# Hw6 - f R R by f x = e ± 1 =x 2 if x 6 = 0 if x = 0 It can be shown that f is di²erentiable in±nitely many times at x = 0 Determine the values

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Math 125A, Fall 2009 Homework 6 Due Date: November 13, 2009 Problem 1: De±ne functions f;g : R ! R by: f ( x ) = 1 X n =0 ( ± 1) n x 2 n +1 (2 n + 1)! and g ( x ) = 1 X n =0 ( ± 1) n x 2 n (2 n )! (a) Prove that f 0 = g and g 0 = ± f . (b) Prove that ( f 2 + g 2 ) 0 = 0. (Hint: Chain rule.) (c) Prove that f 2 + g 2 = 1. Problem 2: Determine whether the statement is true or false. If the statement is true, then prove it. Otherwise, provide a counterexample. (a) If a continuous function f : R ! R is bounded, then f 0 ( x ) exists for all x . (b) If f;g are functions, and f and f + g are di²erentiable, then g is also di²erentiable. (c) If f is a di²erentiable function on [0 ; 2] and lim x ! 1 f ( x ) = 7, then f (1) = 7. Problem 3: De±ne functions f;g : R ! R by: f ( x ) = ( x sin( 1 x ) ; if x 6 = 0 0 ; if x = 0 and g ( x ) = ( x 2 sin( 1 x ) ; if x 6 = 0 0 ; if x = 0 (a) Prove that f 0 (0) does not exist. (b) Prove that g 0 (0) exists and g 0 (0) = 0. Problem 4: De±ne a function
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Unformatted text preview: f : R ! R by: f ( x ) = ( e ± 1 =x 2 ; if x 6 = 0 ; if x = 0 It can be shown that f is di²erentiable in±nitely many times at x = 0. Determine the values of f (0) and f 00 (0). Problem 5: Let f;g be functions which are both di²erentiable at x = c . Moreover, assume that f ( c ) = g ( c ) = 0 and g ( c ) 6 = 0. Prove that: lim x ! c f ( x ) g ( x ) = f ( c ) g ( c ) (Note that this is not L’Hospital’s rule.) Problem 6: Let f be any function on the real line and suppose that: j f ( x ) ± f ( y ) j ² j x ± y j 2 for all x;y 2 R . Prove that f is a constant function....
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## 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.

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