hw13 - Suppose that f is differentiable on a,c and c,b(a...

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MATH 444: ELEMENTARY REAL ANALYSIS HOMEWORK 13 Due date: Dec 3 (Wed) Exercises from the textbook. 6.2: 7; 8; 12; 15 7.1: 6; 8; 10; 11 Out-of-textbook exercises. 1. (a) Using the fact that ( sin x ) = cos x show that lim x 0 sin x x = 1. (b) Using part (a), carefully prove that sin x is not differentiable at 0. 2. Using the Mean Value Theorem, prove the following inequalities: (a) sin x x , for all x R . (b) ( 1 + x ) α 1 + αx , for all α > 1 and x > - 1. 3. Let f [ a,b ] R be continuous and let c ( a,b ) . Suppose that
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Unformatted text preview: ) . Suppose that f is differentiable on ( a,c ) and ( c,b ) . (a) Prove the following test for determining whether c is a relative minimum: If there is a neighborhood ( c-δ,c + δ ) ⊆ [ a,b ] such that ● for all x ∈ ( c-δ,c ) , f ′ ( x ) ≤ 0, ● and, for all x ∈ ( c,c + δ ) , f ′ ( x ) ≥ 0, then f has a relative minimum at c . (b) Write down the analogue for a relative maximum....
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