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Cubic with three real roots depending on k

Cubic with three real roots depending on k

IP 4
8 marks

The function \(\mathrm f\) is defined by \(\mathrm f(x)=x^3-3kx+2\), where \(k\) is a real constant.

  1. Show that the curve \(y=\mathrm f(x)\) has two stationary points if and only if \(k>0\).[2]
  2. For \(k>0\), find the coordinates of the stationary points in terms of \(k\), and determine their nature.[3]
  3. Hence find the range of values of \(k\) for which the equation \(\mathrm f(x)=0\) has three distinct real roots. Justify your answer.[3]

Solution:

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Answer:(a) \(x^2=k\) has two solutions only when \(k>0\) (b) minimum \(\left(\sqrt k,\,2-2k\sqrt k\right)\), maximum \(\left(-\sqrt k,\,2+2k\sqrt k\right)\) (c) \(k>1\)

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