Stationary point of a product of powers

A curve has equation \(y=x^m(1-x)^n\), where \(m\) and \(n\) are positive integers.

1. Show that \(\dfrac{\mathrm dy}{\mathrm dx}=x^{m-1}(1-x)^{n-1}\left[m-(m+n)x\right]\).[3]
2. Hence show that the curve has exactly one stationary point with \(0<x<1\), and state its \(x\)-coordinate in terms of \(m\) and \(n\).[2]
3. The curve \(y=x^2(1-x)^k\), where \(k\) is a positive integer, has a stationary point at \(x=\dfrac14\). Find \(k\) and the \(y\)-coordinate of this stationary point.[3]