Asymptotes and derivative of a rational function and the point on one branch nearest a fixed point
Asymptotes and derivative of a rational function and the point on one branch nearest a fixed point
13 marks
The function \(\mathrm{f}\) is defined by \(\mathrm{f}(x)=\dfrac{2x-5}{x+1}\) for \(x\in\mathbb{R}\), \(x\neq-1\).
- Write down the equation of the
- vertical asymptote of the graph of \(y=\mathrm{f}(x)\)
- horizontal asymptote of the graph of \(y=\mathrm{f}(x)\).[2]
- Find \(\mathrm{f}'(x)\), simplifying your answer.[4]
- Hence, explain why \(\mathrm{f}\) is not a decreasing function on any interval of its domain.[2]
The point \(P\big(x,\mathrm{f}(x)\big)\), where \(x<-1\), lies on the graph of \(y=\mathrm{f}(x)\). The distance from \(P\) to the point \(R(0,5)\) is \(d\).
- Find an expression for \(d\) in terms of \(x\) only.[2]
- Hence, find the coordinates of the point \(P\) for which \(d\) takes its least value.[3]
Solution:
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Answer:(a)(i) \(x=-1\) (a)(ii) \(y=2\) (b) \(\mathrm{f}'(x)=\dfrac{7}{(x+1)^2}\) (c) \(\mathrm{f}'(x)>0\) for all \(x\neq-1\) (d) \(d=\sqrt{x^2+\left(\dfrac{3x+10}{x+1}\right)^2}\) (e) \(P(-2.70,\ 6.12)\)
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