2026 MAY TZA P1 Q8

Consider the function \(f(x) = ax^3 + bx^2 + cx + d\), where \(x \in \mathbb{R}\) and where \(a, b, c\) and \(d\) are real constants.
The graph of \(f\) has a point of inflexion at \( (-2, -2c + d + 24)\).

1. Show that \(a = \frac{3}{2}\) and \(b = 9\).

   [6]

It is given that the tangents to the graph of \(f\) at \(x = -3\) and \(x = k\) are horizontal.

2. [7]

   1. Show that \(c = \frac{27}{2}\).

   2. Find the value of \(k\).

   3. State whether the \(f\) has a local maximum or a local minimum at \(x = k\), justifying your answer.

The graph of \(f\) intersects the \(y\)-axis at the point \(\mathrm{P}\).

3. Show that the tangent to the graph of \(f\) at \(x = -3\) passes through \(\mathrm{P}\).

   [2]