2025 Nov TZ3 P1 Q9

The function \(f\) is defined by \(f(x) = \frac{5\mathrm{e}^x - \mathrm{e}^{2x}}{\mathrm{e}^x +1}\). The following diagram shows part of the graph of \(f\). The graph intersects the \(y\)-axis at point \(\mathrm{P}\) and intersects the \(x\)-axis at point \(\mathrm{R}\).
Point \(\mathrm{Q}\) is a local maximum point with coordinates \( \left(q , 7 - 2\sqrt{6} \right) \).

1. [6]

   1. Find the coordinates of point \(\mathrm{P}\).

   2. Find the coordinates of point \(\mathrm{R}\).

2. State the range of \(f\).

   [1]

3. Show that \(f'(x) = \frac{-\mathrm{e}^{3x} - 2\mathrm{e}^{2x} + 5\mathrm{e}^x}{(e^x + 1)^2}\).

   [3]

4. Hence, show that \(q = \ln{\left(\sqrt{6} - 1 \right) }\).

   [6]