The function \(\mathrm{f}\) is defined by \(\mathrm{f}(x)=\dfrac{12+16x-x^2}{x-6},\ x\neq6\).
Express \(\mathrm{f}\) in the form \(\mathrm{f}(x)=A+Bx+\dfrac C{x-6}\), where \(A,\ B\) and \(C\) are constants to be determined.[4]
Sketch the graph of \(y=\mathrm{f}(x)\), indicating clearly the coordinates of any turning points, axes intercepts and asymptotes.[5]
Show that \(\mathrm{f}'(x)<0\) for all \(x\in\mathbb R,\ x\neq6\).
Explain whether \(\mathrm{f}(x)\) is a decreasing function.[4]
Let \(\mathrm{g}(x)=kx\), where \(k\in\mathbb R\).
Find the range of values of \(k\) for which the graphs of \(y=\mathrm{f}(x)\) and \(y=\mathrm{g}(x)\) do not intersect.[5]