hw7sol - Math 125A, Fall 2009 Solutions for Homework 7...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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.
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 03/18/2012 for the course MATH 67 taught by Professor Schilling during the Fall '07 term at UC Davis.

Page1 / 2

hw7sol - Math 125A, Fall 2009 Solutions for Homework 7...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online