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Nabla Tuition TMUA Paper 1 · Q20

Paper 1Functions & graphsAlgebra

For a real constant kk, the function ff is defined by

f(x)=x3−3x+k.f(x)=x^3-3x+k.

The equation

f(f(x))=kf(f(x))=k

has exactly seven distinct real solutions.

Which of the following gives the complete set of possible values of kk?

A−2<k<2-2<k<2
B−2<k<3−2 or 2−3<k<2-2<k<\sqrt3-2\text{ or }2-\sqrt3<k<2
C−2<k≤3−2 or 2−3≤k<2-2<k\le\sqrt3-2\text{ or }2-\sqrt3\le k<2
D3−2<k<2−3\sqrt3-2<k<2-\sqrt3
Ek<3−2 or k>2−3k<\sqrt3-2\text{ or }k>2-\sqrt3

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