A function is defined on the open interval and is known to be continuous on that interval
MATHEMATICS · Multi-part · AP · Calculus AB · Limits and Continuity · Limits and Continuity
Problem
A function q is defined on the open interval (0,3) and is known to be continuous on that interval. The following function values are given:
q(0.2)=4, q(2)=−1, and q(2.9)=5. No information is provided about any values of q at x=0 or x=3.
(1) Prove that there exists at least one number c in (0.2,2) such that q(c)=0. Present your argument in a logically ordered sequence of statements.
(2) Discuss whether the existence of a number c in (2,2.9) with q(c)=0 can be guaranteed from the information provided. Give a reasoned explanation, clearly indicating any additional information that would be necessary to reach a definitive conclusion.
(3) Explain how extending the domain of q to include the endpoints x=0 and x=3 could affect the conclusions in parts (a) and (b). Be explicit about which new data points would be required and why.