hw12 - MATH 444 ELEMENTARY REAL ANALYSIS HOMEWORK 12 Due...

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MATH 444: ELEMENTARY REAL ANALYSIS HOMEWORK 12 Due date: Nov 19 (Wed) Exercises from the textbook. 6.1: 13; 16; 17 6.2: 1(b)(d); 3(b)(d) Out-of-textbook exercises. 1. Show that f ( x ) = x 3 is differentiable at every x 0 R by finding the Carath´ eodory function φ R R for f at x 0 . 2. Let I = [ 1 , ) and f I R be defined by f ( x ) = x 4 + 2 x 3 - x + 1. Let J = [ 3 , ) and you may take as granted that f ( I ) = J and f has an inverse g J
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Unformatted text preview: g ∶ J → R . Explain why g must be diﬀerentiable at every y ∈ J and ﬁnd g ′ ( f ( x )) for all x ∈ I . 3. Prove the following, indicating every theorem used in your proof. (a) For every n ∈ N , ( x 1 n ) ′ = 1 n x 1 n-1 . (b) For every r ∈ Q , ( x r ) ′ = rx r-1 . Hint : Write r = m n ....
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