2021 May TZ1 P1 Q5

Consider the functions \(f(x) = -(x - h)^2 + 2k\) and \(g(x) = e^{x - 2} + k\) where \(h , k \in \mathbb{R}\).

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

   [1]

The graphs of \(f\) and \(g\) have a common tangent at \(x = 3\).

2. Show that \(h = \frac{\mathrm{e} + 6}{2}\).

   [3]

3. Hence, show that \(k = \mathrm{e} + \frac{\mathrm{e}^2}{4}\).

   [3]