Consider the one-parameter family defined on

MATHEMATICS · Subjective · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation

Problem

Consider the one-parameter family pk(x)=k x3−xp_k(x)=k\,x^{3}-x defined on [−1,1][-1,1], where kk is a real constant. The average rate of change of pkp_k over [−1,1][-1,1] equals k−1k-1. Determine all real values of kk for which the equation pk′(c)=k−1p_k'(c)=k-1 has exactly two distinct solutions cc in (−1,1)(-1,1).

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