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hw7 - find a function which is uniformly continuous but...

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Math 125A, Fall 2009 Homework 7 Due Date: November 20, 2009 Problem 1: Let f be a function satisfying f 00 ( x ) = 0 for all x R . Prove that f has the form f ( x ) = ax + b for some constants a, b . (Integration is not an acceptable answer since we have not yet covered the fundamental theorem of calculus.) 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 is differentiable, and f ( n ) = 0 for all integers n , then there are infinitely many real numbers x such that f 0 ( x ) = 0. (b) If f is an infinitely differentiable function, and T is the Taylor series of f centered at x = 0, then f ( x ) = T ( x ) for all x . (c) If f is an infinitely differentiable function, T is the Taylor series of f centered at x = 0, and | f ( n ) | ≤ 1000 for all n , then f ( x ) = T ( x ) for all x . Problem 3: Define a function g : R R by: g ( x ) = ± x - 1 , if x < 0 x + 1 , if x 0 Prove that there is no function whose derivative is g ( x ). Problem 4: Let f be a differentiable function. (a) If f 0 is bounded, then prove that f is uniformly continuous. (b) Prove, by means of a counterexample, that the converse is not true. In other words,
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Unformatted text preview: find a function which is uniformly continuous but whose derivative is unbounded. Problem 5: Define a function f : R → R by: f ( x ) = ± e-1 /x 2 , if x 6 = 0 , if x = 0 (a) Prove, by mathematical induction, that f ( n ) (0) = 0 for all n . (It may help to review Problem 4 of Homework 6.) (b) Find the Taylor series of f centered at x = 0. Problem 6: Let f be a twice differentiable function on (0 , ∞ ) and let M , M 1 , M 2 denote the least upper bounds of | f ( x ) | , | f ( x ) | , | f 00 ( x ) | respectively on (0 , ∞ ). (a) Let h > 0. By using the Taylor series expansion of f centered at x = c , show that: f ( c ) = 1 2 h ² f ( c + 2 h )-f ( c ) ³-hf 00 ( ξ ) for some ξ ∈ ( c, c + 2 h ). Explain why | f | ≤ hM 2 + M /h for all h > 0. (b) Prove that M 2 1 ≤ 4 M M 2 when M 2 > 0 ....
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