Intercepts, range and derivative of an exponential quotient, and the exact position of its maximum point

The function \(f\) is defined by \(f(x)=\frac{4\mathrm{e}^x-\mathrm{e}^{2x}}{\mathrm{e}^x+3}\), for \(x\in\mathbb{R}\). Part of the graph of \(y=f(x)\) is shown in the diagram below. The graph meets the \(y\)-axis at \(P\) and the \(x\)-axis at \(R\), and it has a local maximum point \(Q\left(q,\,10-2\sqrt{21}\right)\).

1. 1. Find the coordinates of \(P\).
   2. Find the coordinates of \(R\).[6]
2. State the range of \(f\).[1]
3. Show that \(f'(x)=\frac{-\mathrm{e}^{3x}-6\mathrm{e}^{2x}+12\mathrm{e}^x}{\left(\mathrm{e}^x+3\right)^2}\).[3]
4. Hence, show that \(q=\ln\left(\sqrt{21}-3\right)\).[6]