A function is continuous on the closed interval with and
MATHEMATICS · Single correct · AP · Calculus AB · Limits and Continuity · Limits and Continuity
Problem
A function g is continuous on the closed interval [2,7] with g(2)=−3 and g(7)=5. Which of the following best explains why the equation g(x)=0 must have at least one solution in (2,7)?
(a) Because the average rate of change of g on [2,7] is positive, g(x)=0 must occur in the interval.
(b) Because g is continuous on [2,7], Rolle’s Theorem guarantees there is a c in (2,7) with g(c)=0.
(c) Because g takes both negative and positive values at the endpoints of the interval and is continuous, it must cross 0 somewhere between 2 and 7.
(d) Because g is continuous on (2,7), it is one-to-one, so it takes every value between −3 and 5 exactly once.