MATHEMATICS · Single correct · AP · Calculus AB · Limits and Continuity · Limits and Continuity
Problem
A function g satisfies g(1)=−5 and g(4)=7. Which statement best justifies that the equation g(x)=2 has at least one real solution in the interval [1,4]?
(a) Because g is differentiable on (1,4) and g(4)−g(1)=0, a real solution must exist by the Mean Value Theorem.
(b) Because g is continuous on [1,4] and 2 lies between g(1) and g(4), the Intermediate Value Theorem guarantees at least one real solution of g(x)=2 in [1,4].
(c) Because g is increasing on [1,4], the equation g(x)=2 must have exactly one solution in that interval.
(d) Because g is continuous on [1,4] and g(1)g(4)<0, the Intermediate Value Theorem ensures g(x)=2 has a solution on [1,4].