MATHEMATICS · Multi-part · CBSE · Mathematics · Continuity and Differentiability · Continuity and Differentiability
Problem
Suppose u and v are continuous functions on [a,b] with u(a)>v(a) and u(b)<v(b). A student claims that the inequality u(x)=v(x) must have at least one solution x∈(a,b).
(a) Describe, in complete sentences, why the continuity of both u and v on [a,b] is critical to evaluating the student’s claim.
(b) Explain how you would use a single auxiliary function to confirm or refute the claim. Your answer should identify the function and outline the logical structure of the argument without invoking any named theorem.
(c) Provide a brief discussion of how the conclusion might fail if either u or v were not continuous at some point in (a,b), supporting your explanation with a specific illustrative example (no computation required).