When a linear fractional function decreases everywhere

The function \(\mathrm f\) is defined by \(\mathrm f(x)=\dfrac{ax+b}{cx+d}\) for \(x\ne-\dfrac dc\), where \(a\), \(b\), \(c\) and \(d\) are constants and \(c\ne0\).

1. Show that \(\mathrm f'(x)=\dfrac{ad-bc}{(cx+d)^2}\).[2]
2. Explain why, if \(ad-bc\ne0\), \(\mathrm f\) is either increasing on every interval of its domain or decreasing on every interval of its domain, and state the condition for each.[2]
3. Find the range of values of \(k\) for which \(\mathrm g(x)=\dfrac{kx+3}{x+k}\) is decreasing for all \(x\ne-k\).[3]
4. Explain why \(\mathrm g\) is neither increasing nor decreasing when \(k=\sqrt3\).[1]