10 - Math 310 hw 10 solutions Monday 30 Nov 2009 29.2 29.14...

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Math 310 - hw 10 solutions 29.2, 29.14, 29.18; Monday, 30 Nov 2009 29.2 Prove that | cos x - cos y | ≤ | x - y | for all x,y R . Let f : R R be defined by f ( x ) := cos x for x R . We will use the facts that f is differentiable on R and that f 0 ( x ) = - sin x for all x R . Let x,y R . If x = y there is nothing to prove, so we may assume that x 6 = y . Since f is differentiable on R we may apply the Mean Value Theorem to conclude that f ( x ) - f ( y ) = f 0 ( c )( x - y ) for some c between x and y . Since | f 0 ( c ) | = | - sin c | ≤ 1. Hence, | cos x - cos y | = | f ( x ) - f ( y ) | = | f 0 ( c ) | · | x - y | ≤ 1 · | x - y | = | x - y | , as desired. ± 29.14 Suppose that f is differentiable on R , that 1 f 0 ( x ) 2 for x R , and that f (0) = 0. Prove that x f ( x ) 2 x for all x 0. Let x 0. Since f (0) = 0, the desired condition holds for x = 0; so we may assume that x > 0. Since f is continuous on [0 ,x ] and differentiable on (0 ,x ), we may invoke the Mean Value Theorem to conclude that there
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