Specimen Paper SPEC/5/TZ0 P2 Q9

Consider a function \(f\), such that \(f(x) = 5.8\sin{\left( \frac{\pi}{6} (x + 1) \right)} + b\) , \(0 \leq x \leq 10\), \(b \in \mathbb{R}\).

1. Find the period of \(f\).

   [2]

The function \(f\) has a local maximum at the point \( (2, 21.8)\), and a local minimum at \( (8, 10.2)\).

2. 1. Find the value of \(b\).

   2. Hence, find the value of \(f(6)\).

      [4]

A second function \(g\) is given by \(g(x) = p\sin{ \left( \frac{2\pi}{9}(x - 3.75) \right)} + q\), \(0 \leq x \leq 10\); \(p, q \in \mathbb{R}\).
The function \(g\) passes through the points \( (3, 2.5)\) and \( (6, 15.1)\).

3. Find the values of \(p\) and the value of \(q\).

   [5]

4. Find the value of \(x\) for which the functions have the greatest difference.

   [2]