Let be continuous. The Extreme Value Theorem guarantees that:
Answer and explanation
Correct answer: A. attains both an absolute maximum and minimum on
- Step 1: The Extreme Value Theorem (Weierstrass) applies to any continuous real function defined on a closed bounded interval . It asserts that there exist points such that for all . **Answer: attains both an absolute maximum and minimum on . Why not the others:** " is strictly monotonic throughout the domain" continuous functions can oscillate infinitely and need not be monotonic; " has a horizontal tangent at some point" requires differentiability and equal endpoints, which is Rolle's theorem; " must vanish at least once in the interval" requires opposite endpoint signs , which is Bolzano's theorem.
Practise more Limits, Continuity and Global Theorems.
Free questions with an explanation for every one, and full timed Panhellenic Examinations (Greece) mock tests.