For each real number

MATHEMATICS · Single correct · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation

Problem

For each real number kk, define fk(x)=x3−3x2+kxf_k(x)=x^{3}-3x^{2}+kx on the interval [0,4][0,4]. For which of the following values of kk does fkf_k have exactly one critical point in (0,4)(0,4), and that point yields the absolute minimum of fkf_k on [0,4][0,4]? (a) k=3k=3 (b) k=6k=6 (c) k=0k=0 (d) k=1k=1

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