Word Count : 250

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Let f be differentiable on [a,b] and suppose f'(x) ≠ 0 for every x in [a,b]. We show f is strictly monotone. First, f' cannot change sign on [a,b]. If there were u,v with f'(u)>0 and f'(v)<0 then by Darboux's theorem f' would take every value between f'(u) and f'(v), so in particular some point c would satisfy f'(c)=0, contradicting the hypothesis. _________ _____ ___ _____ ______ ________ _________ __________.
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