hw2 - Let f : [ a, b ] R be a continuous function. If f ( x...

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Math 125A, Fall 2009 Homework 2 Due Date: October 9, 2009 Problem 1: Prove that x = cos x for some number x [0 , π 2 ]. Problem 2: Determine whether the statement is true or false. If the statement is true, then prove it. Otherwise, provide a counterexample. (a) If f : [ a, b ] R is continuous, then f assumes a maximum value. (b) If f : [ a, b ] R assumes a maximum value, then f is continuous. (c) If f is continuous on [ - 2 , 2], then f ([ - 2 , 2]) is an interval. (d) If f ([ - 2 , 2]) is an interval, then f is continuous. Problem 3: Let f, g : [ a, b ] R be continuous functions. If f ( a ) < g ( a ) and f ( b ) > g ( b ), then f ( c ) = g ( c ) for some c ( a, b ). Problem 4:
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Unformatted text preview: Let f : [ a, b ] R be a continuous function. If f ( x ) > 0 for all x [ a, b ], then there exists a number c > 0 such that f ( x ) c for all x [ a, b ]. (Hint: Use Theorem 18.1.) Problem 5: Let f : R R be a continuous function for which f ( x ) Q for all x R . Prove that f must be a constant function. (Hint: Proof by contradiction.) Problem 6: Using Theorem 19.2, explain why the function f ( x ) = x 17 sin x-e x cos 3 x is uniformly continuous on [0 , ]....
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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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