  Given an infinite word X over an alphabet A a letter b occurring in X, and a
total order \sigma on A, we call the smallest word with respect to \sigma
starting with b in the shift orbit closure of X an extremal word of X. In this
paper we consider the extremal words of morphic words. If X = g(f^{\omega}(a))
for some morphisms f and g, we give two simple conditions on f and g that
guarantees that all extremal words are morphic. This happens, in particular,
when X is a primitive morphic or a binary pure morphic word. Our techniques
provide characterizations of the extremal words of the Period-doubling word and
the Chacon word and give a new proof of the form of the lexicographically least
word in the shift orbit closure of the Rudin-Shapiro word.
