SPS SPS FM 2020 June — Question 3

Exam BoardSPS
ModuleSPS FM (SPS FM)
Year2020
SessionJune
TopicFactor & Remainder Theorem
TypeMultiple unknowns with derivative condition

3. $$\mathrm { g } ( x ) = 4 x ^ { 3 } + a x ^ { 2 } + 4 x + b$$ where \(a\) and \(b\) are constants.
Given that
  • ( \(2 x + 1\) ) is a factor of \(\mathrm { g } ( x )\)
  • the curve with equation \(y = \mathrm { g } ( x )\) has a point of inflection at \(x = \frac { 1 } { 6 }\)
    1. find the value of \(a\) and the value of \(b\)
    2. Show that there are no stationary points on the curve with equation \(y = \mathrm { g } ( x )\).