Find stationary points with unknown constants

Questions where the polynomial contains unknown constants (a, b, etc.) that must first be determined from given conditions before finding stationary points.

1 questions · Standard +0.3

1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.07n Stationary points: find maxima, minima using derivatives
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Edexcel AS Paper 1 2021 November Q16
11 marks Standard +0.3
  1. The curve \(C\) has equation \(y = \mathrm { f } ( x )\) where
$$f ( x ) = a x ^ { 3 } + 15 x ^ { 2 } - 39 x + b$$ and \(a\) and \(b\) are constants.
Given
  • the point \(( 2,10 )\) lies on \(C\)
  • the gradient of the curve at \(( 2,10 )\) is - 3
    1. (i) show that the value of \(a\) is - 2
      (ii) find the value of \(b\).
    2. Hence show that \(C\) has no stationary points.
    3. Write \(\mathrm { f } ( x )\) in the form \(( x - 4 ) \mathrm { Q } ( x )\) where \(\mathrm { Q } ( x )\) is a quadratic expression to be found.
    4. Hence deduce the coordinates of the points of intersection of the curve with equation
$$y = \mathrm { f } ( 0.2 x )$$ and the coordinate axes.