Consider the continuous function on the interval .

MATHEMATICS · Multi-part · AP · Calculus AB · Limits and Continuity · Limits and Continuity

Problem

Consider the continuous function

f(x)=x^{3}-4x+1

on the interval

[0,2]

. (a) Calculate

f(0)

and

f(2)

. State the signs of these values. (b) Explain why at least one value

c

(0,2)

satisfies

f(c)=0

. (c) Let

m=1

be the midpoint of the interval. Evaluate

f(m)

. Using the signs found in parts (a) and (c), identify the subinterval of length

1

that must still contain a zero of

f

and justify your choice.