IB AA HL May 2026 P1 TZA Q10

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 \(a=\frac32\) and \(b=9\).[6]
2. It is given that the tangents to the graph of \(f\) at \(x=-3\) and \(x=k\) are horizontal.
   1. Show that \(c=\frac{27}{2}\).
   2. Find the value of \(k\).
   3. State whether \(f\) has a local maximum or a local minimum at \(x=k\), justifying your answer.[7]
3. The graph of \(f\) intersects the \(y\)-axis at the point \(P\).
   Show that the tangent to the graph of \(f\) at \(x=-3\) passes through \(P\).[2]