MATH116 Lecture Notes - Lecture 14: Guidonian Hand
9 Dec 2015
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Practice solutions 6: suppose it is given that f(cid:48)(x) = 0 for all x (a, b). Use the mean value theorem to prove that f must be constant on (a, b). Hint: consider two arbitrary points x1 < x2 contained in (a, b) and apply the mean value theorem. Solution: because f(cid:48) is given to be 0 on (a, b), we can conclude that it is di erentiable and continuous on (a, b) (and any subinterval of (a, b) as well), so the mean value theorem applies. Let x1 and x2 be two numbers in (a, b) with x1 < x2. By mvt, there exists some c (x1, x2) such that f(cid:48)(c) = f (x2) f (x1) This means x2 x1 or f (x2) f (x1) x2 x1. = 0 f (x2) f (x1) = 0 f (x2) = f (x1)
