2025 NOV TZ1 P1 Q8

The function \(f\) has a derivative given by \(f'(x) = 3x^2 + 12x - 15\).
The graph of \(y = f(x)\) has horizontal tangents at the points where \(x = a\) and \(x = b\), \(a < b\).

1. Find the value of \(a\) and the value of \(b\).

   [3]

The following diagrams shows three intervals along the \(x\)-axis defined by \(a\) and \(b\). The sign of the first derivative of \(f\) is shown in each interval.

2. State, with a reason, whether there is a local maximum point or a local minimum point on the graph of \(y = f(x)\) at \(x = a\).

   [2]

The second derivative \(f''(x)\) is zero at \(x = c\).

3. Find the value of \(c\)

   [3]

The following diagram shows two intervals along the \(x\)-axis defined by \(c\). The sign of the second derivative is shown in each interval.

4. State, with a reason, whether there is a point of inflexion on the graph of \(y = f(x)\) at \(x = c\).

   [2]

5. Given that \(f(-2) = 36\), find \(f(x)\).

   [4]