Coefficients of a cubic from its point of inflexion and horizontal tangents, and the nature of a turning point

The function \(g\) is given by \(g(x)=px^3+qx^2+rx+s\) for \(x\in\mathbb{R}\), with real constants \(p\), \(q\), \(r\) and \(s\).

The point \((1,\,r+s-4)\) on the graph of \(y=g(x)\) is a point of inflexion.

1. Show that \(p=2\) and \(q=-6\).[6]
2. The graph of \(y=g(x)\) has a horizontal tangent at \(x=3\) and another at \(x=m\), where \(m\ne3\).
   1. Show that \(r=-18\).
   2. Find \(m\).
   3. Determine the nature of the stationary point at \(x=m\), justifying your answer.[7]
3. Show that, at the point of inflexion, the tangent to the curve \(y=g(x)\) is parallel to the line \(24x+y=5\).[2]