Cubic with a given point of inflexion: coefficients, the second stationary point and a horizontal tangent through the y-intercept

The curve \(y=f(x)\), \(x\in\mathbb{R}\), has equation \(f(x)=ax^3+bx^2+cx+d\), in which \(a\), \(b\), \(c\) and \(d\) are constants. The point \((2,\ 2c+d+8)\) is a point of inflexion of the curve.

1. Show that \(a=-\dfrac12\) and \(b=3\).[6]

The curve has a horizontal tangent at \(x=3\) and another horizontal tangent at \(x=k\), where \(k\neq3\).

2. 1. Show that \(c=-\dfrac92\).
   2. Find \(k\).
   3. Determine whether the stationary point at \(x=k\) is a maximum or a minimum point. Justify your answer.[7]

The curve meets the \(y\)-axis at \(Q\).

3. Show that \(Q\) lies on the tangent to the curve at \(x=3\).[2]