Stationary points and a point of inflexion from sign diagrams, and a cubic from its derivative

A function \(\mathrm{f}\) has derivative \(\mathrm{f}'(x)=-6x^2+6x+36\). The tangents to the graph of \(y=\mathrm{f}(x)\) are horizontal at \(x=a\) and \(x=b\), where \(a<b\).

1. Find \(a\) and \(b\).[3]

The sign diagram below shows the sign of \(\mathrm{f}'(x)\) in each of the three intervals of the \(x\)-axis separated by \(a\) and \(b\).

2. State, giving a reason, whether the stationary point of the graph of \(y=\mathrm{f}(x)\) at \(x=b\) is a local maximum or a local minimum.[2]

The equation \(\mathrm{f}''(x)=0\) has the single solution \(x=c\).

3. Find \(c\).[3]

The sign diagram below shows the sign of \(\mathrm{f}''(x)\) on each side of \(c\).

4. State, giving a reason, whether the graph of \(y=\mathrm{f}(x)\) has a point of inflexion at \(x=c\).[2]
5. It is given that \(\mathrm{f}(2)=50\). Find an expression for \(\mathrm{f}(x)\).[4]