Consider the one-parameter family of continuous functions

MATHEMATICS · Single correct · AP · Calculus AB · Limits and Continuity · Limits and Continuity

Problem

Consider the one-parameter family of continuous functions fp(x)=x3−3x+pf_{p}(x)=x^{3}-3x+p. A student plans to locate a real zero by repeatedly halving the interval [0,2][0,2]. For which of the following values of the parameter pp is this strategy guaranteed to capture a zero inside (0,2)(0,2)? (a) p<−2p<-2 (b) p∈(0,2)p\in(0,2) (c) p∈(−2,0)p\in(-2,0) (d) p∈(−2,2)p\in(-2,2)

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