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Use the Mean Value Theorem and the fact that for f(x) = cos(x), f′(x) = −sin(x),...

  1. Use the Mean Value Theorem and the fact that for f(x) = cos(x), f′(x) = −sin(x), to prove that, for x, y ∈ R,

    | cos x − cos y| ≤ |x − y|.

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Answer #1

f(x) = cos(x) is continuous on R & differentiable on R, hence, it is continuous and differentiable on any interval in R.

By mean value theorem on the interval [x,y] for the function f, we have,

{f(x) - f(y)}/(x-y) = f'(z) for some z in (x,y)

Now, f'(z) = -sin(z) and, |f'(z)| = |-sin(z)| = |sin(z)| 1

So, |{f(x) - f(y)}/(x-y)| 1

So, |f(x) - f(y)| |x-y|

So, |cos(x) - cos(y)| |x - y| for all x, y in R

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