SPS SPS FM 2024 October — Question 1

Exam BoardSPS
ModuleSPS FM (SPS FM)
Year2024
SessionOctober
TopicDifferentiation from First Principles
  1. Given the function \(f ( x ) = x - x ^ { 2 }\), defined for all real values of \(x\),
    1. Show that \(f ^ { \prime } ( x ) = 1 - 2 x\) by differentiating \(f ( x )\) from first principles.
    2. Find the maximum value of \(f ( x )\).
    3. Explain why \(f ^ { - 1 } ( x )\) does not exist.
      [0pt] [BLANK PAGE]
    4. The quadratic equation \(k x ^ { 2 } + 2 k x + 2 k = 3 x - 1\), where \(k\) is a constant, has no real roots.
    5. Show that \(k\) satisfies the inequality
    $$4 k ^ { 2 } + 16 k - 9 > 0$$
  2. Hence find the set of possible values of \(k\). Give your answer in set notation.
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