Let f(x) be a nonconstant, linear function on the interval [a,b] where a < b. Then one of f(a) and f(b) will be the unique local and absolute maximum and one will be the unique local and absolute minimum (if your definition of local extremum allows for a one-sided extremum). Note that every point between a and b qualifies as one of the c's whose existence is guaranteed by the MVT. If we now consider the constant function on [a,b], then every point between a and b qualifies as all of the following: one of the c's as well as a local max, a local min, an absolute max, a local min, and an absolute min.
-- Professor Jeff Stuart, Chair Department of Mathematics Pacific Lutheran University Tacoma, WA 98447 USA (253) 535 - 7403 firstname.lastname@example.org