Specimen Paper SPEC/5/TZ0 P1 Q8

Let \(f(x) = \frac{1}{3} x^3 + x^2 -15x + 17\).

1. Find \(f'(x)\).

   [2]

The graph of \(f\) has horizontal tangents at the points where \(x = a\) and \(x = b\), \(a < b\).

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

   [3]

3. 1. Sketch the graph of \(y = f'(x)\).

   2. Hence explain why the graph of \(f\) has a local maximum point at \(x = a\).

      [2]

4. 1. Find \(f''(b)\).

   2. Hence, use your answer to part (d)(i) to show that the graph of \(f\) has a local minimum point at \(x = b\).

      [4]

The normal to the graph of \(f\) at \(x = a\) and the tangent to the graph of \(f\) at \(x = b\) intersect at the point \( (p,q) \).

5. Find the value of \(p\) and the value of \(q\).

   [5]