In certain intuitionistic frameworks the extreme value theorem cannot be proved. Depending on the exact framework, counterexamples can be constructed as well; see for example pp. 294-295 in

Troelstra, A. S.; van Dalen, D. Constructivism in mathematics. Vol. I. An introduction. Studies in Logic and the Foundations of Mathematics, 121. North-Holland Publishing Co., Amsterdam, 1988. xx+342+XIV pp. ISBN: 0-444-70266-0; 0-444-70506-6

The Smooth Infinitesimal Analysis (SIA) following Lawvere, Kock, J. Bell, and others is based on a category-theoretic framework where the background logic is intuitionistic. What is the status of the extreme value theorem in SIA?