2026 DSS P1 Q11

2026 DSS P1 Q11

Secondary 4
6 marks

The equation of a curve is \(y=px^3+qx^2+2x-2\), where \(p\) and \(q\) are constants.

  1. Show that \(y\) is always increasing if \(q^2<6p\).[3]

It is now given that \(p=6\) and \(q=1\).

  1. Show that \(x=\dfrac12\) is an \(x\)-intercept of the curve.[1]
  2. Hence explain why the curve \(y=6x^3+x^2+2x-2\) has only one \(x\)-intercept.[2]

Solution:

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Answer:(a) Derivative positive for all real \(x\). (b) \(y(\frac12)=0\). (c) Strict increase and the known root give exactly one intercept.

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