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Let f:(a,b)→R be twice differentiable, and assume that |f '(x)-f '(y)|≤ |x-y| for all x, y ∈ (a,b).

Show that |f(x) - f(y) - f '(x)(x-y)|≤ |x-y|^2 for all x, y ∈ (a,b).

I am stuck and not quite sure how to even begin. Any help would be appreciated greatly!

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Have you tried the mean value theorem? – ploosu2 Apr 28 '14 at 23:01
up vote 0 down vote accepted

Assume $x>y$. By the mean value theorem there is $c\in (y, x)$ s.t.

$$f(x) - f(y) = f'(c)(x-y).$$

Plug this in. Do you see it from there on?

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I see it now. Thank you so much, that was all I needed to get going – MathMajor Apr 28 '14 at 23:38

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