, as it seems like a magical idea to me that came out from nowhere.
Thank you in advance!
Essentially, they are trying to transform the derivative of $p(x)$ onto $Q(x)$. Sure, any $Q(x)$ would do the job but why quadratic function? The reason is simply because you want to take advantage of the assumption that $p(x)$ is a probability density function which integrates to 1, so you really want $Q''$ to be a constant. Then again, why not any quadratic function? I suspect any quadratic function would do the job, but since you are integrating over $[0,1]$, choosing $Q(x)=x(x-1)$ kills the boundary term which simplifies the latter algebra when you are trying to find $p(x)$.