2021 May TZ1 P1 Q7

Let \(f(x) = mx^2 - 2mx\), where \(x \in \mathbb{R}\) and \(m \in \mathbb{R}\). The line \(y = mx - 9\) meets the graph of \(f\) at exactly one point.

1. Show that \(m = 4\).

   [6]

The function \(f\) can be expressed in the form \(f(x) = 4(x - p)(x - q)\), where \(p , q \in \mathbb{R}\).

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

   [2]

The function \(f\) can also be expressed in the form \(f(x) = 4(x - h)^2 + k\), where \(h, k \in \mathbb{R}\).

3. Find the value of \(h\) and the value of \(k\).

   [3]

4. Hence find the values of \(x\) where the graph of \(f\) is both negative and increasing.

   [3]