Originally Posted by
Shanks since f(0)=f(1), by the mean vaule theorem there exist a point s in (0,1) such that f'(s)=0,
similarly there exist a point t in (1,2) such that f'(t)=2.
thus by the Darboux's theorem, there exist a point c in (s,t) such that f'(c)=0.2 QED.