IGNOU BMTC 133 2026 - Let f be a differentiable function whose derivative never vanish

Question

Let f be a differentiable function whose derivative never vanishes on [a, b]. Show that f is either strictly increasing or strictly decreasing.

20 Jan 2026

Answer :

Word Count : 250

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. Hence either f'(x)>0 for all x or f'(x)<0 for all x in [a,b]. Assume f'(x)>0 for all x; the opposite case is analogous. Fix x _________ ____ _______ _______ _____ _______ _________ __________ _____ _______.
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