Stationary points of a cubic classified from the sign of the derivative and the second derivative, and a normal meeting a tangent

Let \(f(x)=\frac{1}{3}x^3-x^2-24x+100\).

1. Find \(f'(x)\).[2]

The tangent to the graph of \(f\) is parallel to the \(x\)-axis at exactly two points, where \(x=a\) and \(x=b\), with \(a<b\).

2. Find \(a\) and \(b\).[3]
3. 1. Sketch the curve \(y=f'(x)\).
   2. Hence explain why \(f\) has a local maximum when \(x=a\).[2]
4. 1. Find \(f''(b)\).
   2. Hence, by using part (d)(i), show that \(f\) attains a local minimum value when \(x=b\).[4]

The normal to the curve \(y=f(x)\) where \(x=a\) intersects the tangent to the same curve where \(x=b\) at \((p,q)\).

5. Find \(p\) and \(q\).[5]