Edexcel C3 2015 June — Question 9

Exam BoardEdexcel
ModuleC3 (Core Mathematics 3)
Year2015
SessionJune
TopicPolynomial Division & Manipulation

9. Given that \(k\) is a negative constant and that the function \(\mathrm { f } ( x )\) is defined by $$f ( x ) = 2 - \frac { ( x - 5 k ) ( x - k ) } { x ^ { 2 } - 3 k x + 2 k ^ { 2 } } , \quad x \geqslant 0$$
  1. show that \(\mathrm { f } ( x ) = \frac { x + k } { x - 2 k }\)
  2. Hence find \(\mathrm { f } ^ { \prime } ( x )\), giving your answer in its simplest form.
  3. State, with a reason, whether \(\mathrm { f } ( x )\) is an increasing or a decreasing function. Justify your answer.