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hw7sol

# hw7sol - Math 125A Fall 2009 Solutions for Homework...

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Math 125A, Fall 2009 Solutions for Homework 7 (Prepared by Matt Low) 1. It follows from Theorem 29.4 that f 0 is a constant function. In other words, f 0 ( x ) = C for some constant C . De±ne a new function g ( x ) = f ( x ) ± Cx . Note that g 0 ( x ) = 0, so Theorem 29.4 again implies that g ( x ) = D for some constant D . Thus f ( x ) = Cx + D for all x 2 R . 2. (a) True. Consider any interval of the form [ n;n + 1] for some integer n . By the mean value theorem (Theorem 29.3), there is a number c 2 ( n;n + 1) such that: f 0 ( c ) = f ( n + 1) ± f ( n ) ( n + 1) ± n = 0 ± 0 1 = 0 Because there are in±nitely many of these intervals [ n;n + 1], there are also in±nitely many of these points c at which f 0 ( c ) = 0. (b) False. For example, consider the function f de±ned in Problem 5. In Problem 5(b), you showed that T ( x ) = 0 for all x . But f ( x ) 6 = 0 when x 6 = 0. (c) True. This is an immediate consequence of Corollary 31.4. 3. By Theorem 29.8, the derivative of any function satis±es intermediate value theorem. But g ( x ) does not. For example, g ( ± 1) = ± 2 and g (1) = 2, but there is no number c at which g ( c ) = 0.

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