Mathematics

Let \(f(x) = 3 + \frac{20}{x^2 - 4}\), for \(x \neq \pm2\). The graph of \(f\) is given below.

The \(y\)-intercept is at the point \(\mathrm{A}\).

1. 1. Find the coordinates of \(\mathrm{A}\).
   2. Show that \(f'(x) = 0\) at \(\mathrm{A}\).[7]
2. The second derivative \(f''(x) = \frac{40(3x^2 +4)}{(x^2 - 4)^3}\). Use this to[6]
   1. justify that the graph of \(f\) has a local maximum at \(\mathrm{A}\);
   2. explain why the graph of \(f\) does not have a point of inflexion.
3. Describe the behaviour of the graph of \(f\) for large \(|x|\).[1]
4. Write down the range of \(f\).[2]